Design & Analysis of Algorithms Lecture Notes
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The document consists of lecture notes on the design and analysis of algorithms for B.Tech 6th semester students, covering topics such as algorithm introduction, asymptotic notations, recurrences, and divide and conquer methods. It provides a comprehensive overview of various algorithms, their efficiencies, and methods to analyze their performance, including worst-case scenarios and dynamic programming. The notes are structured in modules and contain detailed discussions on specific algorithms and their applications, aimed at enhancing students' understanding of algorithmic principles.
![Time taken=
It is more than 5.5hrs it is under 20 mins.
So choosing a good algorithm (algorithm with slower rate of growth) as used by
computer B affects a lot.
Lecture 2 - Growth of Functions ( Asymptotic notations)
Before going for growth of functions and asymptotic notation let us see how to analyase
an algorithm.
How to analyse an Algorithm
Let us form an algorithm for Insertion sort (which sort a sequence of numbers).The pseudo
code for the algorithm is give below.
Pseudo code:
for j=2 to A length -------------------------------------------------- C1
key=A[j]-----------------------------------------------------------------C2
//Insert A[j] into sorted Array A[1.....j-1]------------------------C3
i=j-1------------------------------------------------------------------------C4
while i>0 & A[j]>key---------------------------------------------------C5
A[i+1]=A[i]---------------------------------------------------------------C6
i=i-1------------------------------------------------------------------------C7
A[i+1]=key----------------------------------------------------------------C8](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-7-320.jpg)
![So,T(n) will never be less than cn2
. But if we will take the assumption of T(k)=c1 k2
-c2k, then we
can find that T(n) = O(n2
)
2. BY ITERATIVE METHOD:
e.g. T(n)=2T(n/2)+n
=> 2[2T(n/4) + n/2 ]+n
=>22
T(n/4)+n+n
=> 22
[2T(n/8)+ n/4]+2n
=>23
T(n/23)
+3n
After k iterations ,T(n)=2k
T(n/2k
)+kn--------------(1)
Sub problem size is 1 after n/2k
=1 => k=logn
So,afterlogn iterations ,the sub-problem size will be 1.
So,when k=logn is put in equation 1
T(n)=nT(1)+nlogn
nc+nlogn ( say c=T(1))
O(nlogn)
3.BY RECURSSION TREE METHOD:
In a recursion tree ,each node represents the cost of a single sub-problem somewhere in the set of
recursive problems invocations .we sum the cost within each level of the tree to obtain a set of
per level cost,and then we sum all the per level cost to determine the total cost of all levels of
recursion .
Constructing a recursion tree for the recurrence T(n)=3T(n/4)+cn2](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-15-320.jpg)
![}
Let a recurrence relation is expressed as T(n)=
ϴ(1), if n<=C
aT(n/b) + D(n)+ C(n) ,otherwise
then n=input size a=no. Of sub-problemsn/b= input size of the sub-problems
Lecture 7: Worst case analysis of merge sort, quick sort
Merge sort
It is one of the well-known divide-and-conquer algorithm. This is a simple and very efficient
algorithm for sorting a list of numbers.
We are given a sequence of n numberswhich we will assume is stored in an array A [1...n].
Theobjective is to output a permutation of this sequence, sorted in increasing order. This is
normally done by permuting the elements within the array A.
How can we apply divide-and-conquer to sorting? Here are the major elements of the Merge
Sort algorithm.
Divide: Split A down the middle into two sub-sequences, each of size roughly n/2 .
Conquer: Sort each subsequence (by calling MergeSort recursively on each).
Combine: Merge the two sorted sub-sequences into a single sorted list.
The dividing process ends when we have split the sub-sequences down to a single item. A
sequence of length one is trivially sorted. The key operation where all the work is done is in the
combine stage,which merges together two sorted lists into a single sorted list. It turns out that
the merging process is quite easy to implement.](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-19-320.jpg)
![The following figure gives a high-level view of the algorithm. The “divide” phase is shown on the
left. It works top-down splitting up the list into smaller sublists. The “conquer and combine”
phases areshown on the right. They work bottom-up, merging sorted lists together into larger
sorted lists.
Merge Sort
Designing the Merge Sort algorithm top-down. We’ll assume that the procedure thatmerges
two sortedlist is available to us. We’ll implement it later. Because the algorithm is called
recursively on sublists,in addition to passing in the array itself, we will pass in two indices,
which indicate the first and lastindices of the subarray that we are to sort. The call
MergeSort(A, p, r) will sort the sub-arrayA [ p..r ] and return the sorted result in the same
subarray.
Here is the overview. If r = p, then this means that there is only one element to sort, and we
may returnimmediately. Otherwise (if p < r) there are at least two elements, and we will invoke
the divide-and-conquer. We find the index q, midway between p and r, namely q = ( p + r ) / 2
(rounded down to thenearest integer). Then we split the array into subarrays A [ p..q ] and A [ q
+ 1 ..r ] . Call Merge Sort recursively to sort each subarray. Finally, we invoke a procedure
(which we have yet to write) whichmerges these two subarrays into a single sorted array.
MergeSort(array A, int p, int r) {
if (p < r) { // we have at least 2 items
q = (p + r)/2
MergeSort(A, p, q) // sort A[p..q]
MergeSort(A, q+1, r) // sort A[q+1..r]](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-20-320.jpg)
![Merge(A, p, q, r) // merge everything together
} }
Merging: All that is left is to describe the procedure that merges two sorted lists. Merge(A, p, q,
r)assumes that the left subarray, A [ p..q ] , and the right subarray, A [ q + 1 ..r ] , have already
been sorted.We merge these two subarrays by copying the elements to a temporary working
array called B. Forconvenience, we will assume that the array B has the same index range A,
that is, B [ p..r ] . We have to indices i and j, that point to the current elements ofeach subarray.
We move the smaller element into the next position of B (indicated by index k) andthen
increment the corresponding index (either i or j). When we run out of elements in one array,
thenwe just copy the rest of the other array into B. Finally, we copy the entire contents of B
back into A.
Merge(array A, int p, int q, int r) { // merges A[p..q] with A[q+1..r]
array B[p..r]
i = k = p //initialize pointers
j = q+1
while (i <= q and j <= r) { // while both subarrays are nonempty
if (A[i] <= A[j]) B[k++] = A[i++] // copy from left subarray
else B[k++] = A[j++] // copy from right subarray
}
while (i <= q) B[k++] = A[i++] // copy any leftover to B
while (j <= r) B[k++] = A[j++]
for i = p to r do A[i] = B[i] // copy B back to A }
Analysis: What remains is to analyze the running time of MergeSort. First let us consider the
running timeof the procedure Merge(A, p, q, r). Let n = r − p + 1 denote the total length of both
the leftand right subarrays. What is the running time of Merge as a function of n? The algorithm
contains fourloops (none nested in the other). It is easy to see that each loop can be executed
at most n times. (Ifyou are a bit more careful you can actually see that all the while-loops](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-21-320.jpg)
![together can only be executed ntimes in total, because each execution copies one new element
to the array B, and B only has space forn elements.) Thus the running time to Merge n items is
Θ ( n ) . Let us write this without the asymptoticnotation, simply as n. (We’ll see later why we
do this.)
Now, how do we describe the running time of the entire MergeSort algorithm? We will do this
throughthe use of a recurrence, that is, a function that is defined recursively in terms of itself.
To avoidcircularity, the recurrence for a given value of n is defined in terms of values that are
strictly smallerthan n. Finally, a recurrence has some basis values (e.g. for n = 1 ), which are
defined explicitly.
Let’s see how to apply this to MergeSort. Let T ( n ) denote the worst case running time of
MergeSort onan array of length n. For concreteness we could count whatever we like: number
of lines of pseudocode,number of comparisons, number of array accesses, since these will only
differ by a constant factor.Since all of the real work is done in the Merge procedure, we will
count the total time spent in theMerge procedure.
First observe that if we call MergeSort with a list containing a single element, then the running
time is aconstant. Since we are ignoring constant factors, we can just write T ( n ) =1 .
When we call MergeSortwith a list of length n >1 , e.g. Merge(A, p, r), where r − p +1 = n, the
algorithm first computes q = ( p + r ) / 2 . The subarray A [ p..q ] , which contains q − p + 1
elements. You can verify that is of size n/ 2 . Thus the remaining subarray A [ q +1 ..r ] has n/ 2
elements in it. How long does it taketo sort the left subarray? We do not know this, but
because n/ 2< n for n >1 , we can express this as T (n/ 2) . Similarly, we can express the time
that it takes to sort the right subarray as T (n/ 2).
Finally, to merge both sorted lists takes n time, by the comments made above. In conclusion we
have
T ( n ) =1 if n = 1 ,
2T (n/ 2) + n otherwise.
Solving the above recurrence we can see that merge sort has a time complexity of Θ (n log n) .](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-22-320.jpg)
![QUICKSORT
Worst-case running time: O (n2).
Expected running time: O (n lgn).
Sorts in place.
Description of quicksort
Quicksort is based on the three-step process of divide-and-conquer.
• To sort the subarrayA[p . . r ]:
Divide: Partition A[p . . r ], into two (possibly empty) subarraysA[p . . q − 1] and
A[q + 1 . . r ], such that each element in the ÞrstsubarrayA[p . . q − 1] is ≤ A[q] and
A[q] is ≤ each element in the second subarrayA[q + 1 . . r ].
Conquer: Sort the two subarrays by recursive calls to QUICKSORT.
Combine: No work is needed to combine the subarrays, because they are sorted in place.
• Perform the divide step by a procedure PARTITION, which returns the index q that marks the
position separating the subarrays.
QUICKSORT (A, p, r)
ifp < r
thenq ←PARTITION(A, p, r )
QUICKSORT (A, p, q − 1)
QUICKSORT (A, q + 1, r)
Initial call is QUICKSORT (A, 1, n)
Partitioning
Partition subarrayA [p . . . r] by the following procedure:
PARTITION (A, p, r)
x ← A[r ]
i ← p –1
for j ← p to r –1
do if A[ j ] ≤ x
theni ← i + 1](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-23-320.jpg)
![exchangeA[i ] ↔ A[ j ]
exchangeA[i + 1] ↔ A[r ]
returni + 1
• PARTITION always selects the last element A[r ] in the subarrayA[p . . r ] as the pivot the
element around which to partition.
• As the procedure executes, the array is partitioned into four regions, some of which may be
empty:
Performance of Quicksort
The running time of Quicksort depends on the partitioning of the subarrays:
• If the subarrays are balanced, then Quicksort can run as fast as mergesort.
• If they are unbalanced, then Quicksort can run as slowly as insertion sort.
Worst case
• Occurs when the subarrays are completely unbalanced.
• Have 0 elements in one subarray and n − 1 elements in the other subarray.](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-24-320.jpg)
![Lecture 8 - Heaps and Heap sort
HEAPSORT
Inplace algorithm
Running Time: O(n log n)
Complete Binary Tree
The (binary) heap data structure is an array object that we can view as a nearly complete binary
tree.Each node of the tree corresponds to an element of the array. The tree is completely filled on
all levels except possibly the lowest, which is filled from the left up to a point.
The root of the tree is A[1], and given the index i of a node, we can easily compute the indices of
its parent, left child, and right child:
PARENT (i) => return [i/2]
LEFT (i) => return 2i
RIGHT (i) => return 2i+ 1
On most computers, the LEFT procedure can compute 2i in one instruction by simply shifting the
binary representation of i left by one bit position.
Similarly, the RIGHT procedure can quickly compute 2i + 1 by shifting the binary
representation of i left by one bit position and then adding in a 1 as the low-order bit.](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-26-320.jpg)
![The PARENT procedure can compute [i/2] by shifting i right one bit position. Good
implementations of heapsort often implement these procedures as "macros" or "inline"
procedures.
There are two kinds of binary heaps: max-heaps and min-heaps.
• In a max-heap,the max-heap property is that for every node i other than the root,
A[PARENT(i)] >= A[i] ,that is, the value of a node is at most the value of its parent.
Thus, the largest element in a max-heap is stored at the root, and the subtree rooted at a
node contains values no larger than that contained at the node itself.
• A min-heap is organized in the opposite way; the min-heap property is that for every
node i other than the root, A[PARENT(i)<=A[i] ,
The smallest element in a min-heap is at the root.
The height of a node in a heap is the number of edges on the longest simple downward
path from the node to a leaf and
The height of the heap is the height of its root.
Height of a heap of n elements which is based on a complete binary tree is O(log n).
Maintaining the heap property
MAX-HEAPIFY lets the value at A[i] "float down" in the max-heap so that the
subtree rooted at index i obeys the max-heap property.
MAX-HEAPIFY(A,i)
1. l LEFT(i)
2. r RIGHT(i)
3. if A[l] > A[i]
4. largest l
5. if A[r] > A[largest]
6. Largest r
7. if largest != i
8. Then exchange A[i] A[largest]](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-27-320.jpg)
![9. MAX-HEAPIFY(A,largest)
At each step, the largest of the elements A[i], A[LEFT(i)], and A[RIGHT(i)] is
determined, and its index is stored in largest. If A[i] is largest, then the subtree rooted at node i is
already a max-heap and the procedure terminates. Otherwise, one of the two children has the
largest element, and A[i ] is swapped with A[largest], which causes node i and its children to
satisfy the max-heap property. The node indexed by largest, however, now has the original value
A[i], and thus the subtree rooted at largest might violate the max-heap property. Consequently,
we call MAX-HEAPIFY recursively on that subtree.
Figure: The action of MAX-HEAPIFY (A, 2), where heap-size = 10. (a) The initial con-
figuration, with A [2] at node i = 2 violating the max-heap property since it is not larger than
both children. The max-heap property is restored for node 2 in (b) by exchanging A [2] with
A[4], which destroys the max-heap property for node 4. The recursive call MAX-HEAPIFY (A,4)](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-28-320.jpg)
![now has i = 4. After swapping A[4] with A[9], as shown in (c), node 4 is fixed up, and the
recursive call MAX-HEAPIFY(A, 9) yields no further change to the data structure.
The running time of MAX-HEAPIFY by the recurrence can be described as
T (n) < = T (2n/3) + O (1)
The solution to this recurrence is T(n)=O(log n)
Building a heap
Build-Max-Heap(A)
1. for i [n/2] to 1
2. do MAX-HEAPIFY(A,i)
4 1 3 2 1 9 1 1 8 7](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-29-320.jpg)
![We can derive a tighter bound by observing that the time for MAX-HEAPIFY to run at a node
varies with the height of the node in the tree, and the heights of most nodes are small. Our
tighter analysis relies on the properties that an n-element heap has height [log n] and at most
[n/2h+1
] nodes of any height h.
The total cost of BUILD-MAX-HEAP as being bounded is T(n)=O(n)
The HEAPSORT Algorithm
HEAPSORT(A)
1. BUILD MAX-HEAP(A)
2. for i=n to 2
3. exchange A[1] with A[i]
4. MAX-HEAPIFY(A,1)](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-30-320.jpg)
![profit piand weight wi. Then we can say that if
n
n
w
p
w
p
w
p ≥≥≥ LL
2
2
1
1 the solution
( )nxxxx LL321 ,, generated by greedy method is an optimal solution. The proof of the above
statement is left as an exercise for the readers. The algorithm 6.1 describes the greedy method for
finding the optimal solution for fractional knapsack problem.
AlgorithmFKNAPSACK (p, w, x, n, M)
// p[1:n] and w[1:n] contains the profit and weight of n objects. Mis the maximum capacity of knapsack
and x[1:n] in the solution vector.//
{
for (i = 1; i<= n; i ++)
x[i] = 0 ; // initialize the solution to 0 //
cu = M // cu is the remaining capacity of the knapsack//
for (i =1; i<= n ; i ++){
if(w[i] >cu )
break;
else{
x[i] = 1 ;
cu = cu – w[i] ;
}
}
if( i<= n){
x[i] = cu/w[i] ;
returnx;
}
Algorithm 1. Greedy algorithm for fractional knapsack problem.
HHHHuffman Codinguffman Codinguffman Codinguffman Coding
Each character is represented in 8 bits when characters are coded using standard codes such as
ASCII. It can be seen that the characters coded using standard codes have fixed-length code word](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-41-320.jpg)
![Algorithm 2. Algorithm of Huffman coding.
Activity Selection ProblemActivity Selection ProblemActivity Selection ProblemActivity Selection Problem
Suppose we have a set of activities S = {a1, a2 ………an} that with to use a common resource. The
objective is to schedule the activities in such a way that maximum number of activities can be
performed. Each activity ai has a start timesi and a finish time fi, where 0 ≤ Si<fi< ∞. The activities ai and
aj are said to compatible if the intervals [si, fi] and [Sj, fj] do not overlap that means si ≥ fj or sj ≥ fi.
For example, let us consider the following set S of activities, which are sorted in monotonically
increasing order of finish time.
i a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11
si 1 3 0 5 3 5 6 8 8 2 12
fi 4 5 6 7 8 9 10 11 72 13 14
For this example, the subsets {a3, a9, a11}, {a1, a4, a8, a11} and {a2, a4, a9, a11} consist of mutually
compatible activities. We have two largest subsets of mutually compatible activities.
Now, we can devise greedy algorithm that works in a top down fashion. We assume that the n
input activities are ordered by monotonically increasing finish time or it can be sorted into this order in
O(nlog2n) time. The greedy algorithm for activity selection problem is given below.
