MSD( Most Significant Digit ) Radix Sort

Last Updated : 15 Jun, 2021

In this article, two types of Radix Sort are discussed:

LSD Radix Sort: It starts sorting from the end of strings (the Least significant digit).
MSD Radix Sort: It starts sorting from the beginning of strings (the Most significant digit).

In this article, the task is to discuss the MSD Radix Sort and compare it with LSD Radix Sort.

Approach: The idea is to perform the following steps for each digit i where the value of i varies from the most significant digit to the least significant digit:

Store elements in different buckets according to their i^th digit.
Recursively sort each bucket that has more than one element.

Most vs Least Significant Digit Radix Sort:

The idea is to sort the fixed-length integers, MSD is more efficient than LSD because it may not have to examine every digit of each integer:

LSD Radix Sort:

MSD Radix Sort:

MSD Radix sort

MSD can be used to sort strings of variable length, unlike LSD. LSD has to be stable in order to work correctly, but MSD can either be made stable or unstable and MSD can work with random strings.

MSD Radix sort variable length string

Time Complexity:
- LSD Radix sort: Best and Worst-Case time complexity is O(N*M) where M = length of the longest string.
  MSD Radix sort: Best Case time complexity is O(N) and the Worst-Case time complexity is O(N*M) where M = the average length of strings.
Auxiliary Space:
- LSD Radix sort: O(N + B)
- MSD Radix sort: O(N + MB), where M = length of the longest string and B = size of radix (B=10 possible numbers or B=256 characters or B=2 for Binary).
MSD uses recursion, so it requires more space than LSD. This means that MSD is much slower than LSD when working with a few inputs.

Implementation of MSD Radix Sort:

Using linked list: This implementation is for integers using linked list. A fixed-length array for every node will take a very large amount of storage.

Below is the implementation of MSD Radix Sort using a linked list:

C++

// C++ program for the implementation
// of MSD Radix Sort using linked list
#include <iostream>
#include <vector>
 
using namespace std;
 
// Linked list node structure
struct node {
    vector<int> arr;
    struct node* nxt[10];
};
 
// Function to create a new node of
// the Linked List
struct node* new_node(void)
{
    struct node* tempNode = new node;
 
    for (int i = 0; i < 10; i++) {
        tempNode->nxt[i] = NULL;
    }
 
    // Return the created node
    return tempNode;
}
 
// Function to sort the given array
// using MSD Radix Sort recursively
void msd_sort(struct node* root, int exp,
              vector<int>& sorted_arr)
{
    if (exp <= 0) {
        return;
    }
 
    int j;
 
    // Stores the numbers in different
    // buckets according their MSD
    for (int i = 0;
         i < root->arr.size();
         i++) {
 
        // Get the MSD in j
        j = (root->arr[i] / exp) % 10;
 
        // If j-th index in the node
        // array is empty create and
        // link a new node in index
        if (root->nxt[j] == NULL) {
            root->nxt[j] = new_node();
        }
 
        // Store the number in j-th node
        root->nxt[j]->arr.push_back(
            root->arr[i]);
    }
 
    // Sort again every child node that
    // has more than one number
    for (int i = 0; i < 10; i++) {
 
        // If root->next is NULL
        if (root->nxt[i] != NULL) {
 
            if (root->nxt[i]->arr.size()
                > 1) {
 
                // Sort recursively
                msd_sort(root->nxt[i],
                         exp / 10,
                         sorted_arr);
            }
 
            // If any node have only
            // one number then it means
            // the number is sorted
            else {
                sorted_arr.push_back(
                    root->nxt[i]->arr[0]);
            }
        }
    }
}
 
// Function to calculate the MSD of the
// maximum  value in the array
int get_max_exp(vector<int> arr)
{
    // Stores the maximum element
    int mx = arr[0];
 
    // Traverse the given array
    for (int i = 1; i < arr.size(); i++) {
 
        // Update the value of maximum
        if (arr[i] > mx) {
            mx = arr[i];
        }
    }
 
    int exp = 1;
 
    while (mx > 10) {
        mx /= 10;
        exp *= 10;
    }
 
    // Return the resultant value
    return exp;
}
 
// Function to print an array
void print(vector<int> arr)
{
    for (int i = 0; i < arr.size(); i++)
        cout << arr[i] << " ";
 
    cout << endl;
}
 
// Driver Code
int main()
{
    // create the root node
    struct node* root = new_node();
 
