| In computer science, merge sort (also commonly spelled mergesort) is an efficient, general-purpose, comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948. |
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Merge sort animation. The elements to sort are represented by dots.
Conceptually, a merge sort works as follows:
- Divide the unsorted list into n sublists, each containing 1 element (a list of 1 element is considered sorted).
- Repeatedly merge sublists to produce new sorted sublists until there is only 1 sublist remaining. This will be the sorted list.
Top-down implementation
Example C-like code using indices for top down merge sort algorithm that recursively splits the list (called runs in this example) into sublists until sublist size is 1, then merges those sublists to produce a sorted list. The copy back step is avoided with alternating the direction of the merge with each level of recursion.
// Array A[] has the items to sort; array B[] is a work array.
TopDownMergeSort(A[ ], B[ ], n){
CopyArray(A, 0, n, B); // duplicate array A[] into B[]
TopDownSplitMerge(B, 0, n, A); // sort data from B[] into A[]
}
// Sort the given run of array A[] using array B[] as a source. // iBegin is inclusive; iEnd is exclusive (A[iEnd] is not in the set).TopDownSplitMerge(B[], iBegin, iEnd, A[])
{
if(iEnd - iBegin < 2) // if run size == 1
return; // consider it sorted
// split the run longer than 1 item into halves
iMiddle = (iEnd + iBegin) / 2; // iMiddle = mid point
// recursively sort both runs from array A[] into B[]
TopDownSplitMerge(A, iBegin, iMiddle, B); // sort the left run TopDownSplitMerge(A, iMiddle, iEnd, B); // sort the right run // merge the resulting runs from array B[] into A[] TopDownMerge(B, iBegin, iMiddle, iEnd, A); }
// Left source half is A[ iBegin:iMiddle-1]. // Right source half is A[iMiddle:iEnd-1 ]. // Result is B[ iBegin:iEnd-1 ]. TopDownMerge(A[], iBegin, iMiddle, iEnd, B[]) { i = iBegin, j = iMiddle; // While there are elements in the left or right runs... for (k = iBegin; k < iEnd; k++) { // If left run head exists and is <= existing right run head. if (i < iMiddle && (j >= iEnd || A[i] <= A[j])) { B[k] = A[i]; i = i + 1; } else { B[k] = A[j]; j = j + 1; } } }
CopyArray(A[], iBegin, iEnd, B[])
{
for(k = iBegin; k < iEnd; k++)
B[k] = A[k];
}
Bottom-up implementation
Example C-like code using indices for bottom up merge sort algorithm which treats the list as an array of n sublists (called runs in this example) of size 1, and iteratively merges sub-lists back and forth between two buffers:
// array A[] has the items to sort; array B[] is a work array
void BottomUpMergeSort(A[], B[], n)
{
// Each 1-element run in A is already "sorted".
// Make successively longer sorted runs of length 2, 4, 8, 16... until whole array is sorted.
for (width = 1; width < n; width = 2 * width)
{
// Array A is full of runs of length width.
for (i = 0; i < n; i = i + 2 * width)
{
// Merge two runs: A[i:i+width-1] and A[i+width:i+2*width-1] to B[]
// or copy A[i:n-1] to B[] ( if(i+width >= n) )
BottomUpMerge(A, i, min(i+width, n), min(i+2*width, n), B);
}
// Now work array B is full of runs of length 2*width.
// Copy array B to array A for next iteration.
// A more efficient implementation would swap the roles of A and B.
CopyArray(B, A, n);
// Now array A is full of runs of length 2*width.
}
}
// Left run is A[iLeft :iRight-1].
// Right run is A[iRight:iEnd-1 ].
void BottomUpMerge(A[], iLeft, iRight, iEnd, B[])
{
i = iLeft, j = iRight;
// While there are elements in the left or right runs...
for (k = iLeft; k < iEnd; k++) {
// If left run head exists and is <= existing right run head.
if (i < iRight && (j >= iEnd || A[i] <= A[j])) {
B[k] = A[i];
i = i + 1;
} else {
B[k] = A[j];
j = j + 1;
}
}
}
void CopyArray(B[], A[], n)
{
for(i = 0; i < n; i++)
A[i] = B[i];
}
Top-down implementation using lists
Pseudocode for top down merge sort algorithm which recursively divides the input list into smaller sublists until the sublists are trivially sorted, and then merges the sublists while returning up the call chain.
function merge_sort(list m)
// Base case. A list of zero or one elements is sorted, by definition.
if length of m ≤ 1 then
return m
// Recursive case. First, divide the list into equal-sized sublists
// consisting of the first half and second half of the list.
