本文概述
考虑一个整数数组A[]并遵循以下两种查询。
- update(l, r, x)：将x添加到从A[l]到A[r]的所有值(包括两者)。
- printArray()：打印当前修改后的数组。
例子 ：
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}.一种简单的解决方案要执行以下操作：
- update(l, r, x)：从l到r运行循环, 并将x添加到从A[l]到A[r]的所有元素中
- printArray()：仅打印A[]。
以上两个操作的时间复杂度为O(n)
一个有效的解决方案是使用差异数组。
差异数组
给定数组A[i]的D [i]定义为D [i] = A[i] -A[i-1](对于0 <i <N)和D [0] = A[0]基于0的索引。差异数组可用于执行范围更新查询” l r x”, 其中l是左索引, r是右索引, x是要添加的值, 在所有查询之后, 你都可以从中返回原始数组。可以以O(1)复杂度执行更新范围操作的地方。
- update(l, r, x)：将x添加到D [l]并将其从D [r + 1]中减去, 即, 我们做D [l] + = x, D [r + 1]-= x
- printArray()：执行A[0] = D [0]并打印。对于其余元素, 执行A[i] = A[i-1] + D [i]并打印它们。
此处更新的时间复杂度提高到O(1)。请注意, printArray()仍需要O(n)时间。
C ++
//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;
}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[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.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[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.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[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.输出如下：
20 15 20 40
20 35 70 60来源：
https://www.srcmini02.com/69113.html