    // Stores the unsorted array
    // in the root node
    root->arr.insert(root->arr.end(),
                     { 9330, 9950, 718,
                       8977, 6790, 95,
                       9807, 741, 8586,
                       5710 });
 
    cout << "Unsorted array : ";
 
    // Print the unsorted array
    print(root->arr);
 
    // Find the optimal longest exponent
    int exp = get_max_exp(root->arr);
 
    // Stores the sorted numbers
    vector<int> sorted_arr;
 
    // Function Call
    msd_sort(root, exp, sorted_arr);
 
    cout << "Sorted array : ";
 
    // Print the sorted array
    print(sorted_arr);
 
    return 0;
}

C

// C program for the implementation
// of MSD Radix Sort using linked list
// Linked list node structure
#include <stdio.h>
#include <stdlib.h> // For using malloc
#include <string.h> // For using memset
 
// Output array filled length
int sorted_array_length = 0;
 
struct node {
    int arr[100];
    int arr_length;
    struct node* nxt[10];
};
 
// Function to create a new node of
// the Linked List
struct node* new_node(void)
{
    struct node* tempNode
        = (struct node*)malloc(sizeof(struct node));
 
    tempNode->arr_length = 0;
 
    for (int i = 0; i < 10; i++) {
        tempNode->nxt[i] = NULL;
    }
 
    // Return the created node
    return tempNode;
}
 
// Function to sort the given array
// using MSD Radix Sort recursively
void msd_sort(struct node* root, int exp, int* sorted_arr)
{
    if (exp <= 0) {
        return;
    }
 
    int j;
 
    // Stores the numbers in different
    // buckets according their MSD
    for (int i = 0; i < root->arr_length; i++) {
 
        // Get the MSD in j
        j = (root->arr[i] / exp) % 10;
 
        // If j-th index in the node
        // array is empty create and
        // link a new node in index
        if (root->nxt[j] == NULL) {
            root->nxt[j] = new_node();
        }
 
        // Store the number in j-th node
        root->nxt[j]->arr[root->nxt[j]->arr_length++]
            = root->arr[i];
    }
 
    // Sort again every child node that
    // has more than one number
    for (int i = 0; i < 10; i++) {
 
        // If root->next is NULL
        if (root->nxt[i] != NULL) {
 
            if (root->nxt[i]->arr_length > 1) {
 
                // Sort recursively
                msd_sort(root->nxt[i], exp / 10,
                         sorted_arr);
            }
 
            // If any node have only
            // one number then it means
            // the number is sorted
            else {
                sorted_arr[sorted_array_length++]
                    = root->nxt[i]->arr[0];
            }
        }
    }
}
 
// Function to calculate the MSD of the
// maximum value in the array
int get_max_exp(int* arr, int n)
{
    // Stores the maximum element
    int mx = arr[0];
 
    // Traverse the given array
    for (int i = 1; i < n; i++) {
 
        // Update the value of maximum
        if (arr[i] > mx) {
            mx = arr[i];
        }
    }
 
    int exp = 1;
 
    while (mx > 10) {
        mx /= 10;
        exp *= 10;
    }
 
    // Return the resultant value
    return exp;
}
 
// Function to print an array
void print(int* arr, int n)
{
    for (int i = 0; i < n; i++)
        printf("%d ", arr[i]);
 
    printf("\n");
}
 
// Driver Code
int main()
{
    // Unsorted array
    int array[] = { 9330, 9950, 718, 8977, 6790,
                    95,   9807, 741, 8586, 5710 };
 
    // Input array length
    int n = sizeof(array) / sizeof(array[0]);
 
    // create the root node
    struct node* root = new_node();
 
    // Stores the unsorted array
    // in the root node and
    // set arr_length
    memcpy(root->arr, array, sizeof(array));
    root->arr_length = n;
 
    printf("Unsorted array : ");
 