AlgorithmACTIVITY SELECTION (S, f)
{
n = LENGTH (S) ; // n is the total number of activities //
A = {a1} ; // A is the set of selected activities and initialized to a1)//
i = 1; // i represents the recently selected activity //
for (j = 2 ; j< = n ; j++)
{
if (sj ≥ fi) {
A = A∪ {am} ;
i = j ;
}](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-45-320.jpg)
![This algorithm starts with a tree that has only one edge, the minimum weight edge. The edges (j, q)
is added one by one such that node j is already included, node q is not included and weight wt(j, q) is the
minimum amongst all the edges (x, y) for which x is in the tree and yis not. In order to execute this
algorithm efficiently, we have a node index near(j) associated with each node j that is not yet included in
the tree. If a node is included in the tree, near(j) = 0. The node near(j) is selected into the tree such that
wt(j, near(j)) in the minimum amongst all possible choices for near(j).
AlgorithmPRIM (E, wt, n, T)
//E is the set of edges, wt(n, n) is the weight adjacency matrix for G, n is the number of nodes and T(n–1,
2) stores the spanning tree.
{
(k, l) = edge with minimum wt.
minwt = wt[k, l] ;
T[1, 1] = k, T[1, 2] = l ;
for(i = 1; i<=n; i++){
if(wt[i, k] <wt [i, l] )
near[i] = k;
else
near[i] = l;
}
near[k] = near[l] = 0;
for(i = 2; i≤=n-1 ; i++)
{
letj be an index such that near[j] ≠ 0 and wt[j, near[j]] is minimum.
T[i, 1] = j; T[i, 2] = near[j];
minwt = minwt + wt[j, near[j]] ;
near[j] = 0 ;
for(k = 1; k<= n ; k++){
if (near[k] ≠ 0 and wt[k, near[k]] >wt[k, j])
near[k] = j;
}](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-47-320.jpg)
![• Is there a path from city A to city B?
• If there is more than one path from A to B, which is the shortest or least cost path?
Let us consider the graph G = (V, E), a weighting function w(e) for the edges in E and a source node
v0. The problem is to determine the shortest path from v0 to all the remaining nodes of G. The solution
to this problem is suggested by E.W. Dijkstra and the algorithm is popularly known as Dijustra’s
algorithm.
This algorithm finds the shortest paths one by one. If we have already constructed i shortest paths,
then the next path to be constructed should be the next shortest path. Let S be the set of vertices to
which the shortest paths have already been generated. For z not in S, let dist[z] be the length of the
shortest path starting form v0, going through only those vertices that are in S and ending at z. Let u is the
vertex in S to which the shortest path has already been found. If dist[z] >dist[u] + w(u,z) then dist[z] is
updated to dist[u] + w(u,z) and the predecessor of z is set to u. The Dijustra’s algorithm is presented in
Algorithm 6.6.
AlgorithmDijkstra (v0, W, dist, n)
// v0 is the source vertex, W is the adjacency matrix to store the weights of the links, dist[k] is the array
to store the shortest path to vertex k, n is the number of vertices//
{
for (i = 1 ; i < = n ; i ++){
S[i] = 0; // Initialize the set S to empty i.e. i is not inserted into the set//
dist[i] = w(v0,i) ; //Initialize the distance to each node
}
S[vo] = 1; dist[v0] = 0;
for (j = 2; j<= n; j++) {
choose a vertex u from those vertices not in S such that dist [u] is minimum.
S[u] = 1;
for (each z adjacent to u with S[z] = 0) {
if(dist[z] >dist [u] + w[u, z])
dist[z] = dist [u] + w[u, z];
}
}
}
Algorithm 6.6. Dijkstra’s algorithm for finding shortest path.](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-52-320.jpg)
![• Bottom-up computation: The DP technique starts with the smallest problem instance and
develops the solution to sub instances of longer size and finally obtains the solution of the
original problem instance.
The strategy can be used when the process of obtaining a solution of a problem can be viewed as a
sequence of decisions. The problems of this type can be solved by taking an optimal sequence of
decisions. An optimal sequence of decisions is found by taking one decision at a time and never making
an erroneous decision. In Dynamic Programming, an optimal sequence of decisions is arrived at by using
the principle of optimality. The principle of optimality states that whatever be the initial state and
decision, the remaining decisions must constitute an optimal decision sequence with regard to the state
resulting form the first decision.
A fundamental difference between the greedy strategy and dynamic programming is that in the
greedy strategy only one decision sequence is generated, wherever in the dynamic programming, a
number of them may be generated. Dynamic programming technique guarantees the optimal solution
for a problem whereas greedy method never gives such guarantee.
Matrix chain MultiplicationMatrix chain MultiplicationMatrix chain MultiplicationMatrix chain Multiplication
Let, we have three matrices A1, A2 and A3, with order (10 x 100), (100 x 5) and (5 x 50) respectively.
Then the three matrices can be multiplied in two ways.
(i) First, multiplying A2 and A3, then multiplying A1 with the resultant matrix i.e. A1(A2 A3).
(ii) First, multiplying A1 and A2, and then multiplying the resultant matrix with A3 i.e. (A1A2) A3.
The number of scalar multiplications required in case 1 is 100 * 5 * 50 + 10 * 100 * 50 = 25000 + 50,000
= 75,000 and the number of scalar multiplications required in case 2 is 10 * 100 * 5 + 10 * 5 * 50 = 5000
+ 2500 = 7500
To find the best possible way to calculate the product, we could simply parenthesize the expression
in every possible fashion and count each time how many scalar multiplications are required. Thus the
matrix chain multiplication problem can be stated as “find the optimal parenthesisation of a chain of
matrices to be multiplied such that the number of scalar multiplications is minimized”.
Dynamic Programming Approach for Matrix Chain Multiplication
Let us consider a chain of n matrices A1, A2……….An, where the matrix Ai has dimensions P[i-1] x P[i].
Let the parenthesisation at k results two sub chains A1…….Ak and Ak+1……..An. These two sub chains must
each be optimal for A1……An to be optimal. The cost of matrix chain (A1….An) is calculated as
cost(A1……Ak) + cost(Ak+1…...An) + cost of multiplying two resultant matrices together i.e.
cost(A1……An)= cost(A1……Ak) + cost(Ak+1…...An) + cost of multiplying two resultant matrices together.
Here, the cost represents the number of scalar multiplications. The sub chain (A1….Ak) has a dimension
P[0] x P[k] and the sub chain (Ak+1……An) has a dimension P[k] x P[n]. The number of scalar
multiplications required to multiply two resultant matrices is P[0] x P[k] x P[n].](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-54-320.jpg)
![][][]1[],[],[],[ jPkPiPjkmkimjim ××−++=
{ }][][]0[],[],1[min],1[
1
nPkPPnkmkmnm
nk
××++=
<≤
Let m[i, j] be the minimum number of scalar multiplications required to multiply the matrix chain
(Ai………..Aj). Then
(i) m[i, j] = 0 if i = j
(ii) m[i, j] = minimum number of scalar multiplications required to multiply (Ai….Ak) + minimum
number of scalar multiplications required to multiply (Ak+1….An) + cost of
multiplying two resultant matrices i.e.
However, we don’t know the value of k, for which m[i, j] is minimum. Therefore, we have to try all j – i
possibilities.
[ ] { }
××−++
=
=
<≤
OtherwisejPkPiPjkmkim
jif i
jim
jki
][][]1[],[],[min
0
,
Therefore, the minimum number of scalar multiplications required to multiply n matrices A1 A2……An is
The dynamic programming approach for matrix chain multiplication is presented in Algorithm 7.2.
AlgorithmMATRIX-CHAIN-MULTIPLICATION (P)
// P is an array of length n+1 i.e. from P[0] to P[n]. It is assumed that the matrix Ai has the dimension P[i-
1] ×P[i].
{
for(i = 1; i<=n; i++)
m[i, i] = 0;
for(l = 2; l<=n; l++){
for(i = 1; i<=n-(l-1); i++){
j = i + (l-1);
m[i, j] = ∞;
for(k = i; k<=j-1; k++)
q = m[i, k] + m[k+1, j] + P[i-1] P[k] P[j] ;
if (q<m [i, j]){](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-55-320.jpg)
![m[i, j] = q;
s[i, j] = k;
}
}
}
}
returnm and s.
}
Algorithm 7.2 Matrix Chain multiplication algorithm.
Now let us discuss the procedure and pseudo code of the matrix chain multiplication. Suppose, we
are given the number of matrices in the chain is n i.e. A1, A2………An and the dimension of matrix Ai is P[i-
1] ×P[i]. The input to the matrix-chain-order algorithm is a sequenceP[n+1] = {P[0], P[1], …….P[n]}. The
algorithm first computes m[i, i] = 0 for i = 1, 2, …….n in lines 2-3. Then, the algorithm computes m[i, j] for
j– i = 1 in the first step to the calculation of m[i, j] for j – i = n -1 in the last step. In lines 3 – 11, the value
of m[i, j] is calculated for j – i = 1 to j –i = n – 1 recursively. At each step of the calculation of m[i, j], a
calculation on m[i, k] and m[k+1, j] for i≤k<j, are required, which are already calculated in the previous
steps.
To find the optimal placement of parenthesis for matrix chain multiplication Ai, Ai+1, …..Aj, we should
test the value of i≤k<j for which m[i, j] is minimum. Then the matrix chain can be divided from (A1 ……Ak)
and (Ak+1 ……. Aj).
Let us consider matrices A1,A2……A5 to illustrate MATRIX-CHAIN-MULTIPLICATIONalgorithm. The matrix
chain order P = {P0, P1, P2, P3, P4, P5} = {5, 10, 3, 12, 5, 50}. The objective is to find the minimum number
of scalar multiplications required to multiply the 5 matrices and also find the optimal sequence of
multiplications.
The solution can be obtained by using a bottom up approach that means first we should calculate mii
for 1≤i ≤ 5. Then mijis calculated for j – i = 1 to j – i = 4. We can fill the table shown in Fig. 7.4 to find the
solution.
Fig. 7.4Table to store the partial solutions of the matrix chain multiplication problem
The value of mii for 1≤i ≤ 5 can be filled as 0 that means the elements in the first row can be assigned 0.
Then
For j – i = 1](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-56-320.jpg)
![Hence, minimum number of scalar multiplications required to multiply the given five matrices in
1655.
To find the optimal parenthesization of A1……….A5, we find the value of k is 4 for which m15 is
minimum. So the matrices can be splitted to (A1….A4) (A5). Similarly, (A1….A4) can be splitted to (A1A2) (A3
A4) because for k = 2, m14 is minimum. No further splitting is required as the subchains (A1A2) and (A3 A4)
has length 1. So the optimal paranthesization of A1 …….A5 in ( (A1 A2) (A3 A4) ) (A5).
Time complexity of multiplying a chain of n matrices
Let T(n) be the time complexity of multiplying a chain of n matrices.
( )
( ) ( )[ ]
( ) ( )[ ]
( ) ( ) [ ]
( ) [ ] )1.7()1()2()1(2)(
)()(11
11)()(1)(
11)()(1
11
)(
1
1
1
1
1
1
LLLLL −++++Θ=⇒
−++−Θ+Θ=
>Θ+−++Θ=⇒
>Θ+−++Θ
=Θ
=
∑
∑
∑
−
=
−
=
−
=
nTTTnnT
knTkTn
nifknTkTnT
nifknTkT
if n
nT
n
k
n
k
n
k
Replacing n by n-1, we get
( ) [ ] )2.7()2()2()1(21)1( LLLLL −++++−Θ=− nTTTnnT
Subtracting equation 7.2 from equation 7.1, we have
( ) ( )
( )
( ) ( )[ ] ( ) ( )
( )[ ]
( )[ ]
( )n
n
n
nn
T
nTnT
nTnT
nTnnnTnT
2
2
13
33311
)1(333311
)2(3131)2(3131
)1(31)(
)1(21)1()(
12
122
2
Ο=
−
=
++++Θ=
+++++Θ=
−+Θ+Θ=−+Θ+Θ=
−+Θ=⇒
−+−Θ−Θ=−−
−
−−
LL
LL
Longest Common SubsequenceLongest Common SubsequenceLongest Common SubsequenceLongest Common Subsequence
The longest common subsequence (LCS) problem can be formulated as follows “Given two
sequences X = 〈x1, x2 ……….xn〉 and Y = 〈y1, y2………yn〉 and the objective is to find the LCS Z = 〈z1, z2
………zn〉 that is common to x and y”.](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-58-320.jpg)
![Let us define c[m, n] to be the length of an LCS of the sequences Xm and Yn. The optimal structure of
the LCS problem gives the recursive formula
{ }
)1.7....(..........
if1],[],1,[max
if11]1,[
0or0if0
][
≠−−
=+−−
==
=
nm
nm
yxnmcnmc
yxnmc
nm
m,nc
Generalizing equation 7.1, we can formulate
{ }
)2.7....(..........
if1],[],1,[max
if11]1,[
0or0if0
][
≠−−
=+−−
==
=
ji
ji
yxjicjic
yxjic
ji
i,jc
Computing the length of an LCS
Based on equation (7.1), we could write an exponential recursive algorithm but there are only m*n
distinct problems. Hence, for the solution of m*n distinct subproblems, we use dynamic programming to
compute the solution using bottom up approach.
The algorithm LCS_length (X, Y) takes two sequences X = 〈x1, x2………..xn〉 and Y = 〈y1, y2………..yn〉 as
inputs and find c[m, n] as the maximum length of the subsequence in X and Y. It stores c[i, j] and b[i, j] in
tables c[m, n] and b[m, n] respectively, which simplifies the construction of optimal solution.
AlgorithmLCS_LENGTH (X, Y)
{
m=length [X]
n=length [Y]
for( i =1; i<=m; i++)
c[i,0] = 0;
for(j=0; j<n; j++)
c[0, j]= 0;
for(i=1; i< m; i++){
for(j = 1; j <= n; j++){
if(x[i] = = y[j]) {
c[i, j] = 1 + c[i-1, j-1];
b[i,j] = ‘ ’;
}](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-60-320.jpg)
![else{
if(c[ i-1, j] ≥ c[i, j-1] )
c[i, j] = c[i-1, j];
b[i,j] = ‘↑↑↑↑’;
else
c[i,j] = c[i, j-1];
b[i, j] = ‘←←←←’
}
return c and b ;
}
Algorithm 7.3 Algorithm for finding Longest common subsequence .
Constructing an LCS
The algorithm LCS_LENGTH returns c and b tables. The b table can be used to construct the LCS of X and Y
quickly.
AlgorithmPRINT_LCS (b, X, i, j)
{
if (i == 0 || j == 0)
return;
if (b[i, j] = = ‘ ‘) {
PRINT_LCS (b, X, i-1, j-1)
Printxi
}
else if (b[i, j] = = ‘↑’)
PRINT_LCS (b, X, i-1, j)
else
PRINT_LCS (b, X, i, j-1)
}](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-61-320.jpg)
![length is −∞. When there are no negaƟve weight cycles, the shortest path between any two nodes in n-
node graph contains n−1 edges.
Let distl
[u] be the length of the shortest path from source node v to the node u under the constraint
that the shortest path contains at most l edges. The dist1
[u]=cost[v,u], for 1≤u≤n. As we discussed
earlier, if there is no negative weight cycle then the shortest path contains atmostn−1 edges. Hence,
distn-1
[u] in the length of the shortest path from v to u. Our goal is to compute distn-1
[u] and this can be
done by using Dynamic programming approach. We can make the following observations
(i) If the shortest path from v to u with at most k, k>1 edges has no more than k−1 edges, then
distk
[u]= distk-1
[u].
(ii) If the shortest path from v to u with at most k, k>1 edges has exactly k edges, then it is made up
of a shortest path from v to some vertex j followed by edge<j, u>. The path from v to j has k−1
edges and its length is distk-1
[j]. All vertices i such that <i, u>∈E are the candidates of j. Since we
are interested in a shortest path, the i that minimizes distk-1
[i]=cost[i,u] is the correct value for j.
Distk
[u]=
( )
[ ] ( ){ }{ }uitidistu k
Eui
,cosmin],[dist 1
,
1-k
min
+−
∈
The Bellman ford algorithm is presented in Algorithm 7.6.
AlgorithmBELLMAN FORD(v,cost,dist,n)
//v is the source, cost is the adjacency matrix representing the cost of edges, dist stores the distance to
all nodes, n is the number of nodes//
{
1 for(i=1;i<=n; i++)
2 dist[i]=cost[v][i];
3 for(k=2; k<= n-1; k++){
4 for each u such that k ≠ v and (j, u)∈E
5 if(dist[u]>dist[j]+cost[j][u])
6 dist[u]=dist[j]+cost[j][u];
7 }//end for k
}
Algorithm 7.6 Bellman FordAlgorithm.
The Bellman ford algorithm starts with computing the path from source to each node i in v. Since there
are n nodes in a graph, the path to any node can contain atmostn-1 nodes. Then for each node u,
remaining n-2 nodes are required to be examined. If <j, u>∈E then the condition dist[u] is compared](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-66-320.jpg)
![with dist[j] + cost[j][u]. If dist[u] is greater than dist[j]+dist[j][u] then dist[u] is updated to
dist[j]+cost[j][u].
Let us consider the example network shown in Fig. 7.12.
The example graph and its adjacency matric
The time complexity of Bellman ford algorithm is O(ne). The lines 1-2 takes O(n) time. If the matrix is
stored in an adjacency list then lines 4-6 takes O(e) time. Hence, the lines 3-7 takes O(ne) time.
