Merge Sort with O(1) extra space merge and O(n log n) time [Unsigned Integers Only]
Last Updated :
11 Jul, 2025
We have discussed Merge sort. How to modify the algorithm so that merge works in O(1) extra space and algorithm still works in O(n Log n) time. We may assume that the input values are integers only.
Examples:
Input : 5 4 3 2 1
Output : 1 2 3 4 5
Input : 999 612 589 856 56 945 243
Output : 56 243 589 612 856 945 999
For integer types, merge sort can be made inplace using some mathematics trick of modulus and division. That means storing two elements value at one index and can be extracted using modulus and division.
First we have to find a value greater than all the elements of the array. Now we can store the original value as modulus and the second value as division. Suppose we want to store arr[i] and arr[j] both at index i(means in arr[i]). First we have to find a 'maxval' greater than both arr[i] and arr[j]. Now we can store as arr[i] = arr[i] + arr[j]*maxval. Now arr[i]%maxval will give the original value of arr[i] and arr[i]/maxval will give the value of arr[j]. So below is the implementation on merge sort.
Approach: (Euclidean Division)
dividend = divisor * quotient +remainder
ex: 5/3 = q:1, r:2 applying euclidean: 3*1+2 =>5 (dividend)
divisor = maxele (absolute max element in the array)+1 (so that we always get non zero remainder)
quotient = min(first, second)
remainder = original element
Note: (when getting current element, assume the current container already has encoded element hence using % divisor)
first = arr[i] % divisor
second = arr[j] % divisor
encoded element = remainder + quotient*divisor
Possible issues in Merging:
1. If current number is Integer.MAX then new encoded value which is usually greater than current element will cause integer overflow and data corruption (In python there is no limit to number size so this issue will not occur).
2. Doesn't handle negative numbers (ie, when encoding a -ve number(current) with another -ve number(chosen smallest) the sign can't be preserved since both numbers have -ve sign. Also absolute values must be used when computing dividend = divisor*quotient+remainder (divisor = maxele, quotient = smallest, remainder = original) and sign must be restored, still it might not work due to sign preservation issue.
3. Only applicable to Unsigned integers, like indexes which are usually non-negative.
4. AUX = O(n) in worst case, assuming in a language like python where there is no limit to word/integer size, when input array elements are almost at Integer.MAX, then encoded value will require possibly 2x bits space to represent new number, the 2x bit space on whole can become +1x array size, which is almost like creating an AUX array but in an indirect way.
5. mod and division operations are the costliest, hence reduces overall performance(upto some extent).
C++
// C++ program to sort an array using merge sort such
// that merge operation takes O(1) extra space.
#include <bits/stdc++.h>
using namespace std;
void merge(int arr[], int beg, int mid, int end, int maxele)
{
int i = beg;
int j = mid + 1;
int k = beg;
while (i <= mid && j <= end) {
if (arr[i] % maxele <= arr[j] % maxele) {
arr[k] = arr[k] + (arr[i] % maxele) * maxele;
k++;
i++;
}
else {
arr[k] = arr[k] + (arr[j] % maxele) * maxele;
k++;
j++;
}
}
while (i <= mid) {
arr[k] = arr[k] + (arr[i] % maxele) * maxele;
k++;
i++;
}
while (j <= end) {
arr[k] = arr[k] + (arr[j] % maxele) * maxele;
k++;
j++;
}
// Obtaining actual values
for (int i = beg; i <= end; i++)
arr[i] = arr[i] / maxele;
}
// Recursive merge sort with extra parameter, naxele
void mergeSortRec(int arr[], int beg, int end, int maxele)
{
if (beg < end) {
int mid = (beg + end) / 2;
mergeSortRec(arr, beg, mid, maxele);
mergeSortRec(arr, mid + 1, end, maxele);
merge(arr, beg, mid, end, maxele);
}
}
// This functions finds max element and calls recursive
// merge sort.