    // Print the unsorted array
    print(root->arr, n);
 
    // Find the optimal longest exponent
    int exp = get_max_exp(root->arr, root->arr_length);
 
    // Stores the sorted numbers
    int output[n];
    int* sorted_arr = &output[0];
 
    // Function Call
    msd_sort(root, exp, sorted_arr);
 
    printf("Sorted array : ");
 
    // Print the sorted array
    print(sorted_arr, n);
 
    return 0;
}

Output

Unsorted array : 9330 9950 718 8977 6790 95 9807 741 8586 5710 
Sorted array : 95 718 741 5710 6790 8586 8977 9330 9807 9950

Using Counting Sort() method: This implementation is for the strings based on the counting sort() method. As C style ASCII character is 1 byte. So, the 256 size array is used to count occurrences of characters, and it sorts the strings lexicographically.

Below is the implementation of MSD Radix Sort using the counting sort() method:

For string:

C++

// C++ implementation of MSD Radix Sort
#include <iostream>
#include <unordered_map>
 
using namespace std;
 
// A utility function to print an array
void print(string* str, int n)
{
    for (int i = 0; i < n; i++) {
        cout << str[i] << " ";
    }
    cout << endl;
}
 
// A utility function to get the ASCII value
// of the character at index d in a string
int char_at(string str, int d)
{
    if (str.size() <= d)
        return -1;
    else
        return str.at(d);
}
 
// The main function to sort array using
// MSD Radix Sort recursively
void MSD_sort(string* str, int lo, int hi, int d)
{
 
    // recursion break condition
    if (hi <= lo) {
        return;
    }
 
    int count[256 + 2] = { 0 };
 
    // temp is craeted to easily swap strings in str[]
    // int temp[n] can also be used but,
    // it will take more space.
    unordered_map<int, string> temp;
 
    // Store occurrences of most significant character
    // from each string in count[]
    for (int i = lo; i <= hi; i++) {
        int c = char_at(str[i], d);
        count++;
    }
 
    // Change count[] so that count[] now contains actual
    //  position of this digits in temp[]
    for (int r = 0; r < 256 + 1; r++)
        count[r + 1] += count[r];
 
    // Build the temp
    for (int i = lo; i <= hi; i++) {
        int c = char_at(str[i], d);
        temp[count++] = str[i];
    }
 
    // Copy all strings of temp to str[], so that str[] now
    // contains partially sorted strings
    for (int i = lo; i <= hi; i++)
        str[i] = temp[i - lo];
 
    // Recursively MSD_sort() on each partially sorted
    // strings set to sort them by their next character
    for (int r = 0; r < 256; r++)
        MSD_sort(str, lo + count[r], lo + count[r + 1] - 1,
                 d + 1);
}
 
int main()
{
    string str[] = { "midnight", "badge",  "bag",
                     "worker",   "banner", "wander" };
 
    int n = sizeof(str) / sizeof(str[0]);
 
    cout << "Unsorted array : ";
    // print the unsorted array
    print(str, n);
 
    // Function call
    MSD_sort(str, 0, n - 1, 0);
 
    cout << "Sorted array : ";
    // print the sorted array
    print(str, n);
 
    return 0;
}

C

// C program for the implementation
// of MSD Radix Sort using counting sort()
#include <stdio.h>
#include <string.h>
 
// A utility function to get the ASCII value
// of the character at index d in a string
int char_at(char* str, int d)
{
    if (sizeof(str) / sizeof(str[0]) <= d)
        return -1;
    else
        return str[d];
}
 
int n;
 
// The main function to sort array using
// MSD Radix Sort recursively
void MSD_sort(char** str, int lo, int hi, int d)
{
 
    // recursion break condition
    if (hi <= lo + 1) {
        return;
    }
 
    int count[256 + 2] = { 0 };
 
    // temp is craeted to easily swap strings in str[]
    char temp[n][100];
 
    // Store occurrences of most significant character
    // from each string in count[]
    for (int i = lo; i <= hi; i++) {
        int c = char_at(str[i], d);
        count++;
    }
 