Therefore, the several time of Bellman ford algorithm is O(ne).
Stepwise working procedure of Bellman Ford AlgorithmStepwise working procedure of Bellman Ford AlgorithmStepwise working procedure of Bellman Ford AlgorithmStepwise working procedure of Bellman Ford Algorithm](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-67-320.jpg)
![Thus the optimal solution to the problem can be found by solving optimal solution of the sub-problems
that in the fastest way to either S1,j-1 or S2,j-1. This is referred as optimal structure of assembly line
scheduling problem.
If we find the fastest way to solve assembly line scheduling problem through station j-1 on either
line 1 or line 2. Thus the fastest way through station S1,j is either
• the fastest way to S1, j-1 and then directly through station S1,j.
• the fastest way to S2,j-1 and a transfer from line 2 to line 1 and then through station S1,j.
Similarly, the fastest way through station S2,j is
• the fastest way to S2,j-1 and then directly through station S2,j.
• the fastest way to S1,j-1, a transfer time from station 1.
Let fi[j] be the fastest possible time to get a chassis from the starting point through station Si,j of the
assembly line i. Then chassis directly goes to the first station of each line.
[ ] 1,111 1 aef +=
[ ] 1,222 1 aef +=
Now, we can compute fi[j], 2<j≤n and 1≤i≤2 as
f1[j]=min{f1[j-1] + a1,j, f2[j-1] + t2,j-1 + a1,j }
f2[j]=min{f2[j-1] + a2,j, f1[j-1] + t1,j-1 + a2,j }
If the chassis goes all the way through station n either line 1 or line 2 and then exits, we have
[ ]
[ ] [ ]{ }
+−+−
=+
=
− Otherwiseatnfanfmin
1nifae
nf
nnn1 ,1,1 ,11,22,1
1,11
1
[ ]
[ ] [ ]{ }
+−+−
=+
=
− Otherwiseatnfanfmin
1nifae
nf
nnn ,11,11,22
1,12
2
,1,1
Let the fastest time to get the chassis all the way through the factory is denoted by f*
. Then
f*
= min{f1[n] + x1, f2[n] + x2 }
Now, we can compute the stations through which the chassis must move to deliver the end product at
minimum time. This can be calculated with a backward approach. Let l*
denote the line whose station n
is used in a fastest way through the entire process. If f1[n] + x1<f2[n] + x2 then l*
=1 else l*
=2. Then let us
denote li[j] be the line number 1 or 2 whose station j-1 is used in a fastest way through station Si,j. li[1] is
not required to be calculated since there is no station proceeding to station Si,1.
If f1[j-1] + a1,j ≤f2[j-1]+t2,j-1 + a1,j then l1[j]=1.Otherwise, l1[j]=2. Similarly, if f2[j-1]+ a2,j≤ f1[j-1]+t1,j-1 + a2,j
then l2[j]=2. Otherwise, l2[j]=1. The algorithm for assembly line scheduling problem is presented in
Algorithm7.7.](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-69-320.jpg)
![Illustration of assembly line scheduling procedure.
f1[1] = e1 + a1,1 = 2 + 4 = 6 f2[1] = e2 + a2,1 = 3 + 7 = 10
f1[2] = min{f1[1] + a1,2, f2[1] + t2,1 + a1,2 }
= min{6 + 7, 10 + 2 + 7} =13
f2[2] = min{f2[1] + a2,2, f1[1] + t1,1 + a2,2 }
= min{10 + 2, 6 + 2 + 2} =10
The time required at all the stations of both theassembly lines are shown above. The minimum time
required to assemble an auto is 33. The assembly lines through which the complete auto is assembled is
shown above.](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-70-320.jpg)
![Lecture – 21 Disjoint Set Data Structure
In computing, a disjoint-set data structure, also called a union–find data structure or merge–
find set, is a data structure that keeps track of a set of elements partitioned into a number
of disjoint (non-overlapping) subsets.
It supports the following useful operations:
• Find: Determine which subset a particular element is in. Find typically returns an item
from this set that serves as its "representative"; by comparing the result of
two Find operations, one can determine whether two elements are in the same subset.
• Union: Join two subsets into a single subset.
• Make Set, which makes a set containing only a given element (a singleton), is generally
trivial. With these three operations, many practical partitioning problems can be solved.
In order to define these operations more precisely, some way of representing the sets is needed.
One common approach is to select a fixed element of each set, called its representative, to
represent the set as a whole. Then, Find(x) returns the representative of the set that x belongs to,
and Union takes two set representatives as its arguments.
Example :
Make Set creates 8 singletons.
After some operations of Union, some sets are grouped together.
Applications :
• partitioning of a set
• Boost Graph Library to implement its Incremental Connected Components functionality.
• implementing Kruskal's algorithm to find the minimum spanning tree of a graph.
• Determine the connected components of an undirected graph.
CONNECTED-COMPONENTS(G)
1. for each vertex v∈V[G]
2. do MAKE-SET(v)
3. for each edge (u,v) ∈E[G]
4. doif FIND-SET(u) ≠ FIND-SET(v)
5. then UNION(u,v)
SAME-COMPONENT(u,v)
1. if FIND-SET(u)=FIND-SET(v)
2. thenreturn TRUE
3. elsereturn FALSE](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-72-320.jpg)
![Time and space complexity
The time complexity can be expressed as [3]
since every vertex and every edge will
be explored in the worst case. Note: may vary between and , depending
on how sparse the input graph is.
When the number of vertices in the graph is known ahead of time, and additional data structures are
used to determine which vertices have already been added to the queue, the space complexity can
be expressed as where is the cardinality of the set of vertices. If the graph is
represented by an Adjacency list it occupies [4]
space in memory, while anAdjacency
matrix representation occupies .
Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures.
One starts at the root(selecting some arbitrary node as the root in the case of a graph) and explores
as far as possible along each branch beforebacktracking.
Historical Background
A version of depth-first search was investigated in the 19th century by French mathematician
Charles Pierre Trémaux
Example
A DFS search will visit the nodes in the following order: A, B, D, F, E, C, G
Pseudo Code
Input: A graph G and a vertex v of G
Output: All vertices reachable from v labeled as discovered
A recursive implementation of DFS
1 procedure DFS(G,v):
2 label v as discovered
3 for all edges from v to winG.adjacentEdges(v) do
4 if vertex w is not labeled as discovered then
5 recursively call DFS(G,w)](https://image.slidesharecdn.com/designanalysisofalgorithmslecturenotes-160413121002/85/Design-Analysis-of-Algorithms-Lecture-Notes-78-320.jpg)
Design & Analysis of Algorithms Lecture Notes
- 1. LECTURE NOTES ON DESIGN AND ANALYSIS OF ALGORITHMS B. Tech. 6th Semester Computer Science & Engineering and Information Technology Prepared by Mr. S.K. Sathua – Module I Dr. M.R. Kabat – Module II Dr. R. Mohanty – Module III VEER SURENDRA SAI UNIVERSITY OF TECHNOLOGY, BURLA SAMBALPUR, ODISHA, INDIA – 768018
- 2. CONTENTS MODULE – I Lecture 1 - Introduction to Design and analysis of algorithms Lecture 2 - Growth of Functions ( Asymptotic notations) Lecture 3 - Recurrences, Solution of Recurrences by substitution Lecture 4 - Recursion tree method Lecture 5 - Master Method Lecture 6 - Worst case analysis of merge sort, quick sort and binary search Lecture 7 - Design and analysis of Divide and Conquer Algorithms Lecture 8 - Heaps and Heap sort Lecture 9 - Priority Queue Lecture 10 - Lower Bounds for Sorting MODULE -II Lecture 11 - Dynamic Programming algorithms Lecture 12 - Matrix Chain Multiplication Lecture 13 - Elements of Dynamic Programming Lecture 14 - Longest Common Subsequence Lecture 15 - Greedy Algorithms Lecture 16 - Activity Selection Problem Lecture 17 - Elements of Greedy Strategy Lecture 18 - Knapsack Problem Lecture 19 - Fractional Knapsack Problem Lecture 20 - Huffman Codes MODULE - III Lecture 21 - Data Structure for Disjoint Sets Lecture 22 - Disjoint Set Operations, Linked list Representation Lecture 23 - Disjoint Forests Lecture 24 - Graph Algorithm - BFS and DFS Lecture 25 - Minimum Spanning Trees Lecture 26 - Kruskal algorithm Lecture 27 - Prim's Algorithm Lecture 28 - Single Source Shortest paths Lecture 29 - Bellmen Ford Algorithm Lecture 30 - Dijkstra's Algorithm MODULE -IV Lecture 31 - Fast Fourier Transform Lecture 32 - String matching Lecture 33 - Rabin-Karp Algorithm Lecture 34 - NP-Completeness Lecture 35 - Polynomial time verification Lecture 36 - Reducibility Lecture 37 - NP-Complete Problems (without proofs) Lecture 38 - Approximation Algorithms Lecture 39 - Traveling Salesman Problem
- 3. MODULE - I • Lecture 1 - Introduction to Design and analysis of algorithms • Lecture 2 - Growth of Functions ( Asymptotic notations) • Lecture 3 - Recurrences, Solution of Recurrences by substitution • Lecture 4 - Recursion tree method • Lecture 5 - Master Method • Lecture 6 - Design and analysis of Divide and Conquer Algorithms • Lecture 7 - Worst case analysis of merge sort, quick sort and binary search • Lecture 8 - Heaps and Heap sort • Lecture 9 - Priority Queue • Lecture 10 - Lower Bounds for Sorting
- 4. Lecture 1 - Introduction to Design and analysis of algorithms What is an algorithm? Algorithm is a set of steps to complete a task. For example, Task: to make a cup of tea. Algorithm: • add water and milk to the kettle, • boilit, add tea leaves, • Add sugar, and then serve it in cup. What is Computer algorithm? ‘’a set of steps to accomplish or complete a task that is described precisely enough that a computer can run it’’. Described precisely: very difficult for a machine to know how much water, milk to be added etc. in the above tea making algorithm. These algorithmsrun on computers or computational devices.Forexample, GPS in our smartphones, Google hangouts. GPS uses shortest path algorithm. Online shopping uses cryptography which uses RSA algorithm. Characteristics of an algorithm:- • Must take an input. • Must give some output(yes/no,valueetc.) • Definiteness –each instruction is clear and unambiguous. • Finiteness –algorithm terminates after a finite number of steps. • Effectiveness –every instruction must be basic i.e. simple instruction.
- 5. Expectation from an algorithm • Correctness:- Correct: Algorithms must produce correct result. Produce an incorrect answer:Even if it fails to give correct results all the time still there is a control on how often it gives wrong result. Eg.Rabin-Miller PrimalityTest (Used in RSA algorithm): It doesn’t give correct answer all the time.1 out of 250 times it gives incorrect result. Approximation algorithm: Exact solution is not found, but near optimal solution can be found out. (Applied to optimization problem.) • Less resource usage: Algorithms should use less resources (time and space). Resource usage: Here, the time is considered to be the primary measure of efficiency .We are also concerned with how much the respective algorithm involves the computer memory.But mostly time is the resource that is dealt with. And the actual running time depends on a variety of backgrounds: like the speed of the Computer, the language in which the algorithm is implemented, the compiler/interpreter, skill of the programmers etc. So, mainly the resource usage can be divided into: 1.Memory (space) 2.Time Time taken by an algorithm? performance measurement or Apostoriori Analysis: Implementing the algorithm in a machine and then calculating the time taken by the system to execute the program successfully. Performance Evaluation or Apriori Analysis. Before implementing the algorithm in a system. This is done as follows
- 6. 1. How long the algorithm takes :-will be represented as a function of the size of the input. f(n)→how long it takes if ‘n’ is the size of input. 2. How fast the function that characterizes the running time grows with the input size. “Rate of growth of running time”. The algorithm with less rate of growth of running time is considered better. How algorithm is a technology ? Algorithms are just like a technology. We all use latest and greatest processors but we need to run implementations of good algorithms on that computer in order to properly take benefits of our money that we spent to have the latest processor. Let’s make this example more concrete by pitting a faster computer(computer A) running a sorting algorithm whose running time on n values grows like n2 against a slower computer (computer B) running asorting algorithm whose running time grows like n lg n. They eachmust sort an array of 10 million numbers. Suppose that computer A executes 10 billion instructions per second (faster than anysingle sequential computer at the time of this writing) and computer B executes only 10 million instructions per second, so that computer A is1000 times faster than computer B in raw computing power. To makethe difference even more dramatic, suppose that the world’s craftiestprogrammer codes in machine language for computer A, and the resulting code requires 2n2 instructions to sort n numbers. Suppose furtherthat just an average programmer writes for computer B, using a high- level language with an inefficient compiler, with the resulting code taking 50n lg n instructions. Computer A (Faster) Computer B(Slower) Running time grows like n2 . Grows innlogn. 10 billion instructions per sec. 10million instruction per sec 2n2 instruction. 50 nlogn instruction.
- 7. Time taken= It is more than 5.5hrs it is under 20 mins. So choosing a good algorithm (algorithm with slower rate of growth) as used by computer B affects a lot. Lecture 2 - Growth of Functions ( Asymptotic notations) Before going for growth of functions and asymptotic notation let us see how to analyase an algorithm. How to analyse an Algorithm Let us form an algorithm for Insertion sort (which sort a sequence of numbers).The pseudo code for the algorithm is give below. Pseudo code: for j=2 to A length -------------------------------------------------- C1 key=A[j]-----------------------------------------------------------------C2 //Insert A[j] into sorted Array A[1.....j-1]------------------------C3 i=j-1------------------------------------------------------------------------C4 while i>0 & A[j]>key---------------------------------------------------C5 A[i+1]=A[i]---------------------------------------------------------------C6 i=i-1------------------------------------------------------------------------C7 A[i+1]=key----------------------------------------------------------------C8
- 8. Let Ci be the cost of ith line. Since comment lines will not incur any cost C3=0. Cost No. Of times Executed C1n C2 n-1 C3=0 n-1 C4n-1 C5 C6 ) C7 C8n-1 Running time of the algorithm is: T(n)=C1n+C2(n-1)+0(n-1)+C4(n-1)+C5 +C6( )+C7( )+ C8(n-1) Best case: It occurs when Array is sorted.
- 9. All tjvalues are 1. T(n)=C1n+C2(n-1)+0 (n-1)+C4(n-1)+C5 +C6( )+C7( )+ C8(n-1) =C1n+C2 (n-1) +0 (n-1) +C4 (n-1) +C5 + C8 (n-1) = (C1+C2+C4+C5+ C8) n-(C2+C4+C5+ C8) • Which is of the forman+b. • Linear function of n.So, linear growth. Worst case: It occurs when Array is reverse sorted, and tj =j T(n)=C1n+C2(n-1)+0 (n-1)+C4(n-1)+C5 +C6( )+C7( )+ C8(n-1) =C1n+C2(n-1)+C4(n-1)+C5 +C6( )+C7( )+ C8(n-1) which is of the form an2 +bn+c Quadratic function. So in worst case insertion set grows in n2 . Why we concentrate on worst-case running time? • The worst-case running time gives a guaranteed upper bound on the runningtime for any input. • For some algorithms, the worst case occurs often. For example, when searching, the worst case often occurs when the item being searched for is not present, and searches for absent items may be frequent. • Why not analyze the average case? Because it’s often about as bad as the worst case. Order of growth: It is described by the highest degree term of the formula for running time. (Drop lower-order terms. Ignore the constant coefficient in the leading term.)
- 10. Example: We found out that for insertion sort the worst-case running time is of the form an2 + bn + c. Drop lower-order terms. What remains is an2 .Ignore constant coefficient. It results in n2 .But we cannot say that the worst-case running time T(n) equals n2 .Rather It grows like n2 . But it doesn’t equal n2 .We say that the running time is Θ (n2 ) to capture the notion that the order of growthis n2 . We usually consider one algorithm to be more efficient than another if its worst-case running time has a smaller order of growth. Asymptotic notation • It is a way to describe the characteristics of a function in the limit. • It describes the rate of growth of functions. • Focus on what’s important by abstracting away low-order terms and constant factors. • It is a way to compare “sizes” of functions: O≈ ≤ Ω≈ ≥ Θ ≈ = o ≈ < ω ≈ >
- 12. Example: n2 /2 − 2n = Θ (n2 ), with c1 = 1/4, c2 = 1/2, and n0 = 8.
