Difference Array | Range update query in O(1)

Consider an array A[] of integers and following two types of queries.

  1. update(l, r, x) : Adds x to all values from A[l] to A[r] (both inclusive).
  2. printArray() : Prints the current modified array.

Examples : 

Input : A [] { 10, 5, 20, 40 }
        update(0, 1, 10)
        printArray()
        update(1, 3, 20)
        update(2, 2, 30)
        printArray()
Output : 20 15 20 40
         20 35 70 60
Explanation : The query update(0, 1, 10) 
adds 10 to A[0] and A[1]. After update,
A[] becomes {20, 15, 20, 40}       
Query update(1, 3, 20) adds 20 to A[1],
A[2] and A[3]. After update, A[] becomes
{20, 35, 40, 60}.
Query update(2, 2, 30) adds 30 to A[2]. 
After update, A[] becomes {20, 35, 70, 60}.

A simple solution is to do following :



  1. update(l, r, x) : Run a loop from l to r and add x to all elements from A[l] to A[r]
  2. printArray() : Simply print A[].

Time complexities of both of the above operations is O(n)

An efficient solution is to use difference array.
Difference array D[i] of a given array A[i] is defined as D[i] = A[i]-A[i-1] (for 0 < i < N ) and D[0] = A[0] considering 0 based indexing. Difference array can be used to perform range update queries "l r x" where l is left index, r is right index and x is value to be added and after all queries you can return original array from it. Where update range operations can be performed in O(1) complexity.

  1. update(l, r, x) : Add x to D[l] and subtract it from D[r+1], i.e., we do D[l] += x, D[r+1] -= x
  2. printArray() : Do A[0] = D[0] and print it. For rest of the elements, do A[i] = A[i-1] + D[i] and print them.

Time complexity of update here is improved to O(1). Note that printArray() still takes O(n) time.

C++

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// C++ code to demonstrate Difference Array
#include <bits/stdc++.h>
using namespace std;
  
// Creates a diff array D[] for A[] and returns
// it after filling initial values.
vector<int> initializeDiffArray(vector<int>& A)
{
    int n = A.size();
  
    // We use one extra space because
    // update(l, r, x) updates D[r+1]
    vector<int> D(n + 1);
  
    D[0] = A[0], D[n] = 0;
    for (int i = 1; i < n; i++)
        D[i] = A[i] - A[i - 1];
    return D;
}
  
// Does range update
void update(vector<int>& D, int l, int r, int x)
{
    D[l] += x;
    D[r + 1] -= x;
}
  
// Prints updated Array
int printArray(vector<int>& A, vector<int>& D)
{
    for (int i = 0; i < A.size(); i++) {
        if (i == 0)
            A[i] = D[i];
  
        // Note that A[0] or D[0] decides
        // values of rest of the elements.
        else
            A[i] = D[i] + A[i - 1];
  
        cout << A[i] << " ";
    }
    cout << endl;
}
  
// Driver Code
int main()
{
    // Array to be updated
    vector<int> A{ 10, 5, 20, 40 };
  
    // Create and fill difference Array
    vector<int> D = initializeDiffArray(A);
  
    // After below update(l, r, x), the
    // elements should become 20, 15, 20, 40
    update(D, 0, 1, 10);
    printArray(A, D);
  
    // After below updates, the
    // array should become 30, 35, 70, 60
    update(D, 1, 3, 20);
    update(D, 2, 2, 30);
    printArray(A, D);
  
    return 0;
}

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Java














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// Java code to demonstrate Difference Array 


class GFG { 


      


    // Creates a diff array D[] for A[] and returns 


    // it after filling initial values. 


    static void initializeDiffArray(int A[], int D[]) 


    { 


          


        int n = A.length; 


  


        D[0] = A[0]; 


        D[n] = 0; 


        for (int i = 1; i < n; i++) 


            D[i] = A[i] - A[i - 1]; 


    } 


  


    // Does range update 


    static void update(int D[], int l, int r, int x) 


    { 


        D[l] += x; 


        D[r + 1] -= x; 


    } 


  


    // Prints updated Array 


    static int printArray(int A[], int D[]) 


    { 


        for (int i = 0; i < A.length; i++) { 


              


            if (i == 0) 


                A[i] = D[i]; 


  


            // Note that A[0] or D[0] decides 


            // values of rest of the elements. 


            else


                A[i] = D[i] + A[i - 1]; 


