![CS 385 Analysis of Algorithms Spring 2011
Our decrease-by-one approach to sorting is the insertion sort.
The insertion sort sorts an array of n elements by first sorting the first n−1 elements, then inserting
the last element into its correct position in the array.
insertion_sort(A[0..n-1]) {
for (i=0 to n-1) {
v = A[i]
j = i-1
while (j >= 0 and A[j] > v) {
A[j+1] = A[j]
j--
}
A[j+1] = v
This is an in-place sort and is stable.
Our basic operation for this algorithm is the comparison of keys in the while loop.
We do have differences in worst, average, and best case behavior. In the worst case, the while loop
always executes as many times as possible. This occurs when each element needs to go all the way
at the start of the sorted portion of the array – exactly when the starting array is in reverse sorted
order.
The worst case number of comparisons:
n−1 i−1 n−1
n(n − 1)
Cworst (n) = 1= i= ∈ Θ(n2 )
i=1 j=0 i=1 2
In the best case, the inner loop needs to do just one comparison, determining that the element is
already in its correct position. This happens when the algorithm is presented with already-sorted
input. Here, the number of comparisons:
n−1
Cbest (n) = 1 = n − 1 ∈ Θ(n)
i=1
This behavior is unusual – after all, how often do we attempt to sort an already-sorted array?
However, we come close in some very important cases. If we have nearly-sorted data, we have
nearly this same performance.
A careful analysis of the average case would result in:
n2
Cavg (n) ≈ ∈ Θ(n2 )
4
Of the simple sorting algorithms (bubble, selection, insertion), insertion sort is considered the best
option in general.
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Sienna 5 decreaseandconquer
- 1. Computer Science 385 Analysis of Algorithms Siena College Spring 2011 Topic Notes: Decrease and Conquer Our next class of algorithms are the decrease-and-conquer group. The idea here: 1. Reduce the problem to a smaller instance 2. Solve the smaller instance 3. Modify the smaller instance solution to be a solution to the original The main variations: • Decrease by a constant (often by 1) • Decrease by a constant factor (often by half) • Variable size decrease For example, consider variations on how to compute the value an . The brute-force approach would involve applying the definition and multiplying a by itself, n times. A decrease-by-one approach would reduce the problem to computing the result for the problem of size n − 1 (computing an−1 ) then modifying that to be a solution to the original (by multiplying that result by a). n A decrease-by-constant-factor (2, in this case) approach would involve computing a 2 and multi- plying that result by itself to compute the answer. The only complication here is that we have to treat odd exponents specially, leading to a rule: (a 2 )2 n if n is even n 2 a = (a n−1 2 ) · a if n > 1 and odd a if n = 1 This approach will lead to O(log n) multiplications. Insertion Sort
- 2. CS 385 Analysis of Algorithms Spring 2011 Our decrease-by-one approach to sorting is the insertion sort. The insertion sort sorts an array of n elements by first sorting the first n−1 elements, then inserting the last element into its correct position in the array. insertion_sort(A[0..n-1]) { for (i=0 to n-1) { v = A[i] j = i-1 while (j >= 0 and A[j] > v) { A[j+1] = A[j] j-- } A[j+1] = v This is an in-place sort and is stable. Our basic operation for this algorithm is the comparison of keys in the while loop. We do have differences in worst, average, and best case behavior. In the worst case, the while loop always executes as many times as possible. This occurs when each element needs to go all the way at the start of the sorted portion of the array – exactly when the starting array is in reverse sorted order. The worst case number of comparisons: n−1 i−1 n−1 n(n − 1) Cworst (n) = 1= i= ∈ Θ(n2 ) i=1 j=0 i=1 2 In the best case, the inner loop needs to do just one comparison, determining that the element is already in its correct position. This happens when the algorithm is presented with already-sorted input. Here, the number of comparisons: n−1 Cbest (n) = 1 = n − 1 ∈ Θ(n) i=1 This behavior is unusual – after all, how often do we attempt to sort an already-sorted array? However, we come close in some very important cases. If we have nearly-sorted data, we have nearly this same performance. A careful analysis of the average case would result in: n2 Cavg (n) ≈ ∈ Θ(n2 ) 4 Of the simple sorting algorithms (bubble, selection, insertion), insertion sort is considered the best option in general. 2
- 3. CS 385 Analysis of Algorithms Spring 2011 Graph Traversals We return to graph structures for our next group of decrease-and-conquer algorithms. In particular, we consider the problem of visiting all vertices in a graph. The two main approaches are the depth-first search (DFS) and the breadth-first search (BFS). See the text for details... Topological Sort Again, see the text for details... 3