    // Change count[] so that count[] now contains actual
    //  position of this digits in temp[]
    for (int r = 0; r < 256 + 1; r++)
        count[r + 1] += count[r];
 
    // Build the temp
    for (int i = lo; i <= hi; i++) {
        int c = char_at(str[i], d);
        memcpy(temp[count++], str[i],
               strlen(str[i]) + 1);
    }
 
    // Copy all strings of temp to str[], so that str[] now
    // contains partially sorted strings
    for (int i = lo; i <= hi; i++) {
        str[i] = strdup(temp[i - lo]);
    }
 
    // Recursively MSD_sort() on each partially sorted
    // strings set to sort them by their next character
    for (int r = 0; r < 256; r++)
        MSD_sort(str, lo + count[r], lo + count[r + 1] - 1,
                 d + 1);
}
 
// Function to print an array
void print(char** str, int n)
{
    for (int i = 0; i < n; i++) {
        printf("%s ", str[i]);
    }
    printf("\n");
}
 
// Driver Code
int main()
{
    // Input String
    char* str[] = { (char*)"midnight", (char*)"badge",
                    (char*)"bag",      (char*)"worker",
                    (char*)"banner",   (char*)"wander" };
 
    // Size of the string
    n = sizeof(str) / sizeof(str[0]);
 
    printf("Unsorted array : ");
 
    // Print the unsorted array
    print(str, n);
 
    // Function Call
    MSD_sort(str, 0, n - 1, 0);
 
    printf("Sorted array : ");
 
    // Print the sorted array
    print(str, n);
 
    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.lang.*;
import java.util.*;
 
public class GFG {
 
    // Utility function to get the ASCII
    // value of the character at index d
    // in the string
    static int char_at(String str, int d)
    {
        if (str.length() <= d)
            return -1;
        else
            return (int)(str.charAt(d));
    }
 
    // Function to sort the array using
    // MSD Radix Sort recursively
    static void MSD_sort(String str[], int lo, int hi,
                         int d)
    {
        // Recursive break condition
        if (hi <= lo) {
            return;
        }
 
        // Stores the ASCII Values
        int count[] = new int[256 + 1];
 
        // Temp is created to easily
        // swap strings in str[]
        HashMap<Integer, String> temp = new HashMap<>();
 
        // Store the occurrences of the most
        // significant character from
        // each string in count[]
        for (int i = lo; i <= hi; i++) {
            int c = char_at(str[i], d);
            count++;
        }
 
        // Change count[] so that count[]
        // now contains actual position
        // of this digits in temp[]
        for (int r = 0; r < 256; r++)
            count[r + 1] += count[r];
 
        // Build the temp
        for (int i = lo; i <= hi; i++) {
            int c = char_at(str[i], d);
            temp.put(count++, str[i]);
        }
 
        // Copy all strings of temp to str[],
        // so that str[] now contains
        // partially sorted strings
        for (int i = lo; i <= hi; i++)
            str[i] = temp.get(i - lo);
 
        // Recursively MSD_sort() on each
        // partially sorted strings set to
        // sort them by their next character
        for (int r = 0; r < 256; r++)
            MSD_sort(str, lo + count[r],
                     lo + count[r + 1] - 1, d + 1);
    }
 
    // Function to print an array
    static void print(String str[], int n)
    {
        for (int i = 0; i < n; i++) {
            System.out.print(str[i] + " ");
        }
        System.out.println();
    }
 
    // Driver Code
    public static void main(String[] args)
    {
 
        // Input String
        String str[] = { "midnight", "badge",  "bag",
                         "worker",   "banner", "wander" };
 
        // Size of the string
        int n = str.length;
 
        System.out.print("Unsorted array : ");
 
        // Print the unsorted array
        print(str, n);
 
        // Function Call
        MSD_sort(str, 0, n - 1, 0);
 
        System.out.print("Sorted array : ");
 
        // Print the sorted array
        print(str, n);
    }
}
 
// This code is contributed by Kingash.

