算法-差异数组（O(1)时间范围更新查询） _数据结构

本文概述

C ++
Java
Python3
C#

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。

【算法-差异数组（O(1)时间范围更新查询）】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] = 0for 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# 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：