- 13. Lecture 3-5: Recurrences, Solution of Recurrences by substitution, Recursion Tree and Master Method Recursion is a particularly powerful kind of reduction, which can be described loosely as follows: • If the given instance of the problem is small or simple enough, just solve it. • Otherwise, reduce the problem to one or more simpler instances of the same problem. Recursion is generally expressed in terms of recurrences. In other words, when an algorithm calls to itself, we can often describe its running time by a recurrence equation which describes the overall running time of a problem of size n in terms of the running time on smaller inputs. E.g.the worst case running time T(n) of the merge sort procedure by recurrence can be expressed as T(n)= ϴ(1) ; if n=1 2T(n/2) + ϴ(n) ;if n>1 whose solution can be found as T(n)=ϴ(nlog n) There are various techniques to solve recurrences. 1. SUBSTITUTION METHOD: The substitution method comprises of 3 steps i. Guess the form of the solution ii. Verify by induction iii. Solve for constants We substitute the guessed solution for the function when applying the inductive hypothesis to smaller values. Hence the name “substitution method”. This method is powerful, but we must be able to guess the form of the answer in order to apply it. e.g.recurrence equation: T(n)=4T(n/2)+n
- 14. step 1: guess the form of solution T(n)=4T(n/2) F(n)=4f(n/2) F(2n)=4f(n) F(n)=n2 So, T(n) is order of n2 Guess T(n)=O(n3 ) Step 2: verify the induction Assume T(k)<=ck3 T(n)=4T(n/2)+n <=4c(n/2)3 +n <=cn3 /2+n <=cn3 -(cn3 /2-n) T(n)<=cn3 as (cn3 /2 –n) is always positive So what we assumed was true. T(n)=O(n3 ) Step 3: solve for constants Cn3 /2-n>=0 n>=1 c>=2 Now suppose we guess that T(n)=O(n2 ) which is tight upper bound Assume,T(k)<=ck2 so,we should prove that T(n)<=cn2 T(n)=4T(n/2)+n 4c(n/2)2 +n cn2 +n
- 15. So,T(n) will never be less than cn2 . But if we will take the assumption of T(k)=c1 k2 -c2k, then we can find that T(n) = O(n2 ) 2. BY ITERATIVE METHOD: e.g. T(n)=2T(n/2)+n => 2[2T(n/4) + n/2 ]+n =>22 T(n/4)+n+n => 22 [2T(n/8)+ n/4]+2n =>23 T(n/23) +3n After k iterations ,T(n)=2k T(n/2k )+kn--------------(1) Sub problem size is 1 after n/2k =1 => k=logn So,afterlogn iterations ,the sub-problem size will be 1. So,when k=logn is put in equation 1 T(n)=nT(1)+nlogn nc+nlogn ( say c=T(1)) O(nlogn) 3.BY RECURSSION TREE METHOD: In a recursion tree ,each node represents the cost of a single sub-problem somewhere in the set of recursive problems invocations .we sum the cost within each level of the tree to obtain a set of per level cost,and then we sum all the per level cost to determine the total cost of all levels of recursion . Constructing a recursion tree for the recurrence T(n)=3T(n/4)+cn2
- 16. Constructing a recursion tree for the recurrence T (n)= 3T (n=4) + cn2 .. Part (a) shows T (n), which progressively expands in (b)–(d) to form the recursion tree. The fully expanded tree in part (d) has height log4n (it has log4n + 1 levels). Sub problem size at depth i =n/4i Sub problem size is 1 when n/4i =1 => i=log4n So, no. of levels =1+ log4n Cost of each level = (no. of nodes) x (cost of each node)
- 17. No. Of nodes at depth i=3i Cost of each node at depth i=c (n/4i )2 Cost of each level at depth i=3i c (n/4i )2 = (3/16)i cn2 T(n)= i=0∑log 4 n cn2 (3/16)i T(n)= i=0∑log 4 n -1 cn2 (3/16)i + cost of last level Cost of nodes in last level =3i T(1) c3log 4 n (at last level i=log4n) cnlog 4 3 T(n)= + c nlog 4 3 <= cn2 <= cn2 *(16/13)+ cnlog 4 3 => T(n)=O(n2 ) 4.BY MASTER METHOD: The master method solves recurrences of the form T(n)=aT(n/b)+f(n) where a>=1 and b>1 are constants and f(n) is a asymptotically positive function . To use the master method, we have to remember 3 cases: 1. If f(n)=O(nlog b a - Ɛ ) for some constants Ɛ >0,then T(n)=ϴ(nlog b a ) 2. If f(n)=ϴ( nlog b a ) then T(n)=ϴ(nlog b a logn) 3. If f(n)=Ὠ(nlog b a+Ɛ ) for some constant Ɛ>0 ,and if a*f(n/b)<=c*f(n) for some constant c<1 and all sufficiently large n,then T(n)=ϴ(f(n))
- 18. e.g. T(n)=2T(n/2)+nlogn ans: a=2 b=2 f(n)=nlogn using 2nd formula f(n)=ϴ( nlog 2 2 logk n) =>ϴ(n1 logk n)=nlogn =>K=1 T(n)=ϴ( nlog 2 2 log1 n) =>ϴ(nlog2 n) Lecture 6 - Design and analysis of Divide and Conquer Algorithms DIVIDE AND CONQUER ALGORITHM • In this approach ,we solve a problem recursively by applying 3 steps 1. DIVIDE-break the problem into several sub problems of smaller size. 2. CONQUER-solve the problem recursively. 3. COMBINE-combine these solutions to create a solution to the original problem. CONTROL ABSTRACTION FOR DIVIDE AND CONQUER ALGORITHM Algorithm D and C (P) { if small(P) then return S(P) else { divide P into smaller instances P1 ,P2 .....Pk Apply D and C to each sub problem Return combine (D and C(P1)+ D and C(P2)+.......+D and C(Pk)) }
- 19. } Let a recurrence relation is expressed as T(n)= ϴ(1), if n<=C aT(n/b) + D(n)+ C(n) ,otherwise then n=input size a=no. Of sub-problemsn/b= input size of the sub-problems Lecture 7: Worst case analysis of merge sort, quick sort Merge sort It is one of the well-known divide-and-conquer algorithm. This is a simple and very efficient algorithm for sorting a list of numbers. We are given a sequence of n numberswhich we will assume is stored in an array A [1...n]. Theobjective is to output a permutation of this sequence, sorted in increasing order. This is normally done by permuting the elements within the array A. How can we apply divide-and-conquer to sorting? Here are the major elements of the Merge Sort algorithm. Divide: Split A down the middle into two sub-sequences, each of size roughly n/2 . Conquer: Sort each subsequence (by calling MergeSort recursively on each). Combine: Merge the two sorted sub-sequences into a single sorted list. The dividing process ends when we have split the sub-sequences down to a single item. A sequence of length one is trivially sorted. The key operation where all the work is done is in the combine stage,which merges together two sorted lists into a single sorted list. It turns out that the merging process is quite easy to implement.
- 20. The following figure gives a high-level view of the algorithm. The “divide” phase is shown on the left. It works top-down splitting up the list into smaller sublists. The “conquer and combine” phases areshown on the right. They work bottom-up, merging sorted lists together into larger sorted lists. Merge Sort Designing the Merge Sort algorithm top-down. We’ll assume that the procedure thatmerges two sortedlist is available to us. We’ll implement it later. Because the algorithm is called recursively on sublists,in addition to passing in the array itself, we will pass in two indices, which indicate the first and lastindices of the subarray that we are to sort. The call MergeSort(A, p, r) will sort the sub-arrayA [ p..r ] and return the sorted result in the same subarray. Here is the overview. If r = p, then this means that there is only one element to sort, and we may returnimmediately. Otherwise (if p < r) there are at least two elements, and we will invoke the divide-and-conquer. We find the index q, midway between p and r, namely q = ( p + r ) / 2 (rounded down to thenearest integer). Then we split the array into subarrays A [ p..q ] and A [ q + 1 ..r ] . Call Merge Sort recursively to sort each subarray. Finally, we invoke a procedure (which we have yet to write) whichmerges these two subarrays into a single sorted array. MergeSort(array A, int p, int r) { if (p < r) { // we have at least 2 items q = (p + r)/2 MergeSort(A, p, q) // sort A[p..q] MergeSort(A, q+1, r) // sort A[q+1..r]
- 21. Merge(A, p, q, r) // merge everything together } } Merging: All that is left is to describe the procedure that merges two sorted lists. Merge(A, p, q, r)assumes that the left subarray, A [ p..q ] , and the right subarray, A [ q + 1 ..r ] , have already been sorted.We merge these two subarrays by copying the elements to a temporary working array called B. Forconvenience, we will assume that the array B has the same index range A, that is, B [ p..r ] . We have to indices i and j, that point to the current elements ofeach subarray. We move the smaller element into the next position of B (indicated by index k) andthen increment the corresponding index (either i or j). When we run out of elements in one array, thenwe just copy the rest of the other array into B. Finally, we copy the entire contents of B back into A. Merge(array A, int p, int q, int r) { // merges A[p..q] with A[q+1..r] array B[p..r] i = k = p //initialize pointers j = q+1 while (i <= q and j <= r) { // while both subarrays are nonempty if (A[i] <= A[j]) B[k++] = A[i++] // copy from left subarray else B[k++] = A[j++] // copy from right subarray } while (i <= q) B[k++] = A[i++] // copy any leftover to B while (j <= r) B[k++] = A[j++] for i = p to r do A[i] = B[i] // copy B back to A } Analysis: What remains is to analyze the running time of MergeSort. First let us consider the running timeof the procedure Merge(A, p, q, r). Let n = r − p + 1 denote the total length of both the leftand right subarrays. What is the running time of Merge as a function of n? The algorithm contains fourloops (none nested in the other). It is easy to see that each loop can be executed at most n times. (Ifyou are a bit more careful you can actually see that all the while-loops
- 22. together can only be executed ntimes in total, because each execution copies one new element to the array B, and B only has space forn elements.) Thus the running time to Merge n items is Θ ( n ) . Let us write this without the asymptoticnotation, simply as n. (We’ll see later why we do this.) Now, how do we describe the running time of the entire MergeSort algorithm? We will do this throughthe use of a recurrence, that is, a function that is defined recursively in terms of itself. To avoidcircularity, the recurrence for a given value of n is defined in terms of values that are strictly smallerthan n. Finally, a recurrence has some basis values (e.g. for n = 1 ), which are defined explicitly. Let’s see how to apply this to MergeSort. Let T ( n ) denote the worst case running time of MergeSort onan array of length n. For concreteness we could count whatever we like: number of lines of pseudocode,number of comparisons, number of array accesses, since these will only differ by a constant factor.Since all of the real work is done in the Merge procedure, we will count the total time spent in theMerge procedure. First observe that if we call MergeSort with a list containing a single element, then the running time is aconstant. Since we are ignoring constant factors, we can just write T ( n ) =1 . When we call MergeSortwith a list of length n >1 , e.g. Merge(A, p, r), where r − p +1 = n, the algorithm first computes q = ( p + r ) / 2 . The subarray A [ p..q ] , which contains q − p + 1 elements. You can verify that is of size n/ 2 . Thus the remaining subarray A [ q +1 ..r ] has n/ 2 elements in it. How long does it taketo sort the left subarray? We do not know this, but because n/ 2< n for n >1 , we can express this as T (n/ 2) . Similarly, we can express the time that it takes to sort the right subarray as T (n/ 2). Finally, to merge both sorted lists takes n time, by the comments made above. In conclusion we have T ( n ) =1 if n = 1 , 2T (n/ 2) + n otherwise. Solving the above recurrence we can see that merge sort has a time complexity of Θ (n log n) .
- 23. QUICKSORT Worst-case running time: O (n2). Expected running time: O (n lgn). Sorts in place. Description of quicksort Quicksort is based on the three-step process of divide-and-conquer. • To sort the subarrayA[p . . r ]: Divide: Partition A[p . . r ], into two (possibly empty) subarraysA[p . . q − 1] and A[q + 1 . . r ], such that each element in the ÞrstsubarrayA[p . . q − 1] is ≤ A[q] and A[q] is ≤ each element in the second subarrayA[q + 1 . . r ]. Conquer: Sort the two subarrays by recursive calls to QUICKSORT. Combine: No work is needed to combine the subarrays, because they are sorted in place. • Perform the divide step by a procedure PARTITION, which returns the index q that marks the position separating the subarrays. QUICKSORT (A, p, r) ifp < r thenq ←PARTITION(A, p, r ) QUICKSORT (A, p, q − 1) QUICKSORT (A, q + 1, r) Initial call is QUICKSORT (A, 1, n) Partitioning Partition subarrayA [p . . . r] by the following procedure: PARTITION (A, p, r) x ← A[r ] i ← p –1 for j ← p to r –1 do if A[ j ] ≤ x theni ← i + 1
- 24. exchangeA[i ] ↔ A[ j ] exchangeA[i + 1] ↔ A[r ] returni + 1 • PARTITION always selects the last element A[r ] in the subarrayA[p . . r ] as the pivot the element around which to partition. • As the procedure executes, the array is partitioned into four regions, some of which may be empty: Performance of Quicksort The running time of Quicksort depends on the partitioning of the subarrays: • If the subarrays are balanced, then Quicksort can run as fast as mergesort. • If they are unbalanced, then Quicksort can run as slowly as insertion sort. Worst case • Occurs when the subarrays are completely unbalanced. • Have 0 elements in one subarray and n − 1 elements in the other subarray.
- 25. • Get the recurrence T (n) = T (n − 1) + T (0) + Θ (n) = T (n − 1) + Θ (n) = O (n2) . • Same running time as insertion sort. • In fact, the worst-case running time occurs when Quicksort takes a sorted array as input, but insertion sort runs in O(n) time in this case. Best case • Occurs when the subarrays are completely balanced every time. • Each subarray has ≤ n/2 elements. • Get the recurrence T (n) = 2T (n/2) + Θ (n) = O(n lgn). Balanced partitioning • QuickPort’s average running time is much closer to the best case than to the worst case. • Imagine that PARTITION always produces a 9-to-1 split. • Get the recurrence T (n) ≤ T (9n/10) + T (n/10) + _ (n) = O (n lgn). • Intuition: look at the recursion tree. • It’s like the one for T (n) = T (n/3) + T (2n/3) + O (n). • Except that here the constants are different; we get log10 n full levels and log10/9 n levels that are nonempty. • As long as it’s a constant, the base of the log doesn’t matter in asymptotic notation. • Any split of constant proportionality will yield a recursion tree of depth O (lgn).
- 26. Lecture 8 - Heaps and Heap sort HEAPSORT Inplace algorithm Running Time: O(n log n) Complete Binary Tree The (binary) heap data structure is an array object that we can view as a nearly complete binary tree.Each node of the tree corresponds to an element of the array. The tree is completely filled on all levels except possibly the lowest, which is filled from the left up to a point. The root of the tree is A[1], and given the index i of a node, we can easily compute the indices of its parent, left child, and right child: PARENT (i) => return [i/2] LEFT (i) => return 2i RIGHT (i) => return 2i+ 1 On most computers, the LEFT procedure can compute 2i in one instruction by simply shifting the binary representation of i left by one bit position. Similarly, the RIGHT procedure can quickly compute 2i + 1 by shifting the binary representation of i left by one bit position and then adding in a 1 as the low-order bit.
- 27. The PARENT procedure can compute [i/2] by shifting i right one bit position. Good implementations of heapsort often implement these procedures as "macros" or "inline" procedures. There are two kinds of binary heaps: max-heaps and min-heaps. • In a max-heap,the max-heap property is that for every node i other than the root, A[PARENT(i)] >= A[i] ,that is, the value of a node is at most the value of its parent. Thus, the largest element in a max-heap is stored at the root, and the subtree rooted at a node contains values no larger than that contained at the node itself. • A min-heap is organized in the opposite way; the min-heap property is that for every node i other than the root, A[PARENT(i)<=A[i] , The smallest element in a min-heap is at the root. The height of a node in a heap is the number of edges on the longest simple downward path from the node to a leaf and The height of the heap is the height of its root. Height of a heap of n elements which is based on a complete binary tree is O(log n). Maintaining the heap property MAX-HEAPIFY lets the value at A[i] "float down" in the max-heap so that the subtree rooted at index i obeys the max-heap property. MAX-HEAPIFY(A,i) 1. l LEFT(i) 2. r RIGHT(i) 3. if A[l] > A[i] 4. largest l 5. if A[r] > A[largest] 6. Largest r 7. if largest != i 8. Then exchange A[i] A[largest]
- 28. 9. MAX-HEAPIFY(A,largest) At each step, the largest of the elements A[i], A[LEFT(i)], and A[RIGHT(i)] is determined, and its index is stored in largest. If A[i] is largest, then the subtree rooted at node i is already a max-heap and the procedure terminates. Otherwise, one of the two children has the largest element, and A[i ] is swapped with A[largest], which causes node i and its children to satisfy the max-heap property. The node indexed by largest, however, now has the original value A[i], and thus the subtree rooted at largest might violate the max-heap property. Consequently, we call MAX-HEAPIFY recursively on that subtree. Figure: The action of MAX-HEAPIFY (A, 2), where heap-size = 10. (a) The initial con- figuration, with A [2] at node i = 2 violating the max-heap property since it is not larger than both children. The max-heap property is restored for node 2 in (b) by exchanging A [2] with A[4], which destroys the max-heap property for node 4. The recursive call MAX-HEAPIFY (A,4)
- 29. now has i = 4. After swapping A[4] with A[9], as shown in (c), node 4 is fixed up, and the recursive call MAX-HEAPIFY(A, 9) yields no further change to the data structure. The running time of MAX-HEAPIFY by the recurrence can be described as T (n) < = T (2n/3) + O (1) The solution to this recurrence is T(n)=O(log n) Building a heap Build-Max-Heap(A) 1. for i [n/2] to 1 2. do MAX-HEAPIFY(A,i) 4 1 3 2 1 9 1 1 8 7
- 30. We can derive a tighter bound by observing that the time for MAX-HEAPIFY to run at a node varies with the height of the node in the tree, and the heights of most nodes are small. Our tighter analysis relies on the properties that an n-element heap has height [log n] and at most [n/2h+1 ] nodes of any height h. The total cost of BUILD-MAX-HEAP as being bounded is T(n)=O(n) The HEAPSORT Algorithm HEAPSORT(A) 1. BUILD MAX-HEAP(A) 2. for i=n to 2 3. exchange A[1] with A[i] 4. MAX-HEAPIFY(A,1)
- 31. A TheHEAPSORT procedure takes time O(n log n), since the call to BUILD-MAX- HEAP takes time O(n) and each of the n - 1 calls to MAX-HEAPIFY takes time O(log n). Lecture 10: Lower Bounds For Sorting Review of Sorting: So far we have seen a number of algorithms for sorting a list of numbers in ascendingorder. Recall that an in-place sorting algorithm is one that uses no additional array storage (however,we allow Quicksort to be called in-place even though they need a stack of size O(log n) for keepingtrack of the recursion). A sorting algorithm is stable if duplicate elements remain in the same relativeposition after sorting. Slow Algorithms: Include BubbleSort, InsertionSort, and SelectionSort. These are all simple Θ (n2 )in-place sorting algorithms. BubbleSort and InsertionSort can be implemented as stable algorithms,but SelectionSort cannot (without significant modifications). 1 2 3 4 7 8 9 10 14 16
- 32. Mergesort: Mergesort is a stable Θ(n log n) sorting algorithm. The downside is that MergeSort isthe only algorithm of the three that requires additional array storage, implying that it is not anin-place algorithm. Quicksort: Widely regarded as the fastest of the fast algorithms. This algorithm is O(n log n) in theexpected case, and O(n2) in the worst case. The probability that the algorithm takes asymptoticallylonger (assuming that the pivot is chosen randomly) is extremely small for large n. It is an(almost) in-place sorting algorithm but is not stable. Heapsort: Heapsort is based on a nice data structure, called a heap, which is a fast priority queue.Elements can be inserted into a heap in O(log n) time, and the largest item can be extracted inO(log n) time. (It is also easy to set up a heap for extracting the smallest item.) If you only wantto extract the k largest values, a heap can allow you to do this is O(n + k log n) time. It is anin-place algorithm, but it is not stable. Lower Bounds for Comparison-Based Sorting: Can we sort faster than O(n log n) time? We will give anargument that if the sorting algorithm is based solely on making comparisons between the keys in thearray, then it is impossible to sort more efficiently than (n log n) time. Such an algorithm is called acomparison-based sorting algorithm, and includes all of the algorithms given above.Virtually all known general purpose sorting algorithms are based on making comparisons, so this isnot a very restrictive assumption. This does not preclude the possibility of a sorting algorithm whoseactions are determined by other types of operations, for example, consulting the individual bits ofnumbers, performing arithmetic operations, indexing into an array based on arithmetic operations onkeys.We will show that any comparison-based sorting algorithm for a input sequence ha1; a2; : : : ; animust make at least (n log n) comparisons in the worst-case. This is still a difficult task if you think about it.It is easy to show that a problem can be solved fast (just give an algorithm). But to show that a problemcannot be solved fast you need to reason in some way about all the possible algorithms that might everbe written. In fact, it seems surprising that you could even hope to prove such a thing. The catch hereis that we are limited to using comparison-based algorithms, and there is a clean mathematical way ofcharacterizing all such algorithms.