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
 
class GFG{
 
// Utility function to get the ASCII
// value of the character at index d
// in the string
static int char_at(String str, int d)
{
    if (str.Length <= d)
        return -1;
    else
        return(int)(str[d]);
}
 
// Function to sort the array using
// MSD Radix Sort recursively
static void MSD_sort(String []str, int lo,
                           int hi, int d)
{
     
    // Recursive break condition
    if (hi <= lo)
    {
        return;
    }
 
    // Stores the ASCII Values
    int []count = new int[256 + 1];
 
    // Temp is created to easily
    // swap strings in []str
    Dictionary<int,
               String> temp = new Dictionary<int,
                                             String>();
 
    // Store the occurrences of the most
    // significant character from
    // each string in []count
    for(int i = lo; i <= hi; i++)
    {
        int c = char_at(str[i], d);
        count++;
    }
 
    // Change []count so that []count
    // now contains actual position
    // of this digits in []temp
    for(int r = 0; r < 256; r++)
        count[r + 1] += count[r];
 
    // Build the temp
    for(int i = lo; i <= hi; i++)
    {
        int c = char_at(str[i], d);
        temp.Add(count++, str[i]);
    }
 
    // Copy all strings of temp to []str,
    // so that []str now contains
    // partially sorted strings
    for(int i = lo; i <= hi; i++)
        str[i] = temp[i - lo];
 
    // Recursively MSD_sort() on each
    // partially sorted strings set to
    // sort them by their next character
    for(int r = 0; r < 256; r++)
        MSD_sort(str, lo + count[r],
                      lo + count[r + 1] - 1,
                       d + 1);
}
 
// Function to print an array
static void print(String []str, int n)
{
    for(int i = 0; i < n; i++)
    {
        Console.Write(str[i] + " ");
    }
    Console.WriteLine();
}
 
// Driver Code
public static void Main(String[] args)
{
 
    // Input String
    String []str = { "midnight", "badge", "bag",
                     "worker", "banner", "wander" };
 
    // Size of the string
    int n = str.Length;
 
    Console.Write("Unsorted array : ");
 
    // Print the unsorted array
    print(str, n);
 
    // Function Call
    MSD_sort(str, 0, n - 1, 0);
 
    Console.Write("Sorted array : ");
 
    // Print the sorted array
    print(str, n);
}
}
 
// This code is contributed by shikhasingrajput

Output

Unsorted array : midnight badge bag worker banner wander 
Sorted array : badge bag banner midnight wander worker

For Integer:

C++

// C++ implementation of MSD Radix Sort
// of MSD Radix Sort using counting sort()
#include <iostream>
#include <math.h>
#include <unordered_map>
 
using namespace std;
 
// A utility function to print an array
void print(int* arr, int n)
{
    for (int i = 0; i < n; i++) {
        cout << arr[i] << " ";
    }
    cout << endl;
}
 
// A utility function to get the digit
// at index d in a integer
int digit_at(int x, int d)
{
    return (int)(x / pow(10, d - 1)) % 10;
}
 
// The main function to sort array using
// MSD Radix Sort recursively
void MSD_sort(int* arr, int lo, int hi, int d)
{
 
    // recursion break condition
    if (hi <= lo) {
        return;
    }
 
    int count[10 + 2] = { 0 };
 
    // temp is craeted to easily swap strings in arr[]
    unordered_map<int, int> temp;
 
    // Store occurrences of most significant character
    // from each integer in count[]
    for (int i = lo; i <= hi; i++) {
        int c = digit_at(arr[i], d);
        count++;
    }
 
    // Change count[] so that count[] now contains actual
    //  position of this digits in temp[]
    for (int r = 0; r < 10 + 1; r++)
        count[r + 1] += count[r];
 
    // Build the temp
    for (int i = lo; i <= hi; i++) {
        int c = digit_at(arr[i], d);
        temp[count++] = arr[i];
    }
 
    // Copy all integers of temp to arr[], so that arr[] now
    // contains partially sorted integers
    for (int i = lo; i <= hi; i++)
        arr[i] = temp[i - lo];
 
    // Recursively MSD_sort() on each partially sorted
    // integers set to sort them by their next digit
    for (int r = 0; r < 10; r++)
        MSD_sort(arr, lo + count[r], lo + count[r + 1] - 1,
                 d - 1);
}
 