void mergeSort(int arr[], int n)
{
int maxele = *max_element(arr, arr+n) + 1;
mergeSortRec(arr, 0, n-1, maxele);
}
int main()
{
int arr[] = { 999, 612, 589, 856, 56, 945, 243 };
int n = sizeof(arr) / sizeof(arr[0]);
mergeSort(arr, n);
cout << "Sorted array \n";
for (int i = 0; i < n; i++)
cout << arr[i] << " ";
return 0;
}
// Java program to sort an array
// using merge sort such that
// merge operation takes O(1)
// extra space.
import java.util.Arrays;
class GFG
{
static void merge(int[] arr, int beg,
int mid, int end,
int maxele)
{
int i = beg;
int j = mid + 1;
int k = beg;
while (i <= mid && j <= end)
{
if (arr[i] % maxele <=
arr[j] % maxele)
{
arr[k] = arr[k] + (arr[i]
% maxele) * maxele;
k++;
i++;
}
else
{
arr[k] = arr[k] +
(arr[j] % maxele)
* maxele;
k++;
j++;
}
}
while (i <= mid)
{
arr[k] = arr[k] + (arr[i]
% maxele) * maxele;
k++;
i++;
}
while (j <= end)
{
arr[k] = arr[k] + (arr[j]
% maxele) * maxele;
k++;
j++;
}
// Obtaining actual values
for (i = beg; i <= end; i++)
{
arr[i] = arr[i] / maxele;
}
}
// Recursive merge sort
// with extra parameter, naxele
static void mergeSortRec(int[] arr, int beg,
int end, int maxele)
{
if (beg < end)
{
int mid = (beg + end) / 2;
mergeSortRec(arr, beg,
mid, maxele);
mergeSortRec(arr, mid + 1,
end, maxele);
merge(arr, beg, mid,
end, maxele);
}
}
// This functions finds
// max element and calls
// recursive merge sort.
static void mergeSort(int[] arr, int n)
{
int maxele = Arrays.stream(arr).max().getAsInt() + 1;
mergeSortRec(arr, 0, n - 1, maxele);
}
// Driver code
public static void main(String[] args)
{
int[] arr = {999, 612, 589,
856, 56, 945, 243};
int n = arr.length;
mergeSort(arr, n);
System.out.println("Sorted array ");
for (int i = 0; i < n; i++)
{
System.out.print(arr[i] + " ");
}
}
}
// This code is contributed by 29AjayKumar
# Python3 program to sort an array using
# merge sort such that merge operation
# takes O(1) extra space.
def merge(arr, beg, mid, end, maxele):
i = beg
j = mid + 1
k = beg
while (i <= mid and j <= end):
if (arr[i] % maxele <= arr[j] % maxele):
arr[k] = arr[k] + (arr[i] %
maxele) * maxele
k += 1
i += 1
else:
arr[k] = arr[k] + (arr[j] %
maxele) * maxele
k += 1
j += 1
while (i <= mid):
arr[k] = arr[k] + (arr[i] %
maxele) * maxele
k += 1
i += 1
while (j <= end):
arr[k] = arr[k] + (arr[j] %
maxele) * maxele
k += 1
j += 1
# Obtaining actual values
for i in range(beg, end + 1):
arr[i] = arr[i] // maxele
# Recursive merge sort with extra
# parameter, naxele
def mergeSortRec(arr, beg, end, maxele):
if (beg < end):
mid = (beg + end) // 2
mergeSortRec(arr, beg, mid, maxele)
mergeSortRec(arr, mid + 1, end, maxele)
merge(arr, beg, mid, end, maxele)
# This functions finds max element and
# calls recursive merge sort.
def mergeSort(arr, n):
maxele = max(arr) + 1
mergeSortRec(arr, 0, n - 1, maxele)
# Driver Code
if __name__ == '__main__':
arr = [ 999, 612, 589, 856, 56, 945, 243 ]
n = len(arr)
mergeSort(arr, n)
print("Sorted array")
for i in range(n):
print(arr[i], end = " ")
# This code is contributed by mohit kumar 29
// C# program to sort an array
// using merge sort such that
// merge operation takes O(1)
// extra space.