- 33. Decision Tree Argument: In order to prove lower bounds, we need an abstract way of modeling “any possible”comparison-based sorting algorithm, we model such algorithms in terms of an abstract modelcalled a decision tree.In a comparison-based sorting algorithm only comparisons between the keys are used to determinethe action of the algorithm. Let ha1; a2; : : : ; anibe the input sequence. Given two elements, aiandaj, their relative order can only be determined by the results of comparisons likeai<aj, ai<=aj,ai=aj, ai>=aj, and ai>aj.A decision tree is a mathematical representation of a sorting algorithm (for a fixed value of n). Eachnode of the decision tree represents a comparison made in the algorithm (e.g., a4 : a7), and the twobranches represent the possible results, for example, the left subtree consists of the remaining comparisonsmade under the assumption that a4 _ a7 and the right subtree for a4 > a7. (Alternatively, onemight be labeled with a4 < a7 and the other with a4 _ a7.)Observe that once we know the value of n, then the “action” of the sorting algorithm is completelydetermined by the results of its comparisons. This action may involve moving elements around in thearray, copying them to other locations in memory, performing various arithmetic operations on non- keydata. But the bottom-line is that at the end of the algorithm, the keys will be permuted in the final array in some way. Each leaf in the decision tree is labeled with the final permutation that the algorithmgenerates after making all of its comparisons.To make this more concrete, let us assume that n = 3, and let’s build a decision tree for SelectionSort.Recall that the algorithm consists of two phases. The first finds the smallest element of the entire list,and swaps it with the first element. The second finds the smaller of the remaining two items, and swapsit with the second element. Here is the decision tree (in outline form). The first comparison is betweena1 and a2. The possible results are: a1 <= a2: Then a1 is the current minimum. Next we compare a1 with a3 whose results might be either: a1 <=a3: Then we know that a1 is the minimum overall, and the elements remain in their originalpositions. Then we pass to phase 2 and compare a2 with a3. The possible results are: a2 <=a3: Final output is ha1; a2; a3i. a2 > a3: These two are swapped and the final output is ha1; a3; a2i.
- 34. a1 > a3: Then we know that a3 is the minimum is the overall minimum, and it is swapped witha1. The we pass to phase 2 and compare a2 with a1 (which is now in the third position ofthe array) yielding either: a2 <=a1: Final output is ha3; a2; a1i. a2 > a1: These two are swapped and the final output is ha3; a1; a2i. a1 > a2: Then a2 is the current minimum. Next we compare a2 with a3 whose results might be either: a2 <=a3: Then we know that a2 is the minimum overall. We swap a2 with a1, and then pass to phase 2, and compare the remaining items a1 and a3. The possible results are: a1 <=a3: Final output is ha2; a1; a3i. a1 > a3: These two are swapped and the final output is ha2; a3; a1i. a2 > a3: Then we know that a3 is the minimum is the overall minimum, and it is swapped witha1. We pass to phase 2 and compare a2 with a1 (which is now in the third position of thearray) yielding either: a2<= a1: Final output is ha3; a2; a1i. a2 > a1: These two are swapped and the final output is ha3; a1; a2i. The final decision tree is shown below. Note that there are some nodes that are unreachable. For example,in order to reach the fourth leaf from the left it must be that a1 _ a2 and a1 > a2, which cannotboth be true. Can you explain this? (The answer is that virtually all sorting algorithms, especiallyinefficient ones like selection sort, may make comparisons that are redundant, in the sense that theiroutcome has already been determined by earlier comparisons.) As you can see, converting a complexsorting algorithm like HeapSort into a decision tree for a large value of n will be very tedious andcomplex, but I hope you are convinced by this exercise that it can be done in a simple mechanical way.
- 35. (Decision Tree for SelectionSort on 3 keys.) Using Decision Trees for Analyzing Sorting: Consider any sorting algorithm. Let T(n) be the maximumnumber of comparisons that this algorithm makes on any input of size n. Notice that the running timefo the algorithm must be at least as large as T(n), since we are not counting data movement or othercomputations at all. The algorithm defines a decision tree. Observe that the height of the decisiontree is exactly equal to T(n), because any path from the root to a leaf corresponds to a sequence ofcomparisons made by the algorithm. As we have seen earlier, any binary tree of height T (n) has at most 2T(n) leaves. This means that thissorting algorithm can distinguish between at most 2T (n) different final actions. Let’s call this quantityA (n), for the number of different final actions the algorithm can take. Each action can be thought of asa specific way of permuting the original input to get the sorted output.How many possible actions must any sorting algorithm distinguish between? If the input consists of ndistinct numbers, then those numbers could be presented in any of n! different permutations. For eachdifferent permutation, the algorithm must “unscramble” the numbers in an essentially different way,that is it must take a different action, implying that A(n) >= n!. (Again, A (n) is usually not exactlyequal to n! because most algorithms contain some redundant unreachable leaves.) Since A(n) ≤ 2T(n) we have 2T(n) ≥ n!, implying that T(n) ≥ lg(n!):
- 36. We can use Stirling’s approximation for n! yielding: n! ≥ T(n) ≥ = Thus we have the following theorem. Theorem: Any comparison-based sorting algorithm has worst-case running time (n log n). This can be generalized to show that the average-case time to sort is also (n log n) (by arguing aboutthe average height of a leaf in a tree with at least n! leaves). The lower bound on sorting can begeneralized to provide lower bounds to a number of other problems as well.
- 37. MODULE -II • Lecture 11 - Dynamic Programming algorithms • Lecture 12 - Matrix Chain Multiplication • Lecture 13 - Elements of Dynamic Programming • Lecture 14 - Longest Common Subsequence • Lecture 15 - Greedy Algorithms • Lecture 16 - Activity Selection Problem • Lecture 17 - Elements of Greedy Strategy • Lecture 18 - Knapsack Problem • Lecture 19 - Fractional Knapsack Problem • Lecture 20 - Huffman Codes
- 38. Greedy MethodGreedy MethodGreedy MethodGreedy Method IntroductionIntroductionIntroductionIntroduction Let we are given a problem to sort the array a = {5, 3, 2, 9}. Someone says the array after sorting is {1, 3, 5, 7}. Can we consider the answer is correct? The answer is definitely “no” because the elements of the output set are not taken from the input set. Let someone says the array after sorting is {2, 5, 3, 9}. Can we admit the answer? The answer is again “no” because the output is not satisfying the objective function that is the first element must be less than the second, the second element must be less than the third and so on. Therefore, the solution is said to be a feasible solution if it satisfies the following constraints. (i) Explicit constraints: - The elements of the output set must be taken from the input set. (ii) Implicit constraints:-The objective function defined in the problem. The best of all possible solutions is called the optimal solution. In other words we need to find the solution which has the optimal (maximum or minimum) value satisfying the given constraints.
- 39. .101 1 ≤≤≤≤ ≤∑= i n i ii xandni Mxw The Greedy approach constructs the solution through a sequence of steps. Each step is chosen such that it is the best alternative among all feasible choices that are available. The choice of a step once made cannot be changed in subsequent steps. Let us consider the problem of coin change. Suppose a greedy person has some 25p, 20p, 10p, 5paise coins. When someone asks him for some change then be wants to given the change with minimum number of coins. Now, let someone requests for a change of top then he first selects 25p. Then the remaining amount is 45p. Next, he selects the largest coin that is less than or equal to 45p i.e. 25p. The remaining 20p is paid by selecting a 20p coin. So the demand for top is paid by giving total 3 numbers of coins. This solution is an optimal solution. Now, let someone requests for a change of 40p then the Greedy approach first selects 25p coin, then a 10p coin and finally a 5p coin. However, the some could be paid with two 20p coins. So it is clear from this example that Greedy approach tries to find the optimal solution by selecting the elements one by one that are locally optimal. But Greedy method never gives the guarantee to find the optimal solution. The choice of each step is a greedy approach is done based in the following: • It must be feasible • It must be locally optimal • It must be unalterable Fractional Knapsack ProblemFractional Knapsack ProblemFractional Knapsack ProblemFractional Knapsack Problem Let there are n number of objects and each object is having a weight and contribution to profit. The knapsack of capacity M is given. The objective is to fill the knapsack in such a way that profit shall be maximum. We allow a fraction of item to be added to the knapsack. Mathematically, we can write ∑= n i ii xp 1 maximize Subject to
- 40. Where pi and wi are the profit and weight of ith object and xi is the fraction of ith object to be selected. For example Given n = 3, (p1, p2, p3) = {25, 24, 15} ( w1, w2, w3) = {18, 15, 10} M = 20 Solution Some of the feasible solutions are shown in the following table. Solution No x1 x2 x3 ∑wi xi ∑pi xi 1 1 2/15 0 20 28.2 2 0 2/3 1 20 31.0 3 0 1 1/2 20 31.5 These solutions are obtained by different greedy strategies. Greedy strategy I: In this case, the items are arranged by their profit values. Here the item with maximum profit is selected first. If the weight of the object is less than the remaining capacity of the knapsack then the object is selected full and the profit associated with the object is added to the total profit. Otherwise, a fraction of the object is selected so that the knapsack can be filled exactly. This process continues from selecting the highest profitable object to the lowest profitable object till the knapsack is exactly full. Greedy strategy II: In this case, the items are arranged by fair weights. Here the item with minimum weight in selected first and the process continues like greedy strategy-I till the knapsack is exactly full. Greedy strategy III: In this case, the items are arranged by profit/weight ratio and the item with maximum profit/weight ratio is selected first and the process continues like greedy strategy-I till the knapsack is exactly full. Therefore, it is clear from the above strategies that the Greedy method generates optimal solution if we select the objects with respect to their profit to weight ratios that means the object with maximum profit to weight ratio will be selected first. Let there are n objects and the object i is associated with
- 41. profit piand weight wi. Then we can say that if n n w p w p w p ≥≥≥ LL 2 2 1 1 the solution ( )nxxxx LL321 ,, generated by greedy method is an optimal solution. The proof of the above statement is left as an exercise for the readers. The algorithm 6.1 describes the greedy method for finding the optimal solution for fractional knapsack problem. AlgorithmFKNAPSACK (p, w, x, n, M) // p[1:n] and w[1:n] contains the profit and weight of n objects. Mis the maximum capacity of knapsack and x[1:n] in the solution vector.// { for (i = 1; i<= n; i ++) x[i] = 0 ; // initialize the solution to 0 // cu = M // cu is the remaining capacity of the knapsack// for (i =1; i<= n ; i ++){ if(w[i] >cu ) break; else{ x[i] = 1 ; cu = cu – w[i] ; } } if( i<= n){ x[i] = cu/w[i] ; returnx; } Algorithm 1. Greedy algorithm for fractional knapsack problem. HHHHuffman Codinguffman Codinguffman Codinguffman Coding Each character is represented in 8 bits when characters are coded using standard codes such as ASCII. It can be seen that the characters coded using standard codes have fixed-length code word
- 42. representation. In this fixed-length coding system the total code length is more. For example, let we have six characters (a, b, c, d, e, f) and their frequency of occurrence in a message is {45, 13, 12, 16, 9, 5}. In fixed-length coding system we can use three characters to represent each code. Then the total code length of the message is (45+13+12+16+9+5) x 3 = 100 x 3 = 300. Let us encode the characters with variable-length coding system. In this coding system, the character with higher frequency of occurrence is assigned fewer bits for representation while the characters having lower frequency of occurrence in assigned more bits for representation. The variable length code for the characters are shown in the following tableThe total code length in variable length coding system is 1 × 45 + 3 × 12 + 3 × 16 × 4 × 9 + 4 × 5 = 224. Hence fixed length code requires 300 bits while variable code requires only 224 bits. a b c d e f 0 101 100 111 1101 1100 Prefix (Free) Codes We have seen that using variable-length code word we minimize the overall encoded string length. But the question arises whether we can decode the string. If a is encoded 1 instead of 0 then the encoded string “111” can be decoded as “d” or “aaa”. It can be seen that we get ambiguous string. The key point to remove this ambiguity is to use prefix codes. Prefix codes is the code in which there is no codeword that is a prefix of other codeword. The representation of “decoding process” is binary tree whose leaves are characters. We interpret the binary codeword for a character as path from the root to that character, where ⇒ “0” means “go to the left child” ⇒ “1” means “go to the right child” Greedy Algorithm for Huffman CodeGreedy Algorithm for Huffman CodeGreedy Algorithm for Huffman CodeGreedy Algorithm for Huffman Code: According to Huffman algorithm, a bottom up tree is built starting from the leaves. Initially, there are n singleton trees in the forest, as each tree is a leaf. The greedy strategy first finds two trees having minimum frequency of occurrences. Then these two trees are merged in a single tree where the frequency of this tree is the total sum of two merged trees. The whole process is repeated until there in only one tree in the forest. Let us consider a set of characters S=<a, b, c, d, e, f> with the following frequency of occurrences P = < 45, 13, 12, 16, 5, 9 >. Initially, these six characters with their frequencies are considered six singleton trees in the forest. The step wise merging these trees to a single tree is shown in Fig. 6.3. The merging is done by selecting two trees with minimum frequencies till there is only one tree in the forest. a : 45 b : 13 c : 12 d : 16 e : 5 f : 9
- 43. Step wise merging of the singleton trees. Now the left branch is assigned a code “0” and right branch is assigned a code “1”. The decode tree after assigning the codes to the branches. The binary codeword for a character is interpreted as path from the root to that character; Hence, the codes for the characters are as follows a = 0 b = 101 c = 100 d = 111
- 44. e = 1100 f = 1101 Therefore, it is seen that no code is the prefix of other code. Suppose we have a code 01111001101. To decode the binary codeword for a character, we traverse the tree. The first character is 0 and the character at which the tree traversal terminates is a. Then, the next bit is 1 for which the tree is traversed right. Since it has not reached at the leaf node, the tree is next traversed right for the next bit 1. Similarly, the tree is traversed for all the bits of the code string. When the tree traversal terminates at a leaf node, the tree traversal again starts from the root for the next bit of the code string. The character string after decoding is “adcf”. AlgorithmHUFFMAN(n, S) { // n is the number of symbols and S in the set of characters, for each character c∈ S, the frequency of occurrence in f(c) // Initialize the priority queue; Q = S ; // Inilialize the priority Q with the frequencies of all the characters of set S// for(i =1 ; i<= n-i, i++){ z = CREAT _NODE ( ); // create a node pointed by z; // // Delete the character with minimum frequency from the Q and store in node x// x = DELETE _MIN (Q); // Delete the character with next minimum frequency from the Q and store in node y// y = DELETE_MIN (Q); z→left = x; // Place x as the left child of z// z→right = y; // Place y as the right child of z// //The value of node z is the sum of values at node x and node y// f(z) = f(x) + f(y); //insert z into the priority Q// INSERT (Q, z); } returnDELETE_MIN(Q) }