  


            System.out.print(A[i] + " "); 


        } 


          


        System.out.println(); 


          


        return 0; 


    } 


      


    // Driver Code 


    public static void main(String[] args) 


    { 


        // Array to be updated 


        int A[] = { 10, 5, 20, 40 }; 


        int n = A.length; 


        // Create and fill difference Array 


        // We use one extra space because 


        // update(l, r, x) updates D[r+1] 


        int D[] = new int[n + 1]; 


        initializeDiffArray(A, D); 


  


        // After below update(l, r, x), the 


        // elements should become 20, 15, 20, 40 


        update(D, 0, 1, 10); 


        printArray(A, D); 


  


        // After below updates, the 


        // array should become 30, 35, 70, 60 


        update(D, 1, 3, 20); 


        update(D, 2, 2, 30); 


          


        printArray(A, D); 


    } 


} 


  


// This code is contributed by Anant Agarwal. 



















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Python3














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# Python3 code to demonstrate Difference Array 


  


# Creates a diff array D[] for A[] and returns 


# it after filling initial values. 


def initializeDiffArray( A): 


    n = len(A) 


  


    # We use one extra space because 


    # update(l, r, x) updates D[r+1] 


    D = [0 for i in range(0 , n + 1)] 


  


    D[0] = A[0]; D[n] = 0


      


    for i in range(1, n ): 


        D[i] = A[i] - A[i - 1] 


    return D 


  


  


# Does range update 


def update(D, l, r, x): 


  


    D[l] += x 


    D[r + 1] -= x 


  


  


# Prints updated Array 


def printArray(A, D): 


  


    for i in range(0 , len(A)): 


        if (i == 0): 


            A[i] = D[i] 


  


        # Note that A[0] or D[0] decides 


        # values of rest of the elements. 


        else: 


            A[i] = D[i] + A[i - 1] 


  


        print(A[i], end = " ") 


      


    print ("") 


  


  


# Driver Code 


A = [ 10, 5, 20, 40 ] 


  


# Create and fill difference Array 


D = initializeDiffArray(A) 


  


# After below update(l, r, x), the 


# elements should become 20, 15, 20, 40 


update(D, 0, 1, 10) 


printArray(A, D) 


  


# After below updates, the 


# array should become 30, 35, 70, 60 


update(D, 1, 3, 20) 


update(D, 2, 2, 30) 


printArray(A, D) 


  


# This code is contributed by Gitanjali. 



















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C#














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// C# code to demonstrate Difference Array 


using System; 


  


class GFG { 


      


    // Creates a diff array D[] for A[] and returns 


    // it after filling initial values. 


    static void initializeDiffArray(int []A, int []D) 


    { 


          


        int n = A.Length; 


  


        D[0] = A[0]; 


        D[n] = 0; 


        for (int i = 1; i < n; i++) 


            D[i] = A[i] - A[i - 1]; 


    } 


  


    // Does range update 


    static void update(int []D, int l, int r, int x) 


    { 


        D[l] += x; 


        D[r + 1] -= x; 


    } 


  


    // Prints updated Array 


    static int printArray(int []A, int []D) 


    { 


        for (int i = 0; i < A.Length; i++) { 


              


            if (i == 0) 


                A[i] = D[i]; 


  


            // Note that A[0] or D[0] decides 


            // values of rest of the elements. 


            else


                A[i] = D[i] + A[i - 1]; 


  


            Console.Write(A[i] + " "); 


        } 


          


        Console.WriteLine(); 


          


        return 0; 


    } 


      


    // Driver Code 


    public static void Main() 


    { 


        // Array to be updated 


        int []A = { 10, 5, 20, 40 }; 


        int n = A.Length; 


        // Create and fill difference Array 


        // We use one extra space because 


        // update(l, r, x) updates D[r+1] 


        int []D = new int[n + 1]; 


        initializeDiffArray(A, D); 


  


        // After below update(l, r, x), the 


        // elements should become 20, 15, 20, 40 


        update(D, 0, 1, 10); 


        printArray(A, D); 


  


        // After below updates, the 


        // array should become 30, 35, 70, 60 


        update(D, 1, 3, 20); 


        update(D, 2, 2, 30); 


          


        printArray(A, D); 


    } 


} 


  


// This code is contributed by vt_m. 



















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Output:


20 15 20 40 
20 35 70 60 





    

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