Divide and Conquer Merge Sort Divide and Conquer

Divide and Conquer Merge. Sort

Divide and Conquer Divide up the problem into at least two sub-problems Recursively solve the sub-problems “Patch up” the solutions to the sub-problems for the final solution

Improvements on a smaller scale Greedy algorithms: exponential poly time (Typical) Divide and Conquer: O(n 2) asymptotically smaller running time

Sorting Given n numbers order them from smallest to largest Works for any set of elements on which there is a total order

Insertion Sort Make sure that all the processed numbers are sorted Input: a 1, a 2, …. , an Output: b 1, b 2, …, bn O(n 2) overall b 1= a 1 for i =2 … n Find 1 ≤ j ≤ i s. t. ai lies between bj-1 and bj Move bj to bi-1 one cell “down” a b bj=ai 4 2 431 3 2 2 43 4 3 1 4 O(log n) O(n)

Other O(n 2) sorting algorithms Selection Sort: In every round pick the min among remaining numbers Bubble sort: The smallest number “bubbles” up

Divide and Conquer Divide up the problem into at least two sub-problems Recursively solve the sub-problems “Patch up” the solutions to the sub-problems for the final solution

Mergesort Algorithm Divide up the numbers in the middle Unless n=2 Sort each half recursively Merge the two sorted halves into one sorted output

How fast can sorted arrays be merged?

Mergesort algorithm Input: a 1, a 2, …, an Output: Numbers in sorted order Merge. Sort( a, n ) If n = 2 return the order min(a 1, a 2); max(a 1, a 2) a. L = a 1, …, an/2 a. R = an/2+1, …, an return MERGE ( Merge. Sort(a. L, n/2), Merge. Sort(a. R, n/2) )

An example run 51 1 100 19 2 8 1 51 19 100 2 8 1 19 51 100 2 1 2 3 4 8 4 3 3 19 Merge. Sort( a, n ) If n = 2 return the order min(a 1, a 2); max(a 1, a 2) a. L = a 1, …, an/2 a. R = an/2+1, …, an return MERGE ( Merge. Sort(a. L, n/2), Merge. Sort(a. R, n/2) ) 3 4 4 51 8 100

Correctness Input: a 1, a 2, …, an Output: Numbers in sorted order Merge. Sort( a, n ) If n = 1 return the order a 1 If n = 2 return the order min(a 1, a 2); max(a 1, a 2) a. L = a 1, …, an/2 a. R = an/2+1, …, an return MERGE ( Merge. Sort(a. L, n/2), Merge. Sort(a. R, n/2) ) Inductive step follows from correctness of MERGE By induction on n

Run time recurrence T(n) ≤ c if n ≤ 2 2*T(n/2) + c*n otherwise