using System;
using System.Linq;
class GFG
{
static void merge(int []arr, int beg,
int mid, int end,
int maxele)
{
int i = beg;
int j = mid + 1;
int k = beg;
while (i <= mid && j <= end)
{
if (arr[i] %
maxele <= arr[j] % maxele)
{
arr[k] = arr[k] + (arr[i] %
maxele) * maxele;
k++;
i++;
}
else
{
arr[k] = arr[k] +
(arr[j] % maxele) *
maxele;
k++;
j++;
}
}
while (i <= mid)
{
arr[k] = arr[k] + (arr[i] %
maxele) * maxele;
k++;
i++;
}
while (j <= end)
{
arr[k] = arr[k] + (arr[j] %
maxele) * maxele;
k++;
j++;
}
// Obtaining actual values
for ( i = beg; i <= end; i++)
arr[i] = arr[i] / maxele;
}
// Recursive merge sort
// with extra parameter, naxele
static void mergeSortRec(int []arr, int beg,
int end, int maxele)
{
if (beg < end)
{
int mid = (beg + end) / 2;
mergeSortRec(arr, beg,
mid, maxele);
mergeSortRec(arr, mid + 1,
end, maxele);
merge(arr, beg, mid,
end, maxele);
}
}
// This functions finds
// max element and calls
// recursive merge sort.
static void mergeSort(int []arr, int n)
{
int maxele = arr.Max() + 1;
mergeSortRec(arr, 0, n - 1, maxele);
}
//Driver code
public static void Main ()
{
int []arr = {999, 612, 589,
856, 56, 945, 243};
int n = arr.Length;
mergeSort(arr, n);
Console.WriteLine("Sorted array ");
for (int i = 0; i < n; i++)
Console.Write( arr[i] + " ");
}
}
// This code is contributed
// by inder_verma.
<script>
// Javascript program to sort an array
// using merge sort such that
// merge operation takes O(1)
// extra space.
function merge(arr,beg,mid,end,maxele)
{
let i = beg;
let j = mid + 1;
let k = beg;
while (i <= mid && j <= end)
{
if (arr[i] % maxele <=
arr[j] % maxele)
{
arr[k] = arr[k] + (arr[i]
% maxele) * maxele;
k++;
i++;
}
else
{
arr[k] = arr[k] +
(arr[j] % maxele)
* maxele;
k++;
j++;
}
}
while (i <= mid)
{
arr[k] = arr[k] + (arr[i]
% maxele) * maxele;
k++;
i++;
}
while (j <= end)
{
arr[k] = arr[k] + (arr[j]
% maxele) * maxele;
k++;
j++;
}
// Obtaining actual values
for (i = beg; i <= end; i++)
{
arr[i] = Math.floor(arr[i] / maxele);
}
}
// Recursive merge sort
// with extra parameter, naxele
function mergeSortRec(arr,beg,end,maxele)
{
if (beg < end)
{
let mid = Math.floor((beg + end) / 2);
mergeSortRec(arr, beg,
mid, maxele);
mergeSortRec(arr, mid + 1,
end, maxele);
merge(arr, beg, mid,
end, maxele);
}
}
// This functions finds
// max element and calls
// recursive merge sort.
function mergeSort(arr,n)
{
let maxele = Math.max(...arr) + 1;
mergeSortRec(arr, 0, n - 1, maxele);
}
// Driver code
let arr=[999, 612, 589,
856, 56, 945, 243];
let n = arr.length;
mergeSort(arr, n);
document.write("Sorted array <br>");
for (let i = 0; i < n; i++)
{
document.write(arr[i] + " ");
}
// This code is contributed by patel2127
</script>
Output: Sorted array
56 243 589 612 856 945 999
Time Complexity: O(n*log(n))
Space Complexity: O(1)
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem