lecture 23
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The document discusses dynamic programming and amortized analysis. It reviews how dynamic tables use amortized analysis to achieve an overall cost of O(1) per insertion by occasionally doubling the table size and reinserting all elements. This results in a worst case cost of O(n) for a single insertion but averages to O(1) over many insertions. It also discusses using an accounting method with a $3 charge per insertion to pay for future table resizes, achieving an amortized cost of O(1) per operation. Finally, it introduces dynamic programming and uses the longest common subsequence problem to illustrate how it breaks problems into optimal subrecurring subproblems.
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![Dynamic Programming Example: Longest Common Subsequence Longest common subsequence ( LCS ) problem: Given two sequences x[1..m] and y[1..n], find the longest subsequence which occurs in both Ex: x = {A B C B D A B }, y = {B D C A B A} {B C} and {A A} are both subsequences of both What is the LCS? Brute-force algorithm: For every subsequence of x, check if it’s a subsequence of y How many subsequences of x are there? What will be the running time of the brute-force alg?](https://image.slidesharecdn.com/23-1224487945718578-8/85/lecture-23-8-320.jpg)
![Finding LCS Length Define c[ i,j ] to be the length of the LCS of x[1.. i ] and y[1.. j ] What is the length of LCS of x and y? Theorem: What is this really saying?](https://image.slidesharecdn.com/23-1224487945718578-8/85/lecture-23-10-320.jpg)
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SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docxanhlodge SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
More instructions for the lab write-up:
1) You are not obligated to use the 'diary' function. It was presented only for you convenience. You
should be copying and pasting your code, plots, and results into some sort of "Word" type editor that
will allow you to import graphs and such. Make sure you always include the commands to generate
what is been asked and include the outputs (from command window and plots), unless the pr.
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lecture 30sajinsc The document reviews concepts related to NP-completeness, including reductions between problems. It provides examples of reducing the directed Hamiltonian cycle problem to the undirected version. It also reduces 3-SAT to the clique problem by transforming a Boolean formula to a graph, then further reduces clique to vertex cover. Hundreds of problems have been shown to be NP-complete through relatively simple reductions like these that leverage previous results.
lecture 29
lecture 29sajinsc The document discusses reductions between problems to prove NP-completeness. It first reviews P, NP, reductions, NP-hard and NP-complete problems. It then walks through reducing directed Hamiltonian cycle to undirected Hamiltonian cycle to traveling salesman problem (TSP) to prove that TSP is NP-complete in 3 steps or less.
lecture 28
lecture 28sajinsc This document discusses NP-completeness and algorithms. It begins by reviewing tractable versus intractable problems, and the complexity classes P and NP. P contains problems solvable in polynomial time, while NP includes problems verifiable in polynomial time. The document then discusses NP-complete problems, which are the hardest problems in NP. It notes that reducing one problem to another shows the first problem is no harder than the second. The document concludes by outlining the steps to prove a problem is NP-complete: choosing a known NP-complete problem, reducing it to the target problem in polynomial time, and showing the target is in NP.
lecture 27
lecture 27sajinsc The document discusses NP-completeness and algorithms. It introduces dynamic programming and greedy algorithms. It then discusses the activity selection problem and shows it can be solved greedily by choosing the activity with the earliest finish time at each step. Finally, it defines the classes P and NP, introduces the concept of reductions to show problems are NP-complete, and states that if any NP-complete problem can be solved in polynomial time, then P=NP.
lecture 26
lecture 26sajinsc The document discusses various algorithms techniques including greedy algorithms, dynamic programming, and their application to problems like the activity selection problem and knapsack problem. It provides examples of optimal subproblems, overlapping subproblems, and how dynamic programming and greedy algorithms can be used to solve problems exhibiting these properties.
lecture 25
lecture 25sajinsc The document describes the 0-1 knapsack problem and how to solve it using dynamic programming. The 0-1 knapsack problem involves packing items of different weights and values into a knapsack of maximum capacity to maximize the total value without exceeding the weight limit. A dynamic programming algorithm is presented that breaks the problem down into subproblems and uses optimal substructure and overlapping subproblems to arrive at the optimal solution in O(nW) time, improving on the brute force O(2^n) time. An example is shown step-by-step to illustrate the algorithm.
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lecture 21sajinsc The document discusses algorithms for solving the single-source shortest path problem, including Dijkstra's algorithm, Bellman-Ford algorithm, and shortest paths in directed acyclic graphs (DAGs). It reviews the key ideas behind relaxation and how algorithms like Bellman-Ford and Dijkstra's use relaxation to iteratively improve the upper bound on shortest path distances. Running times of the algorithms are analyzed, with Dijkstra's algorithm taking O(ElogV) time and Bellman-Ford taking O(VE) time.
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lecture 20sajinsc The document describes Prim's algorithm for finding a minimum spanning tree (MST) of a weighted undirected graph. It shows the pseudocode for Prim's algorithm and step-by-step workings on an example graph to build up the MST. It notes that the key operation is decreasing the key value, which is called once for each edge in the MST, and the running time depends on the priority queue implementation, being O(ElgV) for a binary heap and O(VlgV+E) for a Fibonacci heap.
lecture 19
lecture 19sajinsc The document discusses depth-first search (DFS) graph algorithms. DFS explores a graph by going as deep as possible at each step and backtracking when no further progress can be made. It distinguishes between different types of edges discovered: tree edges connect newly found nodes, back edges connect to ancestors, and cross/forward edges connect between subtrees or ancestors and descendants. DFS runs in O(V+E) time and can detect cycles in O(V) time by checking for back edges.
lecture 18
lecture 18sajinsc The document discusses the algorithms of breadth-first search (BFS) and depth-first search (DFS) on graphs. It provides pseudocode for BFS and DFS, and examples of running the algorithms on sample graphs. Key points include:
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lecture 16
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lecture 15
lecture 15sajinsc The document discusses two algorithms for matrix multiplication and finding the median of an unsorted list:
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lecture 14sajinsc The document discusses red-black trees and their operations. It reviews the properties of red-black trees that guarantee logarithmic height. It then describes that inserting nodes can violate properties and requires recoloring nodes or rotating the tree. The key steps of insertion involve adding the new node as red, then fixing any violations by moving them up the tree through recoloring and rotations.
lecture 13
lecture 13sajinsc The document discusses red-black trees, which are binary search trees augmented with node colors to guarantee a height of O(log n). It describes the properties that red-black trees must satisfy, including that every node is red or black, leaves are black, and if a node is red its children are black. It then proves that these properties ensure the height is O(log n) by showing a subtree has at least 2^bh - 1 nodes, where bh is the black-height. Finally, it notes that common operations like search, insert and delete run in O(log n) time on red-black trees.
lecture 12
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lecture 11
lecture 11sajinsc This document discusses binary search trees (BSTs) and their use for dynamic sets and sorting. It covers BST operations like search, insert, find minimum/maximum, and delete. It explains that BST sorting runs in O(n log n) time like quicksort. Maintaining a height of O(log n) can lead to efficient implementations of priority queues and other dynamic set applications using BSTs.
lecture 10
lecture 10sajinsc The document discusses various sorting algorithms and their time complexities:
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2. Counting sort runs in O(n+k) time where k is the range of input values, and is not a comparison sort.
3. Radix sort treats input as d-digit numbers in some base k and uses counting sort to sort on each digit, achieving O(dn+dk) time which is O(n) when d and k are constants.
4. A randomized selection algorithm finds the ith order statistic in expected O(n) time using randomized partition.
lecture 9
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lecture 23
- 1. CS 332: Algorithms Dynamic Programming
- 2. Review: Amortized Analysis To illustrate amortized analysis we examined dynamic tables 1. Init table size m = 1 2. Insert elements until number n > m 3. Generate new table of size 2 m 4. Reinsert old elements into new table 5. (back to step 2) What is the worst-case cost of an insert? What is the amortized cost of an insert?
- 3. Review: Analysis Of Dynamic Tables Let c i = cost of i th insert c i = i if i-1 is exact power of 2, 1 otherwise Example: Operation Table Size Cost Insert(1) 1 1 1 Insert(2) 2 1 + 1 2 Insert(3) 4 1 + 2 Insert(4) 4 1 Insert(5) 8 1 + 4 Insert(6) 8 1 Insert(7) 8 1 Insert(8) 8 1 Insert(9) 16 1 + 8 1 2 3 4 5 6 7 8 9
- 4. Review: Aggregate Analysis n Insert() operations cost Average cost of operation = (total cost)/(# operations) < 3 Asymptotically, then, a dynamic table costs the same as a fixed-size table Both O(1) per Insert operation
- 5. Review: Accounting Analysis Charge each operation $3 amortized cost Use $1 to perform immediate Insert() Store $2 When table doubles $1 reinserts old item, $1 reinserts another old item We’ve paid these costs up front with the last n /2 Insert()s Upshot: O(1) amortized cost per operation
- 6. Review: Accounting Analysis Suppose must support insert & delete, table should contract as well as expand Table overflows double it (as before) Table < 1/4 full halve it Charge $3 for Insert (as before) Charge $2 for Delete Store extra $1 in emptied slot Use later to pay to copy remaining items to new table when shrinking table What if we halve size when table < 1/8 full?
- 7. Dynamic Programming Another strategy for designing algorithms is dynamic programming A metatechnique, not an algorithm (like divide & conquer) The word “programming” is historical and predates computer programming Use when problem breaks down into recurring small subproblems
- 8. Dynamic Programming Example: Longest Common Subsequence Longest common subsequence ( LCS ) problem: Given two sequences x[1..m] and y[1..n], find the longest subsequence which occurs in both Ex: x = {A B C B D A B }, y = {B D C A B A} {B C} and {A A} are both subsequences of both What is the LCS? Brute-force algorithm: For every subsequence of x, check if it’s a subsequence of y How many subsequences of x are there? What will be the running time of the brute-force alg?
- 9. LCS Algorithm Brute-force algorithm: 2 m subsequences of x to check against n elements of y: O( n 2 m ) We can do better: for now, let’s only worry about the problem of finding the length of LCS When finished we will see how to backtrack from this solution back to the actual LCS Notice LCS problem has optimal substructure Subproblems: LCS of pairs of prefixes of x and y
- 10. Finding LCS Length Define c[ i,j ] to be the length of the LCS of x[1.. i ] and y[1.. j ] What is the length of LCS of x and y? Theorem: What is this really saying?
- 11. The End