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. Fir 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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