- 45. Algorithm 2. Algorithm of Huffman coding. Activity Selection ProblemActivity Selection ProblemActivity Selection ProblemActivity Selection Problem Suppose we have a set of activities S = {a1, a2 ………an} that with to use a common resource. The objective is to schedule the activities in such a way that maximum number of activities can be performed. Each activity ai has a start timesi and a finish time fi, where 0 ≤ Si<fi< ∞. The activities ai and aj are said to compatible if the intervals [si, fi] and [Sj, fj] do not overlap that means si ≥ fj or sj ≥ fi. For example, let us consider the following set S of activities, which are sorted in monotonically increasing order of finish time. i a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 si 1 3 0 5 3 5 6 8 8 2 12 fi 4 5 6 7 8 9 10 11 72 13 14 For this example, the subsets {a3, a9, a11}, {a1, a4, a8, a11} and {a2, a4, a9, a11} consist of mutually compatible activities. We have two largest subsets of mutually compatible activities. Now, we can devise greedy algorithm that works in a top down fashion. We assume that the n input activities are ordered by monotonically increasing finish time or it can be sorted into this order in O(nlog2n) time. The greedy algorithm for activity selection problem is given below. AlgorithmACTIVITY SELECTION (S, f) { n = LENGTH (S) ; // n is the total number of activities // A = {a1} ; // A is the set of selected activities and initialized to a1)// i = 1; // i represents the recently selected activity // for (j = 2 ; j< = n ; j++) { if (sj ≥ fi) { A = A∪ {am} ; i = j ; }
- 46. } returnA ; } Algorithm 3. Algorithm of activity selection problem. The algorithm takes the start and finish times of the activities, represented as arrays s and f , length (s) gives the total number of activities. The set A stores the selected activities.Since the activities are ordered with respect to their finish times the set A is initialized to contain just the first activity a1. The variable i stores the index of the recently selected activity. The for loop considers each activityaj and adds to the set A if it is mutually compatible with the previously selected activities. To see whether activity aj is compatible with every activity assumingly in A, it needs to check whether the short time of aj is greater or equal to the finish time of the recently selected activity ai. Minimum Cost Spanning TreeMinimum Cost Spanning TreeMinimum Cost Spanning TreeMinimum Cost Spanning Tree Let G = (V, E) be the graph where V is the set of vertices, E is the set of edges and |V|= n. The spanning tree G′= (V, E′) is a sub graph of G in which all the vertices of graph G are connected with minimum number of edges. The minimum number of edges required to correct all the vertices of a graph G in n – 1. Spanning tree plays a very important role in designing efficient algorithms. Let us consider a graph shown in Fig 6.6(a). There are a number of possible spanning trees that is shown in Fig 6.6(b). If we consider a weighted graph then all the spanning trees generated from the graph have different weights. The weight of the spanning tree is the sum of its edges weights. The spanning tree with minimum weight is called minimum spanning tree (MST). Fig. 6.7 shows a weighted graph and the minimum spanning tree. A greedy method to obtain the minimum spanning tree would construct the tree edge by edge, where each edge is chosen accounting to some optimization criterion. An obvious criterion would be to choose an edge which adds a minimum weight to the total weight of the edges selected so far. There are two ways in which this criterion can be achieved. 1. The set of edges selected so far always forms a tree, the next edge to be added is such that not only it adds a minimum weight, but also forms a tree with the previous edges; it can be shown that the algorithm results in a minimum cost tree; this algorithm is called Prim’s algorithm. 2. The edges are considered in non decreasing order of weight; the set T of edges at each stage is such that it is possible to complete T into a tree; thus T may not be a tree at all stages of the algorithm; this also results in a minimum cost tree; this algorithm is called Krusleat’s algorithm. Prim’s AlgorithmPrim’s AlgorithmPrim’s AlgorithmPrim’s Algorithm
- 47. This algorithm starts with a tree that has only one edge, the minimum weight edge. The edges (j, q) is added one by one such that node j is already included, node q is not included and weight wt(j, q) is the minimum amongst all the edges (x, y) for which x is in the tree and yis not. In order to execute this algorithm efficiently, we have a node index near(j) associated with each node j that is not yet included in the tree. If a node is included in the tree, near(j) = 0. The node near(j) is selected into the tree such that wt(j, near(j)) in the minimum amongst all possible choices for near(j). AlgorithmPRIM (E, wt, n, T) //E is the set of edges, wt(n, n) is the weight adjacency matrix for G, n is the number of nodes and T(n–1, 2) stores the spanning tree. { (k, l) = edge with minimum wt. minwt = wt[k, l] ; T[1, 1] = k, T[1, 2] = l ; for(i = 1; i<=n; i++){ if(wt[i, k] <wt [i, l] ) near[i] = k; else near[i] = l; } near[k] = near[l] = 0; for(i = 2; i≤=n-1 ; i++) { letj be an index such that near[j] ≠ 0 and wt[j, near[j]] is minimum. T[i, 1] = j; T[i, 2] = near[j]; minwt = minwt + wt[j, near[j]] ; near[j] = 0 ; for(k = 1; k<= n ; k++){ if (near[k] ≠ 0 and wt[k, near[k]] >wt[k, j]) near[k] = j; }
- 48. } if(minwt == ∞) print(“No spanning tree”); returnminwt; } Algorithm 4. Prim’s algorithm for finding MST. Fig 6. The weighted undirected graph to illustrate Prim’s algorithm Let us consider the weighted undirected graph shown in Fig.6.8 and the objective is to construct a minimum spanning tree. The step wise operation of Prim’s algorithm is described as follows. Step 1 The minimum weight edge is (2, 3) with weight 5. Hence, the edge (2, 3) is added to the tree. near(2) and near(3) are set 0. Step 2 Find near of all the nodes that are not yet selected into the tree and its cost. near(1) = 2 weight = 16 near(4) = 2 weight = 6 near (5) = - weight = ∞ near (6) = 2 weight = 11 The node 4 is selected and the edge (2, 4) is added to the tree because weight(4, near(4) ) is minimum. Then near(4) is set 0. Step 3 near(1) = 2 weight = 16 near(5) = 4 weight = 18 near(6) = 2 weight = 11 As weight(6, near(6)) is minimum, the node 6 is selected and edge (2, 6) is added to the tree. So near(6) is set 0 Step 4 near (1) = 2 weight = 16 near (5) = 4 weight = 18 Next, the edge (2, 1) is added to the tree as weight(1, near(1)) is minimum. So near(1) is set 0.
- 49. Step 5 near (5) = 1 weight 12 The edge (1, 5) is added to the tree. The Fig. 6.9(a) to 6.9(e) show the step wise construction of MST by Prim’s algorithm. Fig. 6.9 Step wise construction of MST by Prim’s algorithm Time complexity of Prim’s Algorithm Prim’s algorithm has three for loops. The first for loop finds the near of all nodes which require O(n) time. The second for loop is to find the remaining n-2 edges and the third for loop updates near of each node after adding a vertex to MST. Since the third for loop is within the second for loop, it requires O(n2 ) time. Hence, the overall time complexity of Prim’s algorithm is O(n2 ). Kruskal’s AlgorithmKruskal’s AlgorithmKruskal’s AlgorithmKruskal’s Algorithm This algorithm starts with a list of edges sorted in non decreasing order of weights. It repeatedly adds the smallest edge to the spanning tree that does not create a cycle. Initially, each vertex is in its own tree in the forest. Then, the algorithm considers each edge ordered by increasing weights. If the edge (u, v) connects two different trees, then (u, v) is added to the set of edges of the MST and two trees connected by an edge (u, v) are merged in to a single tree. If an edge (u, v) connects two vertices in the same tree, then edge (u, v) is discarded. The Krushal’s algorithm for finding the MST is presented as follows. It starts with an empty set A, and selects at every stage the shortest edge that has not been chosen or rejected regardless of where this edge is situated in the graph. The pseudo code of Kruskal’s algorithm is given in Algorithm . The operations an disjoint sets used for Krushal’s algorithm is as follows: Make_set(v) : create a new set whose only member is pointed to v. Find_set(v) : returns a pointer to the set containing v. Union(u, v) : unites the dynamic sets that contain u and v into a new set that is union of these two sets. AlgorithmKRUSKAL (V, E, W) // V is the set of vertices, E is the set of edges and W is the adjacency matrix to store the weights of the links. // { A = Φ ;
- 50. for (each vertex u in V) Make_set(u) Create a min heap from the weights of the links using procedure heapify. for (each least weight edge (u, v) in E) // least weight edge is the root of the heap// if (Find_set(u) ≠Find_set(v)){ // u and v are in two different sets // A = A∪{u, v} Union(u, v) } } return A ; } Algorithm. 5 Kruskal’s algorithm for finding MST. Let us consider the graph shown in Fig.6.10 to illustrate Kruskal’s algorithm. The step wise procedure to construct MST by following the procedure presented given below.
- 51. Step wise construction of MST by Kruskal’s algorithm Time complexity of Kruskal’s Algorithm The Kruskal’s algorithm first creates n trees from n vertices which is done in O(n) time. Then, a heap is created in O(n) time using heapify procedure. The least weight edge is at the root of the heap. Hence, the edges are deleted one by one from the heap and either added to the MST or discarded if it forms a cycle. This deletion process requires O(nlog2n). Hence, the time complexity of Kruskal’s algorithm is O(nlog2n). Shortest Path ProblemShortest Path ProblemShortest Path ProblemShortest Path Problem Let us consider a number of cities connected with roads and a traveler wants to travel form his home city A to the destination B with a minimum cost. So the traveler will be interested to know the following:
- 52. • Is there a path from city A to city B? • If there is more than one path from A to B, which is the shortest or least cost path? Let us consider the graph G = (V, E), a weighting function w(e) for the edges in E and a source node v0. The problem is to determine the shortest path from v0 to all the remaining nodes of G. The solution to this problem is suggested by E.W. Dijkstra and the algorithm is popularly known as Dijustra’s algorithm. This algorithm finds the shortest paths one by one. If we have already constructed i shortest paths, then the next path to be constructed should be the next shortest path. Let S be the set of vertices to which the shortest paths have already been generated. For z not in S, let dist[z] be the length of the shortest path starting form v0, going through only those vertices that are in S and ending at z. Let u is the vertex in S to which the shortest path has already been found. If dist[z] >dist[u] + w(u,z) then dist[z] is updated to dist[u] + w(u,z) and the predecessor of z is set to u. The Dijustra’s algorithm is presented in Algorithm 6.6. AlgorithmDijkstra (v0, W, dist, n) // v0 is the source vertex, W is the adjacency matrix to store the weights of the links, dist[k] is the array to store the shortest path to vertex k, n is the number of vertices// { for (i = 1 ; i < = n ; i ++){ S[i] = 0; // Initialize the set S to empty i.e. i is not inserted into the set// dist[i] = w(v0,i) ; //Initialize the distance to each node } S[vo] = 1; dist[v0] = 0; for (j = 2; j<= n; j++) { choose a vertex u from those vertices not in S such that dist [u] is minimum. S[u] = 1; for (each z adjacent to u with S[z] = 0) { if(dist[z] >dist [u] + w[u, z]) dist[z] = dist [u] + w[u, z]; } } } Algorithm 6.6. Dijkstra’s algorithm for finding shortest path.
- 53. Let us consider the graph. The objective is to find the shortest path from source vertex 0 to the all remaining nodes. Iteration Set of nodes to which the shortest path is found Vertex selected Distance 0 1 2 3 4 Initial {0} - 0 10 ∞ 5 ∞ 1 {0, 3} 3 0 8 14 5 7 2 {0, 3,4} 4 0 8 13 5 7 3 {0, 3, 4, 1} 1 0 8 9 5 7 4 {0, 3, 4, 1, 2} 2 0 8 9 5 7 The time complexity of the algorithm is O(n2 ). Dynamic ProgrammingDynamic ProgrammingDynamic ProgrammingDynamic Programming IntroductionIntroductionIntroductionIntroduction The Dynamic Programming (DP) is the most powerful design technique for solving optimization problems. It was invented by mathematician named Richard Bellman inn 1950s. The DP in closely related to divide and conquer techniques, where the problem is divided into smaller sub-problems and each sub-problem is solved recursively. The DP differs from divide and conquer in a way that instead of solving sub-problems recursively, it solves each of the sub-problems only once and stores the solution to the sub-problems in a table. The solution to the main problem is obtained by the solutions of these sub- problems. The steps of Dynamic Programming technique are: • Dividing the problem into sub-problems: The main problem is divided into smaller sub- problems. The solution of the main problem is expressed in terms of the solution for the smaller sub-problems. • Storing the sub solutions in a table: The solution for each sub-problem is stored in a table so that it can be referred many times whenever required.
- 54. • Bottom-up computation: The DP technique starts with the smallest problem instance and develops the solution to sub instances of longer size and finally obtains the solution of the original problem instance. The strategy can be used when the process of obtaining a solution of a problem can be viewed as a sequence of decisions. The problems of this type can be solved by taking an optimal sequence of decisions. An optimal sequence of decisions is found by taking one decision at a time and never making an erroneous decision. In Dynamic Programming, an optimal sequence of decisions is arrived at by using the principle of optimality. The principle of optimality states that whatever be the initial state and decision, the remaining decisions must constitute an optimal decision sequence with regard to the state resulting form the first decision. A fundamental difference between the greedy strategy and dynamic programming is that in the greedy strategy only one decision sequence is generated, wherever in the dynamic programming, a number of them may be generated. Dynamic programming technique guarantees the optimal solution for a problem whereas greedy method never gives such guarantee. Matrix chain MultiplicationMatrix chain MultiplicationMatrix chain MultiplicationMatrix chain Multiplication Let, we have three matrices A1, A2 and A3, with order (10 x 100), (100 x 5) and (5 x 50) respectively. Then the three matrices can be multiplied in two ways. (i) First, multiplying A2 and A3, then multiplying A1 with the resultant matrix i.e. A1(A2 A3). (ii) First, multiplying A1 and A2, and then multiplying the resultant matrix with A3 i.e. (A1A2) A3. The number of scalar multiplications required in case 1 is 100 * 5 * 50 + 10 * 100 * 50 = 25000 + 50,000 = 75,000 and the number of scalar multiplications required in case 2 is 10 * 100 * 5 + 10 * 5 * 50 = 5000 + 2500 = 7500 To find the best possible way to calculate the product, we could simply parenthesize the expression in every possible fashion and count each time how many scalar multiplications are required. Thus the matrix chain multiplication problem can be stated as “find the optimal parenthesisation of a chain of matrices to be multiplied such that the number of scalar multiplications is minimized”. Dynamic Programming Approach for Matrix Chain Multiplication Let us consider a chain of n matrices A1, A2……….An, where the matrix Ai has dimensions P[i-1] x P[i]. Let the parenthesisation at k results two sub chains A1…….Ak and Ak+1……..An. These two sub chains must each be optimal for A1……An to be optimal. The cost of matrix chain (A1….An) is calculated as cost(A1……Ak) + cost(Ak+1…...An) + cost of multiplying two resultant matrices together i.e. cost(A1……An)= cost(A1……Ak) + cost(Ak+1…...An) + cost of multiplying two resultant matrices together. Here, the cost represents the number of scalar multiplications. The sub chain (A1….Ak) has a dimension P[0] x P[k] and the sub chain (Ak+1……An) has a dimension P[k] x P[n]. The number of scalar multiplications required to multiply two resultant matrices is P[0] x P[k] x P[n].
