本文概述
给定两个大小为m和n的正整数数组, 其中m> n。我们需要通过在第二个数组中插入零来最大化点积, 但是我们不能打扰元素的顺序。
例子:
Input : A[] = {2, 3 , 1, 7, 8}
B[] = {3, 6, 7}
Output : 107
Explanation : We get maximum dot product after
inserting 0 at first and third positions in
second array.
Maximum Dot Product : = A[i] * B[j]
2*0 + 3*3 + 1*0 + 7*6 + 8*7 = 107
Input : A[] = {1, 2, 3, 6, 1, 4}
B[] = {4, 5, 1}
Output : 46
询问:Directi面试
推荐:请在"实践首先, 在继续解决方案之前。
解决此问题的另一种方法是, 对于每对元素a [i]和B [j], 其中j> = i, 我们有两种选择:
- 我们将A [i]和B [j]相乘并加到乘积上(我们包括A [i])。
- 我们从乘积中排除A [i](换句话说, 我们在B []的当前位置插入0)
这个想法是使用动态编程.
1) Given Array A[] of size 'm' and B[] of size 'n'
2) Create 2D matrix 'DP[n + 1][m + 1]' initialize it
with '0'
3) Run loop outer loop for i = 1 to n
Inner loop j = i to m
// Two cases arise
// 1) Include A[j]
// 2) Exclude A[j] (insert 0 in B[])
dp[i][j] = max(dp[i-1][j-1] + A[j-1] * B[i -1], dp[i][j-1])
// Last return maximum dot product that is
return dp[n][m]
以下是上述想法的实现。
C ++
// C++ program to find maximum dot product of two array
#include<bits/stdc++.h>
using namespace std;
// Function compute Maximum Dot Product and
// return it
long long int MaxDotProduct( int A[], int B[], int m, int n)
{
// Create 2D Matrix that stores dot product
// dp[i+1][j+1] stores product considering B[0..i]
// and A[0...j]. Note that since all m > n, we fill
// values in upper diagonal of dp[][]
long long int dp[n+1][m+1];
memset (dp, 0, sizeof (dp));
// Traverse through all elements of B[]
for ( int i=1; i<=n; i++)
// Consider all values of A[] with indexes greater
// than or equal to i and compute dp[i][j]
for ( int j=i; j<=m; j++)
// Two cases arise
// 1) Include A[j]
// 2) Exclude A[j] (insert 0 in B[])
dp[i][j] = max((dp[i-1][j-1] + (A[j-1]*B[i-1])) , dp[i][j-1]);
// return Maximum Dot Product
return dp[n][m] ;
}
// Driver program to test above function
int main()
{
int A[] = { 2, 3 , 1, 7, 8 } ;
int B[] = { 3, 6, 7 } ;
int m = sizeof (A)/ sizeof (A[0]);
int n = sizeof (B)/ sizeof (B[0]);
cout << MaxDotProduct(A, B, m, n);
return 0;
}
Java
// Java program to find maximum
// dot product of two array
import java.util.*;
class GFG
{
// Function to compute Maximum
// Dot Product and return it
static int MaxDotProduct( int A[], int B[], int m, int n)
{
// Create 2D Matrix that stores dot product
// dp[i+1][j+1] stores product considering B[0..i]
// and A[0...j]. Note that since all m > n, we fill
// values in upper diagonal of dp[][]
int dp[][] = new int [n + 1 ][m + 1 ];
for ( int [] row : dp)
Arrays.fill(row, 0 );
// Traverse through all elements of B[]
for ( int i = 1 ; i <= n; i++)
// Consider all values of A[] with indexes greater
// than or equal to i and compute dp[i][j]
for ( int j = i; j <= m; j++)
// Two cases arise
// 1) Include A[j]
// 2) Exclude A[j] (insert 0 in B[])
dp[i][j] =
Math.max((dp[i - 1 ][j - 1 ] +
(A[j - 1 ] * B[i - 1 ])), dp[i][j - 1 ]);
// return Maximum Dot Product
return dp[n][m];
}
// Driver code
public static void main(String[] args) {
int A[] = { 2 , 3 , 1 , 7 , 8 };
int B[] = { 3 , 6 , 7 };
int m = A.length;
int n = B.length;
System.out.print(MaxDotProduct(A, B, m, n));
}
}
// This code is contributed by Anant Agarwal.
Python3
# Python 3 program to find maximum dot
# product of two array
# Function compute Maximum Dot Product
# and return it
def MaxDotProduct(A, B, m, n):
# Create 2D Matrix that stores dot product
# dp[i+1][j+1] stores product considering
# B[0..i] and A[0...j]. Note that since
# all m > n, we fill values in upper
# diagonal of dp[][]
dp = [[ 0 for i in range (m + 1 )]
for j in range (n + 1 )]
# Traverse through all elements of B[]
for i in range ( 1 , n + 1 , 1 ):
# Consider all values of A[] with indexes
# greater than or equal to i and compute
# dp[i][j]
for j in range (i, m + 1 , 1 ):
# Two cases arise
# 1) Include A[j]
# 2) Exclude A[j] (insert 0 in B[])
dp[i][j] = max ((dp[i - 1 ][j - 1 ] +
(A[j - 1 ] * B[i - 1 ])) , dp[i][j - 1 ])
# return Maximum Dot Product
return dp[n][m]
# Driver Code
if __name__ = = '__main__' :
A = [ 2 , 3 , 1 , 7 , 8 ]
B = [ 3 , 6 , 7 ]
m = len (A)
n = len (B)
print (MaxDotProduct(A, B, m, n))
# This code is contributed by
# Sanjit_Prasad
C#
// C# program to find maximum
// dot product of two array
using System;
public class GFG{
// Function to compute Maximum
// Dot Product and return it
static int MaxDotProduct( int []A, int []B, int m, int n)
{
// Create 2D Matrix that stores dot product
// dp[i+1][j+1] stores product considering B[0..i]
// and A[0...j]. Note that since all m > n, we fill
// values in upper diagonal of dp[][]
int [, ]dp = new int [n + 1, m + 1];
// Traverse through all elements of B[]
for ( int i = 1; i <= n; i++)
// Consider all values of A[] with indexes greater
// than or equal to i and compute dp[i][j]
for ( int j = i; j <= m; j++)
// Two cases arise
// 1) Include A[j]
// 2) Exclude A[j] (insert 0 in B[])
dp[i, j] =
Math.Max((dp[i - 1, j - 1] +
(A[j - 1] * B[i - 1])), dp[i, j - 1]);
// return Maximum Dot Product
return dp[n, m];
}
// Driver code
public static void Main() {
int []A = {2, 3, 1, 7, 8};
int []B = {3, 6, 7};
int m = A.Length;
int n = B.Length;
Console.Write(MaxDotProduct(A, B, m, n));
}
}
/*This code is contributed by 29AjayKumar*/
输出如下:
107
时间复杂度:氧(nm)
如果发现任何不正确的地方, 或者想分享有关上述主题的更多信息, 请写评论。