Maximize product of same-indexed elements of same size subsequences

Last Updated : 01 Sep, 2023

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 scalar 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 scalar 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 maximum
int 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 array
int 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 array
int 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 Code
int 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);
}

Java

// Java implementation to maximize
// product of same-indexed elements
// of same size subsequences
class GFG{

static final int INF = 10000000;

// Utility function to find the maximum
static 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 array
static 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 array
static 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 Code
public 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

Python3

# Python3 implementation to maximize 
# product of same-indexed elements 
# of same size subsequences
INF = 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 Code
a = [ -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

// C# implementation to maximize
// product of same-indexed elements
// of same size subsequences
using System;

class GFG{

static readonly int INF = 10000000;

// Utility function to find the maximum
static 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 array
static 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 array
static 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 Code
public 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

JavaScript

<script>
// Javascript implementation to maximize
// product of same-indexed elements
// of same size subsequences

let INF = 10000000;
 
// Utility function to find the maximum
function maximum(A, B, C, 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 array
function maxProductUtil(X, Y, A, B, 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 array
function maxProduct(A, N, B, M)
{
     
    // Initialize a 2-D array
    // for memoization
    let dp =new Array(N);
    for (var i = 0; i < dp.length; i++) {
    dp[i] = new Array(2);
    }
    for(let i = 0; i < N; i++)
    {
       for(let j = 0; j < M; j++)
       {
          dp[i][j] = -1;
       }
    }
     
    return maxProductUtil(N - 1, M - 1,
                          A, B, dp);
}

  // Driver Code
    
    let a = [ -2, 6, -2, -5 ];
    let b = [ -3, 4, -2, 8 ];
 
    let n = a.length;
    let m = b.length;
 
    document.write(maxProduct(a, n, b, m));
             
</script>

Output

Efficient approach: Using DP Tabulation method ( Iterative approach )

The approach to solve this problem is same but DP tabulation(bottom-up) method is better then Dp + memoization(top-down) because memoization method needs extra stack space of recursion calls.

Steps to solve this problem :

Create a DP to store the solution of the subproblems and initialize it with 0.
Initialize the DP with base cases dp[0][0] = A[0] * B[0].
Now Iterate over subproblems to get the value of current problem form previous computation of subproblems stored in DP\
Return the final solution stored in dp[N-1][M-1]

Implementation :

C++

// C++ program for above approach

#include <bits/stdc++.h>
using namespace std;

#define INF 10000000

// Utility function to find the maximum
int 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 array
int maxProduct(int A[], int N, int B[], int M)
{
      // Initialize dp to store 
    // computations of subproblems
    vector<vector<int>> dp(N, vector<int>(M, 0));
    
    // Base case
    dp[0][0] = A[0] * B[0];
    
    // Fill first row
    for (int j = 1; j < M; j++) {
        dp[0][j] = max(A[0] * B[j], dp[0][j-1]);
    }
    
    // Fill first column
    for (int i = 1; i < N; i++) {
        dp[i][0] = max(A[i] * B[0], dp[i-1][0]);
    }
    
    // Fill remaining cells
    for (int i = 1; i < N; i++) {
        for (int j = 1; j < M; j++) {
            dp[i][j] = maximum(A[i] * B[j] + dp[i-1][j-1], A[i] * B[j], dp[i-1][j], dp[i][j-1]);
        }
    }
    
  // return final answer
    return dp[N-1][M-1];
}

// Driver Code
int 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]);
    
      // function call
    cout << maxProduct(a, n, b, m);
}

// this code is contributed by bhardwajji

Java

// Java program for above approach

import java.util.*;

class Main 
{
  
// Utility function to find the maximum
static 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 arrays
static int maxProduct(int[] A, int N, int[] B, int M) 
{
  
    // Initialize dp to store computations of subproblems
    int[][] dp = new int[N][M];

    // Base case
    dp[0][0] = A[0] * B[0];

    // Fill first row
    for (int j = 1; j < M; j++) {
        dp[0][j] = Math.max(A[0] * B[j], dp[0][j - 1]);
    }

    // Fill first column
    for (int i = 1; i < N; i++) {
        dp[i][0] = Math.max(A[i] * B[0], dp[i - 1][0]);
    }