// This assumes lists start at index 0.
var left := empty list
var right := empty list
for each x with index i in m do
if i < (length of m)/2 then
add x to left
else
add x to right
// Recursively sort both sublists.
left := merge_sort(left)
right := merge_sort(right)
// Then merge the now-sorted sublists.
return merge(left, right)
In this example, the merge function merges the left and right sublists.
function merge(left, right)
var result := empty list
while left is not empty and right is not empty do
if first(left) ≤ first(right) then
append first(left) to result
left := rest(left)
else
append first(right) to result
right := rest(right)
// Either left or right may have elements left; consume them.
// (Only one of the following loops will actually be entered.)
while left is not empty do
append first(left) to result
left := rest(left)
while right is not empty do
append first(right) to result
right := rest(right)
return result
Bottom-up implementation using lists
Pseudocode for bottom up merge sort algorithm which uses a small fixed size array of references to nodes, where array[i] is either a reference to a list of size 2 i or 0. node is a reference or pointer to a node. The merge() function would be similar to the one shown in the top down merge lists example, it merges two already sorted lists, and handles empty lists. In this case, merge() would use node for its input parameters and return value.
function merge_sort(node head)
// return if empty list
if (head == nil)
return nil
var node array[32]; initially all nil
var node result
var node next
var int i
result = head
// merge nodes into array
while (result != nil)
next = result.next;
result.next = nil
for(i = 0; (i < 32) && (array[i] != nil); i += 1)
result = merge(array[i], result)
array[i] = nil
// do not go past end of array
if (i == 32)
i -= 1
array[i] = result
result = next
// merge array into single list
result = nil
for (i = 0; i < 32; i += 1)
result = merge(array[i], result)
return result
Natural merge sort
A natural merge sort is similar to a bottom up merge sort except that any naturally occurring runs (sorted sequences) in the input are exploited. Both monotonic and bitonic (alternating up/down) runs may be exploited, with lists (or equivalently tapes or files) being convenient data structures (used as FIFO queues or LIFO stacks). In the bottom up merge sort, the starting point assumes each run is one item long. In practice, random input data will have many short runs that just happen to be sorted. In the typical case, the natural merge sort may not need as many passes because there are fewer runs to merge. In the best case, the input is already sorted (i.e., is one run), so the natural merge sort need only make one pass through the data. In many practical cases, long natural runs are present, and for that reason natural merge sort is exploited as the key component of Timsort. Example:
Start : 3--4--2--1--7--5--8--9--0--6
Select runs : 3--4 2 1--7 5--8--9 0--6
Merge : 2--3--4 1--5--7--8--9 0--6
Merge : 1--2--3--4--5--7--8--9 0--6
Merge : 0--1--2--3--4--5--6--7--8--9
Tournament replacement selection sorts are used to gather the initial runs for external sorting algorithms.
Analysis
In sorting n objects, merge sort has an average and worst-case performance of O(n log n). If the running time of merge sort for a list of length n is T(n), then the recurrence T(n) = 2T(n/2) + n follows from the definition of the algorithm (apply the algorithm to two lists of half the size of the original list, and add the n steps taken to merge the resulting two lists). The closed form follows from the master theorem for divide-and-conquer recurrences.
In the worst case, the number of comparisons merge sort makes is equal to or slightly smaller than (n ⌈lg n⌉ - 2⌈lg n⌉ + 1), which is between (n lg n - n + 1) and (n lg n + n + O(lg n)).
For large n and a randomly ordered input list, merge sort's expected (average) number of comparisons approaches α·n fewer than the worst case where
In the worst case, merge sort does about 39% fewer comparisons than quicksort does in the average case. In terms of moves, merge sort's worst case complexity is O(n log n)—the same complexity as quicksort's best case, and merge sort's best case takes about half as many iterations as the worst case.
Merge sort is more efficient than quicksort for some types of lists if the data to be sorted can only be efficiently accessed sequentially, and is thus popular in languages such as Lisp, where sequentially accessed data structures are very common. Unlike some (efficient) implementations of quicksort, merge sort is a stable sort.