// function find the largest integer
int getMax(int arr[], int n)
{
    int mx = arr[0];
    for (int i = 1; i < n; i++)
        if (arr[i] > mx)
            mx = arr[i];
    return mx;
}
 
// Main function to call MSD_sort
void radixsort(int* arr, int n)
{
    // Find the maximum number to know number of digits
    int m = getMax(arr, n);
 
    // get the length of the largest integer
    int d = floor(log10(abs(m))) + 1;
 
    // function call
    MSD_sort(arr, 0, n - 1, d);
}
 
// Driver Code
int main()
{
    // Input array
    int arr[] = { 9330, 9950, 718, 8977, 6790,
                  95,   9807, 741, 8586, 5710 };
 
    // Size of the array
    int n = sizeof(arr) / sizeof(arr[0]);
 
    printf("Unsorted array : ");
 
    // Print the unsorted array
    print(arr, n);
 
    // Function Call
    radixsort(arr, n);
 
    printf("Sorted array : ");
 
    // Print the sorted array
    print(arr, n);
 
    return 0;
}

C

// C program for the implementation
// of MSD Radix Sort using counting sort()
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
 
// A utility function to print an array
void print(int* arr, int n)
{
    for (int i = 0; i < n; i++) {
        printf("%d, ", arr[i]);
    }
    printf("\n");
}
 
// A utility function to get the digit
// at index d in a integer
int digit_at(int x, int d)
{
    return (int)(x / pow(10, d - 1)) % 10;
}
 
// array length
int n;
 
// function to sort array using
// MSD Radix Sort recursively
void MSD_sort(int* arr, int lo, int hi, int d)
{
 
    // recursion break condition
    if (hi <= lo || d < 1) {
        return;
    }
 
    int count[10 + 2] = { 0 };
 
    // temp is craeted to easily swap strings in arr[]
    int temp[n];
 
    // Store occurrences of most significant character
    // from each integer in count[]
    for (int i = lo; i <= hi; i++) {
        int c = digit_at(arr[i], d);
        count++;
    }
 
    // Change count[] so that count[] now contains actual
    //  position of this digits in temp[]
    for (int r = 0; r < 10 + 1; r++)
        count[r + 1] += count[r];
 
    // Build the temp
    for (int i = lo; i <= hi; i++) {
        int c = digit_at(arr[i], d);
        temp[count++] = arr[i];
    }
 
    // Copy all integer of temp to arr[], so that arr[] now
    // contains partially sorted integers
    for (int i = lo; i <= hi; i++) {
        arr[i] = temp[i - lo];
    }
 
    // Recursively MSD_sort() on each partially sorted
    // integers set to sort them by their next digit
    for (int r = 0; r < 10; r++)
        MSD_sort(arr, lo + count[r], lo + count[r + 1] - 1,
                 d - 1);
}
 
// function find the largest integer
int getMax(int arr[], int n)
{
    int mx = arr[0];
    for (int i = 1; i < n; i++)
        if (arr[i] > mx)
            mx = arr[i];
    return mx;
}
 
// Main function to call MSD_sort
void radixsort(int* arr, int n)
{
    // Find the maximum number to know number of digits
    int m = getMax(arr, n);
 
    // get the length of the largest integer
    int d = floor(log10(abs(m))) + 1;
 
    // function call
    MSD_sort(arr, 0, n - 1, d);
}
 
// Driver Code
int main()
{
    // Input array
    int arr[] = { 9330, 9950, 718, 8977, 6790,
                  95,   9807, 741, 8586, 5710 };
 
    // Size of the array
    n = sizeof(arr) / sizeof(arr[0]);
 
    printf("Unsorted array : ");
 
    // Print the unsorted array
    print(arr, n);
 
    // Function Call
    radixsort(arr, n);
 
    printf("Sorted array : ");
 
    // Print the sorted array
    print(arr, n);
 
    return 0;
}

Output

Unsorted array : 9330 9950 718 8977 6790 95 9807 741 8586 5710 
Sorted array : 95 718 741 5710 6790 8586 8977 9330 9807 9950

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