Chapter 7: Sorting Algorithms Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Chapter 7: Sorting Algorithms Merge Sort Lydia Sinapova, Simpson College

Merge Sort Basic Idea Example Analysis Animation

Idea Given an array to be sorted, Two sorted arrays can be merged in linear time with N comparisons only. Given an array to be sorted, consider separately its left half and its right half, sort them and then merge them.

Characteristics Recursive algorithm. Runs in O(NlogN) worst-case running time. Where is the recursion? Each half is an array that can be sorted using the same algorithm - divide into two, sort separately the left and the right halves, and then merge them.

Example

Merge Sort Code void merge_sort ( int [ ] a, int left, int right) { if(left < right) { int center = (left + right) / 2; merge_sort (a,left, center); merge_sort(a,center + 1, right); merge(a, left, center + 1, right); }

Analysis of Merge Sort time to mergesort N/2 elements + Assumption: N is a power of two. For N = 1 time is constant (denoted by 1) Otherwise: time to mergesort N elements = time to mergesort N/2 elements + time to merge two arrays each N/2 el.

Recurrence Relation Time to merge two arrays each N/2 elements is linear, i.e. O(N)   Thus we have:  (a) T(1) = 1 (b) T(N) = 2T(N/2) + N

Solving the Recurrence Relation T(N) = 2T(N/2) + N divide by N: (1) T(N) / N = T(N/2) / (N/2) + 1 Telescoping: N is a power of two, so we can write (2) T(N/2) / (N/2) = T(N/4) / (N/4) +1 (3) T(N/4) / (N/4) = T(N/8) / (N/8) +1 ……. T(2) / 2 = T(1) / 1 + 1

Adding the Equations The sum of the left-hand sides will be equal to the sum of the right-hand sides: T(N) / N + T(N/2) / (N/2) + T(N/4) / (N/4) + … + T(2)/2 = T(N/2) / (N/2) + T(N/4) / (N/4) + …. + T(2) / 2 + T(1) / 1 + LogN  (LogN is the sum of 1’s in the right-hand sides)

Crossing Equal Terms, Final Formula After crossing the equal terms, we get   T(N)/N = T(1)/1 + LogN   T(1) is 1, hence we obtain   T(N) = N + NlogN = (NlogN)  Hence the complexity of the Merge Sort algorithm is (NlogN).