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 :
- update(l, r, x) : Run a loop from l to r and add x to all elements from A[l] to A[r]
- 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.
- 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
- 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++
#include <bits/stdc++.h>
using namespace std;
vector< int > initializeDiffArray(vector< int >& A)
{
int n = A.size();
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;
}
void update(vector< int >& D, int l, int r, int x)
{
D[l] += x;
D[r + 1] -= x;
}
int printArray(vector< int >& A, vector< int >& D)
{
for ( int i = 0; i < A.size(); i++) {
if (i == 0)
A[i] = D[i];
else
A[i] = D[i] + A[i - 1];
cout << A[i] << " " ;
}
cout << endl;
}
int main()
{
vector< int > A{ 10, 5, 20, 40 };
vector< int > D = initializeDiffArray(A);
update(D, 0, 1, 10);
printArray(A, D);
update(D, 1, 3, 20);
update(D, 2, 2, 30);
printArray(A, D);
return 0;
}
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Java
class GFG {
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 ];
}
static void update( int D[], int l, int r, int x)
{
D[l] += x;
D[r + 1 ] -= x;
}
static int printArray( int A[], int D[])
{
for ( int i = 0 ; i < A.length; i++) {
if (i == 0 )
A[i] = D[i];
else
A[i] = D[i] + A[i - 1 ];
System.out.print(A[i] + " " );
}
System.out.println();
return 0 ;
}
public static void main(String[] args)
{
int A[] = { 10 , 5 , 20 , 40 };
int n = A.length;
int D[] = new int [n + 1 ];
initializeDiffArray(A, D);
update(D, 0 , 1 , 10 );
printArray(A, D);
update(D, 1 , 3 , 20 );
update(D, 2 , 2 , 30 );
printArray(A, D);
}
}
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Python3
def initializeDiffArray( A):
n = len (A)
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
def update(D, l, r, x):
D[l] + = x
D[r + 1 ] - = x
def printArray(A, D):
for i in range ( 0 , len (A)):
if (i = = 0 ):
A[i] = D[i]
else :
A[i] = D[i] + A[i - 1 ]
print (A[i], end = " " )
print ("")
A = [ 10 , 5 , 20 , 40 ]
D = initializeDiffArray(A)
update(D, 0 , 1 , 10 )
printArray(A, D)
update(D, 1 , 3 , 20 )
update(D, 2 , 2 , 30 )
printArray(A, D)
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C#
using System;
class GFG {
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];
}
static void update( int []D, int l, int r, int x)
{
D[l] += x;
D[r + 1] -= x;
}
static int printArray( int []A, int []D)
{
for ( int i = 0; i < A.Length; i++) {
if (i == 0)
A[i] = D[i];
else
A[i] = D[i] + A[i - 1];
Console.Write(A[i] + " " );
}
Console.WriteLine();
return 0;
}
public static void Main()
{
int []A = { 10, 5, 20, 40 };
int n = A.Length;
int []D = new int [n + 1];
initializeDiffArray(A, D);
update(D, 0, 1, 10);
printArray(A, D);
update(D, 1, 3, 20);
update(D, 2, 2, 30);
printArray(A, D);
}
}
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Javascript
<script>
function initializeDiffArray( A){
let n = A.length;
let D= [];
D[0] = A[0], D[n] = 0;
for (let i = 1; i < n; i++)
D[i] = A[i] - A[i - 1];
return D;
}
function update(D, l, r, x){
D[l] += x;
D[r + 1] -= x;
return D;
}
function printArray( A, D){
for (let i = 0; i < A.length; i++) {
if (i == 0)
A[i] = D[i];
else
A[i] = D[i] + A[i - 1];
document.write( A[i]+ " " );
}
document.write( "<br>" );
}
let A = [ 10, 5, 20, 40 ];
let D = initializeDiffArray(A);
D = update(D, 0, 1, 10);
printArray(A, D);
D = update(D, 1, 3, 20);
D = update(D, 2, 2, 30);
printArray(A, D);
</script>
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Output:
20 15 20 40
20 35 70 60
Time complexity: O(n)
Auxiliary Space: O(n)