Merge sort's most common implementation does not sort in place; therefore, the memory size of the input must be allocated for the sorted output to be stored in (see below for versions that need only n/2 extra spaces).
Like QuickSort, Merge Sort is a Divide and Conquer algorithm. It divides input array in two halves, calls itself for the two halves and then merges the two sorted halves. The merge() function is used for merging two halves. The merge(arr, l, m, r) is key process that assumes that arr[l..m] and arr[m+1..r] are sorted and merges the two sorted sub-arrays into one. See following C implementation for details.
MergeSort(arr[], l, r)
If r > l
1. Find the middle point to divide the array into two halves:
middle m = (l+r)/2
2. Call mergeSort for first half:
Call mergeSort(arr, l, m)
3. Call mergeSort for second half:
Call mergeSort(arr, m+1, r)
4. Merge the two halves sorted in step 2 and 3:
Call merge(arr, l, m, r)
MERGE SORT PROGRAM IN JAVA
Output:
/* Java program for Merge Sort */class MergeSort{ // Merges two subarrays of arr[]. // First subarray is arr[l..m] // Second subarray is arr[m+1..r] void merge(int arr[], int l, int m, int r) { // Find sizes of two subarrays to be merged int n1 = m - l + 1; int n2 = r - m; /* Create temp arrays */ int L[] = new int [n1]; int R[] = new int [n2]; /*Copy data to temp arrays*/ for (int i=0; i<n1; ++i) L[i] = arr[l + i]; for (int j=0; j<n2; ++j) R[j] = arr[m + 1+ j]; /* Merge the temp arrays */ // Initial indexes of first and second subarrays int i = 0, j = 0; // Initial index of merged subarry array int k = l; while (i < n1 && j < n2) { if (L[i] <= R[j]) { arr[k] = L[i]; i++; } else { arr[k] = R[j]; j++; } k++; } /* Copy remaining elements of L[] if any */ while (i < n1) { arr[k] = L[i]; i++; k++; } /* Copy remaining elements of R[] if any */ while (j < n2) { arr[k] = R[j]; j++; k++; } } // Main function that sorts arr[l..r] using // merge() void sort(int arr[], int l, int r) { if (l < r) { // Find the middle point int m = (l+r)/2; // Sort first and second halves sort(arr, l, m); sort(arr , m+1, r); // Merge the sorted halves merge(arr, l, m, r); } } /* A utility function to print array of size n */ static void printArray(int arr[]) { int n = arr.length; for (int i=0; i<n; ++i) System.out.print(arr[i] + " "); System.out.println(); } // Driver method public static void main(String args[]) { int arr[] = {12, 11, 13, 5, 6, 7}; System.out.println("Given Array"); printArray(arr); MergeSort ob = new MergeSort(); ob.sort(arr, 0, arr.length-1); System.out.println("\nSorted array"); printArray(arr); }}/* This code is contributed by Onuoha David */ |
Output:
Given array is 12 11 13 5 6 7 Sorted array is 5 6 7 11 12 13
Another Java Code on Merge Sort
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Other Uncompiled code on merge sortimport java.util.*;
public class MergerSort
{
public static void main(String[] args)
{
Integer[] a = {2, 6, 3, 5, 1};
mergeSort(a);
System.out.println(Arrays.toString(a));
}
public static void mergeSort(Comparable [ ] a)
{
Comparable[] tmp = new Comparable[a.length];
mergeSort(a, tmp, 0, a.length - 1);
}
private static void mergeSort(Comparable [ ] a, Comparable [ ] tmp, int left, int right)
{
if( left < right )
{
int center = (left + right) / 2;
mergeSort(a, tmp, left, center);
mergeSort(a, tmp, center + 1, right);
merge(a, tmp, left, center + 1, right);
}
}
private static void merge(Comparable[ ] a, Comparable[ ] tmp, int left, int right, int rightEnd )
{
int leftEnd = right - 1;
int k = left;
int num = rightEnd - left + 1;
while(left <= leftEnd && right <= rightEnd)
if(a[left].compareTo(a[right]) <= 0)
tmp[k++] = a[left++];
else
tmp[k++] = a[right++];
while(left <= leftEnd) // Copy rest of first half
tmp[k++] = a[left++];
while(right <= rightEnd) // Copy rest of right half
tmp[k++] = a[right++];
// Copy tmp back
for(int i = 0; i < num; i++, rightEnd--)
a[rightEnd] = tmp[rightEnd];
}
}
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