## Difference Array | Range update query in O(1) tutorials

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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. `

Output:

`20 15 20 40   20 35 70 60 `

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