Difference Array

Software Development
#1

Hello Everyone,

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

update(l, r, x) : Adds x to all values from A[l] to A[r] (both inclusive).
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++ 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;

}

Output:

20 15 20 40 20 35 70 60