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### Chapter 7: Sorting Algorithms

Mark Allen Weiss: Data Structures and Algorithm Analysis in Java

Merge Sort

Lydia Sinapova, Simpson College

Merge Sort

- Basic Idea
- Example
- Analysis
- Animation

Idea

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.

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

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

(LogNis 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).

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