- 55. ][][]1[],[],[],[ jPkPiPjkmkimjim ××−++= { }][][]0[],[],1[min],1[ 1 nPkPPnkmkmnm nk ××++= <≤ Let m[i, j] be the minimum number of scalar multiplications required to multiply the matrix chain (Ai………..Aj). Then (i) m[i, j] = 0 if i = j (ii) m[i, j] = minimum number of scalar multiplications required to multiply (Ai….Ak) + minimum number of scalar multiplications required to multiply (Ak+1….An) + cost of multiplying two resultant matrices i.e. However, we don’t know the value of k, for which m[i, j] is minimum. Therefore, we have to try all j – i possibilities. [ ] { } ××−++ = = <≤ OtherwisejPkPiPjkmkim jif i jim jki ][][]1[],[],[min 0 , Therefore, the minimum number of scalar multiplications required to multiply n matrices A1 A2……An is The dynamic programming approach for matrix chain multiplication is presented in Algorithm 7.2. AlgorithmMATRIX-CHAIN-MULTIPLICATION (P) // P is an array of length n+1 i.e. from P[0] to P[n]. It is assumed that the matrix Ai has the dimension P[i- 1] ×P[i]. { for(i = 1; i<=n; i++) m[i, i] = 0; for(l = 2; l<=n; l++){ for(i = 1; i<=n-(l-1); i++){ j = i + (l-1); m[i, j] = ∞; for(k = i; k<=j-1; k++) q = m[i, k] + m[k+1, j] + P[i-1] P[k] P[j] ; if (q<m [i, j]){
- 56. m[i, j] = q; s[i, j] = k; } } } } returnm and s. } Algorithm 7.2 Matrix Chain multiplication algorithm. Now let us discuss the procedure and pseudo code of the matrix chain multiplication. Suppose, we are given the number of matrices in the chain is n i.e. A1, A2………An and the dimension of matrix Ai is P[i- 1] ×P[i]. The input to the matrix-chain-order algorithm is a sequenceP[n+1] = {P[0], P[1], …….P[n]}. The algorithm first computes m[i, i] = 0 for i = 1, 2, …….n in lines 2-3. Then, the algorithm computes m[i, j] for j– i = 1 in the first step to the calculation of m[i, j] for j – i = n -1 in the last step. In lines 3 – 11, the value of m[i, j] is calculated for j – i = 1 to j –i = n – 1 recursively. At each step of the calculation of m[i, j], a calculation on m[i, k] and m[k+1, j] for i≤k<j, are required, which are already calculated in the previous steps. To find the optimal placement of parenthesis for matrix chain multiplication Ai, Ai+1, …..Aj, we should test the value of i≤k<j for which m[i, j] is minimum. Then the matrix chain can be divided from (A1 ……Ak) and (Ak+1 ……. Aj). Let us consider matrices A1,A2……A5 to illustrate MATRIX-CHAIN-MULTIPLICATIONalgorithm. The matrix chain order P = {P0, P1, P2, P3, P4, P5} = {5, 10, 3, 12, 5, 50}. The objective is to find the minimum number of scalar multiplications required to multiply the 5 matrices and also find the optimal sequence of multiplications. The solution can be obtained by using a bottom up approach that means first we should calculate mii for 1≤i ≤ 5. Then mijis calculated for j – i = 1 to j – i = 4. We can fill the table shown in Fig. 7.4 to find the solution. Fig. 7.4Table to store the partial solutions of the matrix chain multiplication problem The value of mii for 1≤i ≤ 5 can be filled as 0 that means the elements in the first row can be assigned 0. Then For j – i = 1
- 57. m12 = P0 P1 P2 = 5 x 10 x 3 = 150 m23 = P1 P2 P3 = 10 x 3 x 12 = 360 m34 = P2 P3 P4 = 3 x 12 x 5 = 180 m45 = P3 P4 P5 = 12 x 5 x 50 = 3000 For j – i = 2 m13 = min {m11 + m23 + P0 P1 P3, m12 + m33 + P0 P2 P3} = min {0 + 360 + 5 * 10 * 12, 150 + 0 + 5*3*12} = min {360 + 600, 150 + 180} = min {960, 330} = 330 m24 = min {m22 + m34 + P1 P2 P4, m23 + m44 + P1 P3 P4} = min {0 + 180 + 10*3*5, 360 + 0 +10*12*5} = min {180 + 150, 360 + 600} = min {330, 960} = 330 m35 = min {m33 + m45 + P2 P3 P5, m34 + m55 + P2 P4 P5} = min {0 + 3000 + 3*12*50, 180 + 0 + 3*5*50} = min {3000 + 1800 + 180 + 750} = min {4800, 930} = 930 For j – i = 3 m14 = min {m11 + m24 + P0 P1 P4, m12 + m34 + P0 P2 P4, m13+m44+P0 P3 P4} = min {0 + 330 + 5*10*5, 150 + 180 + 5*3*5, 330+0+5*12*5} = min {330 + 250, 150 + 180 + 75, 330 +300} = min {580, 405, 630} = 405 m25 = min {m22 + m35 + P1 P2 P5, m23 + m45 + P1 P3 P5, m24+m55+P1 P4 P5} = min {0 + 930 +10*3*50, 360+3000+10*12*50, 330+0+10*5*50} = min {930 + 1500, 360 +3000+6000, 330+2500} = min {2430, 9360, 2830} = 2430 For j - i = 4 m15 = min{m11+ m25+ P0 P1 P5, m12+m35+ P0 P2 P5, m13 + m45 +P0 P3 P5, m14+m55+P0 P4 P5 } = min{0+2430+5*10*50, 150+930+5*3*50, 330+3000+5*12*50, 405+0+5*5*50} = min {2430+2500, 150+930+750, 330+3000+3000, 405+1250} = min {4930, 1830, 6330, 1655} = 1655
- 58. Hence, minimum number of scalar multiplications required to multiply the given five matrices in 1655. To find the optimal parenthesization of A1……….A5, we find the value of k is 4 for which m15 is minimum. So the matrices can be splitted to (A1….A4) (A5). Similarly, (A1….A4) can be splitted to (A1A2) (A3 A4) because for k = 2, m14 is minimum. No further splitting is required as the subchains (A1A2) and (A3 A4) has length 1. So the optimal paranthesization of A1 …….A5 in ( (A1 A2) (A3 A4) ) (A5). Time complexity of multiplying a chain of n matrices Let T(n) be the time complexity of multiplying a chain of n matrices. ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) [ ] ( ) [ ] )1.7()1()2()1(2)( )()(11 11)()(1)( 11)()(1 11 )( 1 1 1 1 1 1 LLLLL −++++Θ=⇒ −++−Θ+Θ= >Θ+−++Θ=⇒ >Θ+−++Θ =Θ = ∑ ∑ ∑ − = − = − = nTTTnnT knTkTn nifknTkTnT nifknTkT if n nT n k n k n k Replacing n by n-1, we get ( ) [ ] )2.7()2()2()1(21)1( LLLLL −++++−Θ=− nTTTnnT Subtracting equation 7.2 from equation 7.1, we have ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( )[ ] ( )n n n nn T nTnT nTnT nTnnnTnT 2 2 13 33311 )1(333311 )2(3131)2(3131 )1(31)( )1(21)1()( 12 122 2 Ο= − = ++++Θ= +++++Θ= −+Θ+Θ=−+Θ+Θ= −+Θ=⇒ −+−Θ−Θ=−− − −− LL LL Longest Common SubsequenceLongest Common SubsequenceLongest Common SubsequenceLongest Common Subsequence The longest common subsequence (LCS) problem can be formulated as follows “Given two sequences X = 〈x1, x2 ……….xn〉 and Y = 〈y1, y2………yn〉 and the objective is to find the LCS Z = 〈z1, z2 ………zn〉 that is common to x and y”.
- 59. Given two sequences X and Y, we say Z is a common sub sequence of X and Y if Z is a subsequence of both X and Y. For example, X = 〈 A, B, C, B, D, A, B〉 and Y = 〈 B, D, C, A, B, A〉, the sequence 〈B, C, A〉 is a common subsequence. Similarly, there are many common subsequences in the two sequences X and Y. However, in the longest common subsequence problem, we wish to find a maximum length common subsequence of X and Y, that is 〈 B, C, B, A〉 or 〈 B, D, A, B〉. This section shows that the LCS problem can be solved efficiently using dynamic programming. 4.7.1 Dynamic programming for LCS problem Theorem.4.1.(Optimal Structure of an LCS) Let X = 〈x1, x2 ……….xn〉 and Y = 〈y1, y2………yn〉 be sequences and let Z = 〈z1, z2 ………zn〉 be any LCS of X and Y. Case 1. If xm = yn, then zk = xm = yn and Zk-1 is an LCS of Xm-1 and Yn-1. Case 2. If xm≠yn, then zk≠xm implies that Z is an LCS of Xm-1 and Y. Case 3. If xm≠yn, then zk≠yn implies that Z is an LCS of X and Yn-1. Proof The proof of the theorem is presented below for all three cases. Case 1. If xm = yn and we assume that zk≠xm or zk≠yn then xm = yn can be added to Z at any index after k violating the assumption that Zk is the longest common subsequence. Hence zk = xm = yn. If we do not consider Zk-1 as LCS of Xm-1 and Yn-1, then there may exist another subsequence W whose length is more than k-1. Hence, after adding xm = yn to the subsequence W increases the size of subsequence more than k, which again violates our assumption. Hence, Zk=xm = yn and Zk-1 must be an LCS of Xm-1 and Yn-1. Case 2.If xm≠yn, then zk≠xm implies that Z is an LCS of Xm-1 and Yn. If there were a common subsequence W of Xm-1and Y with length greater then k than W would also be an LCS of Xm and Yn violating our assumption that Zk is an LCS of Xm and Yn. Case 3. The proof is symmetric to case-2. Thus the LCS problem has an optimal structure. Overlapping Sub-problems From theorem 4.1, it is observed that either one or two cases are to be examined to find an LCS of Xmand Yn. If xm = yn, then we must find an LCS of Xm-1 and Yn-1. If xm≠yn, then we must find an LCS of Xm-1 and Yn and LCS of Xm and Yn-1. The LCS of X and Y is the longer of these two LCSs.
- 60. Let us define c[m, n] to be the length of an LCS of the sequences Xm and Yn. The optimal structure of the LCS problem gives the recursive formula { } )1.7....(.......... if1],[],1,[max if11]1,[ 0or0if0 ][ ≠−− =+−− == = nm nm yxnmcnmc yxnmc nm m,nc Generalizing equation 7.1, we can formulate { } )2.7....(.......... if1],[],1,[max if11]1,[ 0or0if0 ][ ≠−− =+−− == = ji ji yxjicjic yxjic ji i,jc Computing the length of an LCS Based on equation (7.1), we could write an exponential recursive algorithm but there are only m*n distinct problems. Hence, for the solution of m*n distinct subproblems, we use dynamic programming to compute the solution using bottom up approach. The algorithm LCS_length (X, Y) takes two sequences X = 〈x1, x2………..xn〉 and Y = 〈y1, y2………..yn〉 as inputs and find c[m, n] as the maximum length of the subsequence in X and Y. It stores c[i, j] and b[i, j] in tables c[m, n] and b[m, n] respectively, which simplifies the construction of optimal solution. AlgorithmLCS_LENGTH (X, Y) { m=length [X] n=length [Y] for( i =1; i<=m; i++) c[i,0] = 0; for(j=0; j<n; j++) c[0, j]= 0; for(i=1; i< m; i++){ for(j = 1; j <= n; j++){ if(x[i] = = y[j]) { c[i, j] = 1 + c[i-1, j-1]; b[i,j] = ‘ ’; }
- 61. else{ if(c[ i-1, j] ≥ c[i, j-1] ) c[i, j] = c[i-1, j]; b[i,j] = ‘↑↑↑↑’; else c[i,j] = c[i, j-1]; b[i, j] = ‘←←←←’ } return c and b ; } Algorithm 7.3 Algorithm for finding Longest common subsequence . Constructing an LCS The algorithm LCS_LENGTH returns c and b tables. The b table can be used to construct the LCS of X and Y quickly. AlgorithmPRINT_LCS (b, X, i, j) { if (i == 0 || j == 0) return; if (b[i, j] = = ‘ ‘) { PRINT_LCS (b, X, i-1, j-1) Printxi } else if (b[i, j] = = ‘↑’) PRINT_LCS (b, X, i-1, j) else PRINT_LCS (b, X, i, j-1) }
- 62. Algorithm 7.4 Algorithm to print the Longest common subsequence . Let us consider two sequences X = 〈C, R, O, S, S〉 and Y = 〈R, O, A, D, S〉 and the objective is to find the LCS and its length. The c and b table are computed by using the algorithm LCS_LENGTH for X and Y that is shown in Fig.7.5. The longest common subsequence of X and Y is 〈R, O, S〉 and the length of LCS is 3. Reliability Design ProblemReliability Design ProblemReliability Design ProblemReliability Design Problem In this section, we present the dynamic programming approach to solve a problem with multiplicative constraints. Let us consider the example of a computer return in which a set of nodes are connected with each other. Let ri be the reliability of a node i.e. the probability at which the node forwards the packets correctly in ri. Then the reliability of the path connecting from one node s to another node d is ∏= k i ir 1 where k is the number of intermediate node. Similarly, we can also consider a system with n devices connected in series, where the reliability of device i is ri. The reliability of the system is ∏= k i ir 1 . For example if there are 5 devices connected in series and the reliability of each device is 0.99 then the reliability of the system is 0.99×0.99×0.99×0.99×0.99=0.951. Hence, it is desirable to connect multiple copies of the same devices in parallel through the use of switching circuits. The switching circuits determine the devices in any group functions properly. Then they make use of one such device at each stage. Let mi be the number of copies of device Di in stage i. Then the probability that all mi have malfunction i.e. (1-ri)m i.Hence, the reliability of stage i becomes 1-(1-ri)m i . Thus, if ri =0.99 and mi=2, the reliability of stage i is 0.9999. However, in practical situations it becomes less because the switching circuits are not fully reliable. Let us assume that the reliability of stage i in φ i(mi), i≤n. Thus the reliability of the system is ∏= k i ii m 1 )(φ . Fig. 7.9 The reliability design problem is to use multiple copies of the devices at each stage to increase reliability. However, this is to be done under a cost constraint. Let ci be the cost of each unit of device Di and let c be the cost constraint. Then the objective is to maximize the reliability under the condition that the total cost of the system mici will be less than c. Mathematically, we can write maximize ∏≤≤ ni ii m 1 )(φ
- 63. subject to cmc ni ii ≤∑≤≤1 ni1andmi ≤≤≥1 . We can assume that each ci>0 and so each mi must be in the range ii um1 ≤≤ , where ∑+ = ≠= i n ijandj j i c cc u 1 . The dynamic programming approach finds the optimal solution m1,m2…mn. An optimal sequence of decision i.e. a decision for each mi can result an optimal solution. Let fn(c) be the maximum reliability of the system i.e. maximum ∏= n i ii m 1 )(φ , subject to the constraint cmc ni ii ≤∑≤≤1 and ii xm1 ≤≤ , ni1 ≤≤ . Let we take a decision on the value of mn from {1,2… un}. Then the value of the remaining mn ni1 ≤≤ can be chosen in such a way that ( )imφ for 1−≤≤≤ nni1 can be maximized under the cost constraint c-cnmn. Thus the principle of optimality holds and we can write fn(c)= ( ) ( ){ }jjjjj um cmxfm nn −− ≤≤ 1 1 max φ (8.1) We can generalized the equation 8.1 and we can write fj(x)= ( ) ( ){ }jjjjj um cmxfm jj −− ≤≤ 1 1 max φ (8.2) It is clear that f0(x)=1 for all x, cx ≤≤0 . Let Si consists of tuples of the form (f,x) where f = fi(x). There is atmost one tuple for each different x that remits from a sequence of decisions on m1,m2…mn. If there are two tuples (f1,x1) and (f2,x2) such that f1 ≥ f2 and x1≤ x2 then (f2,x2) is said to be dominated tuple and discarded from Si . Let us design a three stage system with devices D1,D2 and D3. The costs are Rs 30,Rs 15 and Rs 20 respectively. The cost constraint of the system is Rs 105. The reliability of the devices are 0.9, 0.8 and 0.5 respectively. If stage i has mi devices in parallel then ( ) ( )( )im ii rm −−= 11φ . We can write c1=30, c2 =15,c3=20, c=105, r1=0.9, r2=0.8 and r3=0.5. We can calculate the value of ui, for 31 ≤≤ i
- 64. ( ) ( ) ( ) 3 20 60 20 1530105 3 15 55 15 2030105 2 30 70 30 2015105 3 2 1 = = +− = = = +− = = = +− = x x x Then we start with S0 ={(1,0)}. We can obtain each Si from Si-1 by trying out all possible values for mi and combining the resulting tuples together. 1 1S ={(0.9,30)} 1 2S ={(0.99,60)} 1 S ={(0.9,30),(0.99,60)} Considering 1 device at stage q, we can write 2 1S as follows 2 1S ={(0.9×0.8,30+15),(0.99×0.8,60+15)} ={(0.72,45),(0.792,75)} Considering 2 devices of D2 in stage 2, we can compute the reliability at stage 2 ( )22 mφ =1-(1-0.8)2 =0.96 cost at stage 2=2×15=30 Hence, we can write 2 2S ={(0.9×0.96,30+30),(0.99×0.96,60+30)} ={(0.864,60),(0.9504,90)} The tuple (0.9504, 90) is removed as it left only Rs 15 and the maximum cost of the third stage is 20. Now, we can consider 3 devices of D2 in stage 2 and compute the reliability at stage 2 is ( )22 mφ =1-(1-0.8)3 =1-0.008=0.992. Hence, we can write 2 3S ={(0.9×0.992,30+45),(0.99×0.992,60+45)} ={(0.8928,75),(0.98208,105)} The tuple (0.98208,105) is discarded as there is no cost left for stage 3. Combining 2 1S , 2 2S and 2 3S , we get 2 S ={(0.72,45),(0.792,75),(0.864,60),(0.8928,75)} The tuple (0.792,75) is discarded from 2 S as it is dominated by (0.864,60). Now, we can compute 3 1S assuming 1 device at stage 3.
