Maximize product of same-indexed elements of same size subsequences
Given two integer arrays a[] and b[], the task is to find the maximum possible product of the same indexed elements of two equal length subsequences from the two given arrays.
Examples:
Input: a[] = {-3, 1, -12, 7}, b[] = {3, 2, -6, 7}
Output: 124
Explanation: The subsequence [1, -12, 7] from a[] and subsequence [3, -6, 7] from b[] gives the maximum scaler product, (1*3 + (-12)*(-6) + 7*7) = 124.Input: a[] = {-2, 6, -2, -5}, b[] = {-3, 4, -2, 8}
Output: 54
Explanation: The subsequence [-2, 6] from a[] and subsequence [-3, 8] from b[] gives the maximum scaler product, ((-2)*(-3) + 6*8) = 54.
Approach: The problem is similar to the Longest Common Subsequence problem of Dynamic Programming. A Recursive and Memoization based Top-Down method is explained below:
- Let us define a function F(X, Y), which returns the maximum scalar product between the arrays a[0-X] and b[0-Y].
- Following is the recursive definition:
F(X, Y) = max(a[X] * b[Y] + F(X – 1, Y – 1); Use the last elements from both arrays and check further
a[X] * b[Y]; Only use the last element as the maximum product
F(X – 1, Y); Ignore the last element from the first array
F(X, Y – 1)); Ignore the last element from the second element
- Use a 2-D array for memoization and to avoid recomputation of same sub-problems.
Below is the implementation of the above approach:
C++
// C++ implementation to maximize// product of same-indexed elements// of same size subsequences#include <bits/stdc++.h>using namespace std;#define INF 10000000// Utility function to find the maximumint maximum(int A, int B, int C, int D){ return max(max(A, B), max(C, D));}// Utility function to find// the maximum scalar product// from the equal length sub-sequences// taken from the two given arrayint maxProductUtil( int X, int Y, int* A, int* B, vector<vector<int> >& dp){ // Return a very small number // if index is invalid if (X < 0 or Y < 0) return -INF; // If the sub-problem is already // evaluated, then return it if (dp[X][Y] != -1) return dp[X][Y]; // Take the maximum of all // the recursive cases dp[X][Y] = maximum( A[X] * B[Y] + maxProductUtil( X - 1, Y - 1, A, B, dp), A[X] * B[Y], maxProductUtil( X - 1, Y, A, B, dp), maxProductUtil( X, Y - 1, A, B, dp)); return dp[X][Y];}// Function to find maximum scalar// product from same size sub-sequences// taken from the two given arrayint maxProduct(int A[], int N, int B[], int M){ // Initialize a 2-D array // for memoization vector<vector<int> > dp(N, vector<int>(M, -1)); return maxProductUtil( N - 1, M - 1, A, B, dp);}// Driver Codeint main(){ int a[] = { -2, 6, -2, -5 }; int b[] = { -3, 4, -2, 8 }; int n = sizeof(a) / sizeof(a[0]); int m = sizeof(b) / sizeof(b[0]); cout << maxProduct(a, n, b, m);} |
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Java
// Java implementation to maximize// product of same-indexed elements// of same size subsequencesclass GFG{static final int INF = 10000000;// Utility function to find the maximumstatic int maximum(int A, int B, int C, int D){ return Math.max(Math.max(A, B), Math.max(C, D));}// Utility function to find the // maximum scalar product from // the equal length sub-sequences// taken from the two given arraystatic int maxProductUtil(int X, int Y, int []A, int[] B, int [][]dp){ // Return a very small number // if index is invalid if (X < 0 || Y < 0) return -INF; // If the sub-problem is already // evaluated, then return it if (dp[X][Y] != -1) return dp[X][Y]; // Take the maximum of all // the recursive cases dp[X][Y] = maximum(A[X] * B[Y] + maxProductUtil(X - 1, Y - 1, A, B, dp), A[X] * B[Y], maxProductUtil(X - 1, Y, A, B, dp), maxProductUtil(X, Y - 1, A, B, dp)); return dp[X][Y];}// Function to find maximum scalar// product from same size sub-sequences// taken from the two given arraystatic int maxProduct(int A[], int N, int B[], int M){ // Initialize a 2-D array // for memoization int [][]dp = new int[N][M]; for(int i = 0; i < N; i++) { for(int j = 0; j < M; j++) { dp[i][j] = -1; } } return maxProductUtil(N - 1, M - 1, A, B, dp);}// Driver Codepublic static void main(String[] args){ int a[] = { -2, 6, -2, -5 }; int b[] = { -3, 4, -2, 8 }; int n = a.length; int m = b.length; System.out.print(maxProduct(a, n, b, m));}}// This code is contributed by Amal Kumar Choubey |
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Python3
# Python3 implementation to maximize # product of same-indexed elements # of same size subsequencesINF = 10000000# Utility function to find the maximum def maximum(A, B, C, D): return max(max(A, B), max(C, D))# Utility function to find # the maximum scalar product # from the equal length sub-sequences # taken from the two given array def maxProductUtil(X, Y, A, B, dp): # Return a very small number # if index is invalid if (X < 0 or Y < 0): return -INF # If the sub-problem is already # evaluated, then return it if (dp[X][Y] != -1): return dp[X][Y] # Take the maximum of all # the recursive cases dp[X][Y]= maximum(A[X] * B[Y] + maxProductUtil(X - 1, Y - 1, A, B, dp), A[X] * B[Y], maxProductUtil(X - 1, Y, A, B, dp), maxProductUtil(X, Y - 1, A, B, dp)) return dp[X][Y]# Function to find maximum scalar # product from same size sub-sequences # taken from the two given array def maxProduct(A, N, B, M): # Initialize a 2-D array # for memoization dp = [[-1 for i in range(m)] for i in range(n)] return maxProductUtil(N - 1, M - 1, A, B, dp)# Driver Codea = [ -2, 6, -2, -5 ]b = [ -3, 4, -2, 8 ]n = len(a)m = len(b)print(maxProduct(a, n, b, m))# This code is contributed by avanitrachhadiya2155 |
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C#
// C# implementation to maximize// product of same-indexed elements// of same size subsequencesusing System;class GFG{static readonly int INF = 10000000;// Utility function to find the maximumstatic int maximum(int A, int B, int C, int D){ return Math.Max(Math.Max(A, B), Math.Max(C, D));}// Utility function to find the // maximum scalar product from // the equal length sub-sequences// taken from the two given arraystatic int maxProductUtil(int X, int Y, int []A, int[] B, int [,]dp){ // Return a very small number // if index is invalid if (X < 0 || Y < 0) return -INF; // If the sub-problem is already // evaluated, then return it if (dp[X, Y] != -1) return dp[X, Y]; // Take the maximum of all // the recursive cases dp[X, Y] = maximum(A[X] * B[Y] + maxProductUtil(X - 1, Y - 1, A, B, dp), A[X] * B[Y], maxProductUtil(X - 1, Y, A, B, dp), maxProductUtil(X, Y - 1, A, B, dp)); return dp[X, Y];}// Function to find maximum scalar// product from same size sub-sequences// taken from the two given arraystatic int maxProduct(int []A, int N, int []B, int M){ // Initialize a 2-D array // for memoization int [,]dp = new int[N, M]; for(int i = 0; i < N; i++) { for(int j = 0; j < M; j++) { dp[i, j] = -1; } } return maxProductUtil(N - 1, M - 1, A, B, dp);}// Driver Codepublic static void Main(String[] args){ int []a = { -2, 6, -2, -5 }; int []b = { -3, 4, -2, 8 }; int n = a.Length; int m = b.Length; Console.Write(maxProduct(a, n, b, m));}}// This code is contributed by amal kumar choubey |
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Output:
54