    // Fill remaining cells
    for (int i = 1; i < N; i++) {
        for (int j = 1; j < M; j++) {
            dp[i][j] = maximum(A[i] * B[j] + dp[i - 1][j - 1], A[i] * B[j], dp[i - 1][j], dp[i][j - 1]);
        }
    }

    // return final answer
    return dp[N - 1][M - 1];
}

// Driver Code
public 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;

    // function call
    System.out.println(maxProduct(a, n, b, m));
}
}

Python3

def maximum(A, B, C, D):
    return max(max(A, B), max(C, D))

def maxProduct(A, N, B, M):
    # Initialize dp to store computations of subproblems
    dp = [[0] * M for _ in range(N)]

    # Base case
    dp[0][0] = A[0] * B[0]

    # Fill first row
    for j in range(1, M):
        dp[0][j] = max(A[0] * B[j], dp[0][j-1])

    # Fill first column
    for i in range(1, N):
        dp[i][0] = max(A[i] * B[0], dp[i-1][0])

    # Fill remaining cells
    for i in range(1, N):
        for j in range(1, M):
            dp[i][j] = maximum(A[i] * B[j] + dp[i-1][j-1], A[i] * B[j], dp[i-1][j], dp[i][j-1])

    # Return final answer
    return dp[N-1][M-1]

# Driver Code
a = [-2, 6, -2, -5]
b = [-3, 4, -2, 8]

n = len(a)
m = len(b)

# Function call
print(maxProduct(a, n, b, m))

using System;

class MainClass {
    // Utility function to find the maximum
    static 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 arrays
    static int MaxProduct(int[] A, int N, int[] B, int M)
    {
        // Initialize dp to store computations of
        // subproblems
        int[, ] dp = new int[N, M];

        // Base case
        dp[0, 0] = A[0] * B[0];

        // Fill first row
        for (int j = 1; j < M; j++) {
            dp[0, j] = Math.Max(A[0] * B[j], dp[0, j - 1]);
        }

        // Fill first column
        for (int i = 1; i < N; i++) {
            dp[i, 0] = Math.Max(A[i] * B[0], dp[i - 1, 0]);
        }

        // Fill remaining cells
        for (int i = 1; i < N; i++) {
            for (int j = 1; j < M; j++) {
                dp[i, j] = Maximum(
                    A[i] * B[j] + dp[i - 1, j - 1],
                    A[i] * B[j], dp[i - 1, j],
                    dp[i, j - 1]);
            }
        }

        // return final answer
        return dp[N - 1, M - 1];
    }

    // Driver Code
    public 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;

        // function call
        Console.WriteLine(MaxProduct(a, n, b, m));
    }
}

// The code is contributed by Samim Hossain Mondal.

JavaScript

//Javascript code for this approach

// Utility function to find the maximum

function maximum(A, B, C, 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 arrays
function maxProduct(A, N, B, M) {
  // Initialize dp to store computations of subproblems
  let dp = new Array(N).fill().map(() => new Array(M).fill(0));
  
  // Base case
  dp[0][0] = A[0] * B[0];
  
  // Fill first row
  for (let j = 1; j < M; j++) {
    dp[0][j] = Math.max(A[0] * B[j], dp[0][j-1]);
  }
  
  // Fill first column
  for (let i = 1; i < N; i++) {
    dp[i][0] = Math.max(A[i] * B[0], dp[i-1][0]);
  }
  
  // Fill remaining cells
  for (let i = 1; i < N; i++) {
    for (let j = 1; j < M; j++) {
      dp[i][j] = maximum(A[i] * B[j] + dp[i-1][j-1], A[i] * B[j], dp[i-1][j], dp[i][j-1]);
    }
  }
  
  // return final answer
  return dp[N-1][M-1];
}

// Driver Code
let a = [ -2, 6, -2, -5 ];
let b = [ -3, 4, -2, 8 ];

let n = a.length;
let m = b.length;

// function call
console.log(maxProduct(a, n, b, m));

// This code is contributed by Sundaram

Output

Time complexity: O(N*M)
Auxiliary Space: O(N*M)

Maximize product of same-indexed elements of same size subsequences

rupesh_rao

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