- 65. 3 1S ={(0.72×0.5,45+20),(0.864×0.5,60+20),(0.8928×0.5,75+20)} ={(0.36,65),(0.432,80),(0.4464,95)} If there are 2 devices at stage 3, then ( )33 mφ =(1-(1-0.5)2 )=0.75 We can write 3 2S as follows 3 2S ={(0.72×0.75,45+40),(0.864×0.75,60+40),(0.8928×0.75,75+40)} ={(0.54,85),(0.648,100)} (tuple(0.8928×0.75,115) is discarded as cost constraint is 105). If there are 3 devices at stage 3 then ( )33 mφ =(1-(1-0.5)3 )=1-0.125=0.875 Hence, we can write 3 3S ={(0.72×0.875,45+60)}={(0.63,105)} Combining 3 1S , 3 2S and 3 3S we can write S3 discarding the dominant tuples ass given below S3 ={(0.36,65),(0.432,80),(0.54,85),(0.648,100)} The best design has the reliability 0.648 and a cost of 100. Now, we can track back to find the number of devices at each stage. The tuple(0.648,100) is taken from 3 2S that is with 2 devices at stage 2. Thus m3=2. The tuple (0.648,100) was derived from the tuple (0.864,60) taken from 2 2S and computed with considering 2 devices at stage 2. Thus m2=2. The tuple (0.864,60) is derived from the tuple (0.9,30) taken from 1 1S computed with 1 device at stage 1. Thus m1=1. Bellman Ford AlgorithmBellman Ford AlgorithmBellman Ford AlgorithmBellman Ford Algorithm In the previous chapter, it is observed that the Dijkstra’s algorithm finds the shortest path from one node to another node in a graph G = (V, E) where the weights of the links are positive. However, if there are some negative weight edges then dijkstra’s algorithm may fail to find the shortest path from one node to another. Hence, Bellman-ford algorithm solves the single source shortest paths problem in general case. Let us consider the graph G= (V, E) shown in Fig.7.10. Let us assume that node 1 is the source, node 2 and node 3 are destinations. Then by using Dijkstra’s algorithm, we can compute the shortest path to node 2 and node 3 as 5 and 7 respectively whereas it is not actually the case. The shortest path from 1 to 3 is 123 and the path length is 2. This can be computed by Bellman-ford algorithm. Before applying Bellman-ford algorithm, we assume that the negative weight edges are permitted but there should not be any negative weight cycle. This is necessary to answer that the shortest paths consists of a finite number of edges. In Fig.7.11 , the path from 1 to 3 is 121212…123 and the path
- 66. length is −∞. When there are no negaƟve weight cycles, the shortest path between any two nodes in n- node graph contains n−1 edges. Let distl [u] be the length of the shortest path from source node v to the node u under the constraint that the shortest path contains at most l edges. The dist1 [u]=cost[v,u], for 1≤u≤n. As we discussed earlier, if there is no negative weight cycle then the shortest path contains atmostn−1 edges. Hence, distn-1 [u] in the length of the shortest path from v to u. Our goal is to compute distn-1 [u] and this can be done by using Dynamic programming approach. We can make the following observations (i) If the shortest path from v to u with at most k, k>1 edges has no more than k−1 edges, then distk [u]= distk-1 [u]. (ii) If the shortest path from v to u with at most k, k>1 edges has exactly k edges, then it is made up of a shortest path from v to some vertex j followed by edge<j, u>. The path from v to j has k−1 edges and its length is distk-1 [j]. All vertices i such that <i, u>∈E are the candidates of j. Since we are interested in a shortest path, the i that minimizes distk-1 [i]=cost[i,u] is the correct value for j. Distk [u]= ( ) [ ] ( ){ }{ }uitidistu k Eui ,cosmin],[dist 1 , 1-k min +− ∈ The Bellman ford algorithm is presented in Algorithm 7.6. AlgorithmBELLMAN FORD(v,cost,dist,n) //v is the source, cost is the adjacency matrix representing the cost of edges, dist stores the distance to all nodes, n is the number of nodes// { 1 for(i=1;i<=n; i++) 2 dist[i]=cost[v][i]; 3 for(k=2; k<= n-1; k++){ 4 for each u such that k ≠ v and (j, u)∈E 5 if(dist[u]>dist[j]+cost[j][u]) 6 dist[u]=dist[j]+cost[j][u]; 7 }//end for k } Algorithm 7.6 Bellman FordAlgorithm. The Bellman ford algorithm starts with computing the path from source to each node i in v. Since there are n nodes in a graph, the path to any node can contain atmostn-1 nodes. Then for each node u, remaining n-2 nodes are required to be examined. If <j, u>∈E then the condition dist[u] is compared
- 67. with dist[j] + cost[j][u]. If dist[u] is greater than dist[j]+dist[j][u] then dist[u] is updated to dist[j]+cost[j][u]. Let us consider the example network shown in Fig. 7.12. The example graph and its adjacency matric The time complexity of Bellman ford algorithm is O(ne). The lines 1-2 takes O(n) time. If the matrix is stored in an adjacency list then lines 4-6 takes O(e) time. Hence, the lines 3-7 takes O(ne) time. Therefore, the several time of Bellman ford algorithm is O(ne). Stepwise working procedure of Bellman Ford AlgorithmStepwise working procedure of Bellman Ford AlgorithmStepwise working procedure of Bellman Ford AlgorithmStepwise working procedure of Bellman Ford Algorithm
- 68. AAAAssembly Line Schedulingssembly Line Schedulingssembly Line Schedulingssembly Line Scheduling Assembly lines for automobile manufacturing factory An automobile company has two assembly lines as shown in Fig.7.13.The automobile chassis enters an assembly line and goes through all stations of the assembly line and complete auto exists at the end of the assembly line. Each assembly line has n stations and the jth station of ith line is denoted as Sij. The time required at station Sij is aij and the time required to travel from one station to the next station is negligible. Normally, the chassis enters in a line goes through all the stations of the same line. However, in overload situations, the factory manager has the flexibility to switch partially completed auto from a station of one line to the next station of other line. The time required to transfer the partially completed auto from station Sij to the other line is tij; where i=1,2 and j=1,2…n-1. Now the objective to choose some station from line 1 and some station from line 2 so that the total time to manufacture an auto can be minimized. If there are many stations in the assembly lines the brute force search takes much time to determine the stations through which the auto can be assembled. There are 2n possible ways to chose stations from the assembly line. Thus determining the fastest way for assembling the auto takes O(2n ) time, which is infeasible when n is large. This problem can be efficiently solved by dynamic programming technique. The first step of dynamic programming technique is to characterize the structure of an optimal solution. Since, there are 2 assembly lines with stations in each line, the computation of time to move to the 1st station of any assembly line is straight forward. However, there are two choices for j=2,3… n in each assembly line. First, the chassis may come from station S1,j-1 and then directly move to S1,j since the time to move from one station to the next station in the same assembly line is negligible. The second choice is the chassis can come from station S2,j-1 and then been transferred to station S1,j with a transfer time t2,j-1. Let the fastest way through station S1,j is through station S1,j-1. Then there must be a fastest way through from the starting point through station S1,j-1. Similarly, if there is a fastest way through station S2,j-1 then the chassis must have taken a fastest way from the starting point through station S2,j-1.
- 69. Thus the optimal solution to the problem can be found by solving optimal solution of the sub-problems that in the fastest way to either S1,j-1 or S2,j-1. This is referred as optimal structure of assembly line scheduling problem. If we find the fastest way to solve assembly line scheduling problem through station j-1 on either line 1 or line 2. Thus the fastest way through station S1,j is either • the fastest way to S1, j-1 and then directly through station S1,j. • the fastest way to S2,j-1 and a transfer from line 2 to line 1 and then through station S1,j. Similarly, the fastest way through station S2,j is • the fastest way to S2,j-1 and then directly through station S2,j. • the fastest way to S1,j-1, a transfer time from station 1. Let fi[j] be the fastest possible time to get a chassis from the starting point through station Si,j of the assembly line i. Then chassis directly goes to the first station of each line. [ ] 1,111 1 aef += [ ] 1,222 1 aef += Now, we can compute fi[j], 2<j≤n and 1≤i≤2 as f1[j]=min{f1[j-1] + a1,j, f2[j-1] + t2,j-1 + a1,j } f2[j]=min{f2[j-1] + a2,j, f1[j-1] + t1,j-1 + a2,j } If the chassis goes all the way through station n either line 1 or line 2 and then exits, we have [ ] [ ] [ ]{ } +−+− =+ = − Otherwiseatnfanfmin 1nifae nf nnn1 ,1,1 ,11,22,1 1,11 1 [ ] [ ] [ ]{ } +−+− =+ = − Otherwiseatnfanfmin 1nifae nf nnn ,11,11,22 1,12 2 ,1,1 Let the fastest time to get the chassis all the way through the factory is denoted by f* . Then f* = min{f1[n] + x1, f2[n] + x2 } Now, we can compute the stations through which the chassis must move to deliver the end product at minimum time. This can be calculated with a backward approach. Let l* denote the line whose station n is used in a fastest way through the entire process. If f1[n] + x1<f2[n] + x2 then l* =1 else l* =2. Then let us denote li[j] be the line number 1 or 2 whose station j-1 is used in a fastest way through station Si,j. li[1] is not required to be calculated since there is no station proceeding to station Si,1. If f1[j-1] + a1,j ≤f2[j-1]+t2,j-1 + a1,j then l1[j]=1.Otherwise, l1[j]=2. Similarly, if f2[j-1]+ a2,j≤ f1[j-1]+t1,j-1 + a2,j then l2[j]=2. Otherwise, l2[j]=1. The algorithm for assembly line scheduling problem is presented in Algorithm7.7.
- 70. Illustration of assembly line scheduling procedure. f1[1] = e1 + a1,1 = 2 + 4 = 6 f2[1] = e2 + a2,1 = 3 + 7 = 10 f1[2] = min{f1[1] + a1,2, f2[1] + t2,1 + a1,2 } = min{6 + 7, 10 + 2 + 7} =13 f2[2] = min{f2[1] + a2,2, f1[1] + t1,1 + a2,2 } = min{10 + 2, 6 + 2 + 2} =10 The time required at all the stations of both theassembly lines are shown above. The minimum time required to assemble an auto is 33. The assembly lines through which the complete auto is assembled is shown above.
- 71. MODULE - III • Lecture 21 - Data Structure for Disjoint Sets • Lecture 22 - Disjoint Set Operations, Linked list Representation • Lecture 23 - Disjoint Forests • Lecture 24 - Graph Algorithm - BFS and DFS • Lecture 25 - Minimum Spanning Trees • Lecture 26 - Kruskal algorithm • Lecture 27 - Prim's Algorithm • Lecture 28 - Single Source Shortest paths • Lecture 29 - Bellmen Ford Algorithm • Lecture 30 - Dijkstra's Algorithm
- 72. Lecture – 21 Disjoint Set Data Structure In computing, a disjoint-set data structure, also called a union–find data structure or merge– find set, is a data structure that keeps track of a set of elements partitioned into a number of disjoint (non-overlapping) subsets. It supports the following useful operations: • Find: Determine which subset a particular element is in. Find typically returns an item from this set that serves as its "representative"; by comparing the result of two Find operations, one can determine whether two elements are in the same subset. • Union: Join two subsets into a single subset. • Make Set, which makes a set containing only a given element (a singleton), is generally trivial. With these three operations, many practical partitioning problems can be solved. In order to define these operations more precisely, some way of representing the sets is needed. One common approach is to select a fixed element of each set, called its representative, to represent the set as a whole. Then, Find(x) returns the representative of the set that x belongs to, and Union takes two set representatives as its arguments. Example : Make Set creates 8 singletons. After some operations of Union, some sets are grouped together. Applications : • partitioning of a set • Boost Graph Library to implement its Incremental Connected Components functionality. • implementing Kruskal's algorithm to find the minimum spanning tree of a graph. • Determine the connected components of an undirected graph. CONNECTED-COMPONENTS(G) 1. for each vertex v∈V[G] 2. do MAKE-SET(v) 3. for each edge (u,v) ∈E[G] 4. doif FIND-SET(u) ≠ FIND-SET(v) 5. then UNION(u,v) SAME-COMPONENT(u,v) 1. if FIND-SET(u)=FIND-SET(v) 2. thenreturn TRUE 3. elsereturn FALSE
- 73. Lecture 22 - Disjoint Set Operations, Linked list Representation • A disjoint-set is a collection ={S1, S2,…, Sk} of distinct dynamic sets. • Each set is identified by a member of the set, called representative. Disjoint set operations – MAKE-SET(x): create a new set with only x. assume x is not already in some other set. – UNION(x,y): combine the two sets containing x and y into one new set. A new representative is selected. – FIND-SET(x): return the representative of the set containing x. Linked list Representation • Each set as a linked-list, with head and tail, and each node contains value, next node pointer and back-to-representative pointer. • Example: • MAKE-SET costs O(1): just create a single element list. • FIND-SET costs O(1): just return back-to-representative pointer.
- 74. UNION Implementation • A simple implementation: UNION(x,y) just appends x to the end of y, updates all back-to- representative pointers in x to the head of y. • Each UNION takes time linear in the x’s length. • Suppose n MAKE-SET(xi) operations (O(1) each) followed by n-1 UNION – UNION(x1, x2), O(1), – UNION(x2, x3), O(2), – ….. – UNION(xn-1, xn), O(n-1) • The UNIONs cost 1+2+…+n-1=Θ(n2 ) So 2n-1 operations cost Θ(n2 ), average Θ(n) each Lecture 23 - Disjoint Forests • Three operations – MAKE-SET(x): create a tree containing x. O(1) – FIND-SET(x): follow the chain of parent pointers until to the root. O(height of x’s tree) – UNION(x,y): let the root of one tree point to the root of the other. O(1) • It is possible that n-1 UNIONs results in a tree of height n-1. (just a linear chain of n nodes). • So n FIND-SET operations will cost O(n2 ).
- 75. Union by Rank & Path Compression • Union by Rank: Each node is associated with a rank, which is the upper bound on the height of the node (i.e., the height of subtree rooted at the node), then when UNION, let the root with smaller rank point to the root with larger rank. • Path Compression: used in FIND-SET(x) operation, make each node in the path from x to the root directly point to the root. Thus reduce the tree height.
- 76. Lecture 24 - Graph Algorithm - BFS and DFS In graph theory, breadth-first search (BFS) is a strategy for searching in a graph when search is limited to essentially two operations: (a) visit and inspect a node of a graph; (b) gain access to visit the nodes that neighbor the currently visited node. • The BFS begins at a root node and inspects all the neighboring nodes. • Then for each of those neighbor nodes in turn, it inspects their neighbor nodes which were unvisited, and so on. • Compare BFS with the equivalent, but more memory-efficient. Historical Background • BFS was invented in the late 1950s by E. F. Moore, who used to find the shortest path out of a maze, • discovered independently by C. Y. Lee as a wire routing algorithm (published 1961). Example A BFS search will visit the nodes in the following order: A, B, C, E, D, F, G BFS Algorithm The algorithm uses a queue data structure to store intermediate results as it traverses the graph, as follows: 1. Enqueue the root node 2. Dequeue a node and examine it • If the element sought is found in this node, quit the search and return a result. • Otherwise enqueue any successors (the direct child nodes) that have not yet been discovered. 3. If the queue is empty, every node on the graph has been examined – quit the search and return "not found". 4. If the queue is not empty, repeat from Step 2.
- 77. Applications Breadth-first search can be used to solve many problems in graph theory, for example: • Copying Collection, Cheney's algorithm • Finding the shortest path between two nodes u and v (with path length measured by number of edges) • Testing a graph for bipartiteness • (Reverse) Cuthill–McKee mesh numbering • Ford–Fulkerson method for computing the maximum flow in a flow network • Serialization/Deserialization of a binary tree vs serialization in sorted order, allows the tree to be re-constructed in an efficient manner. Pseudo Code Input: A graph G and a root v of G 1 procedure BFS(G,v) is 2 create a queue Q 3 create a set V 4 add v to V 5 enqueuev onto Q 6 whileQ is not empty loop 7 t ← Q.dequeue() 8 ift is what we are looking for then 9 returnt 10 end if 11 for all edges e in G.adjacentEdges(t) loop 12 u ← G.adjacentVertex(t,e) 13 ifu is not in Vthen 14 add u to V 15 enqueueu onto Q 16 end if 17 end loop 18 end loop 19 return none 20 end BFS
- 78. Time and space complexity The time complexity can be expressed as [3] since every vertex and every edge will be explored in the worst case. Note: may vary between and , depending on how sparse the input graph is. When the number of vertices in the graph is known ahead of time, and additional data structures are used to determine which vertices have already been added to the queue, the space complexity can be expressed as where is the cardinality of the set of vertices. If the graph is represented by an Adjacency list it occupies [4] space in memory, while anAdjacency matrix representation occupies . Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. One starts at the root(selecting some arbitrary node as the root in the case of a graph) and explores as far as possible along each branch beforebacktracking. Historical Background A version of depth-first search was investigated in the 19th century by French mathematician Charles Pierre Trémaux Example A DFS search will visit the nodes in the following order: A, B, D, F, E, C, G Pseudo Code Input: A graph G and a vertex v of G Output: All vertices reachable from v labeled as discovered A recursive implementation of DFS 1 procedure DFS(G,v): 2 label v as discovered 3 for all edges from v to winG.adjacentEdges(v) do 4 if vertex w is not labeled as discovered then 5 recursively call DFS(G,w)
- 79. A non-recursive implementation of DFS 1 procedure DFS-iterative(G,v): 2 let S be a stack 3 S.push(v) 4 whileS is not empty 5 v ← S.pop() 6 ifv is not labeled as discovered: 7 label v as discovered 8 for all edges from v to winG.adjacentEdges(v) do 9 S.push(w) Applications • Finding connected components. • Topological sorting. • Finding the bridges of a graph. • Generating words in order to plot the Limit Set of a Group. • Finding strongly connected components. • Planarity testing • Solving puzzles with only one solution, such as mazes. (DFS can be adapted to find all solutions to a maze by only including nodes on the current path in the visited set.) • Maze generation may use a randomized depth-first search. • Finding bi-connectivity in graphs.