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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 and A. After update,
A[] becomes {20, 15, 20, 40}
Query update(1, 3, 20) adds 20 to A,
A and A. After update, A[] becomes
{20, 35, 40, 60}.
Query update(2, 2, 30) adds 30 to A.
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 = A 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 = D 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++

 `// C++ code to demonstrate Difference Array``#include ``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 = A, 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 or D 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;``}`

## Java

 `// 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 or D 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.`

## Python3

 `# 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 or D 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.`

## C#

 `// 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 = A;``        ``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 or D 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.`

## Javascript

 ``

Output:

```20 15 20 40
20 35 70 60 ```

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