Maximum number of uncrossed lines between two given arrays

Given two arrays A[] and B[], the task is to find the maximum number of uncrossed lines between the elements of the two given arrays.

A straight line can be drawn between two array elements A[i] and B[j] only if:

  • A[i] = B[j]
  • The line does not intersect any other line.

Examples:

Input: A[] = {3, 9, 2}, B[] = {3, 2, 9} 
Output:
Explanation: 
The lines between A[0] to B[0] and A[1] to B[2] does not intersect each other.

Input: A[] = {1, 2, 3, 4, 5}, B[] = {1, 2, 3, 4, 5} 
Output: 5



Naive Approach: The idea is to generate all the subsequences of array A[] and try to find them in array B[] so that the two subsequences can be connected by joining straight lines. The longest such subsequence found to be common in A[] and B[] would have the maximum number of uncrossed lines. So print the length of that subsequence.

Time Complexity: O(M * 2N
Auxiliary Space: O(1)

Efficient Approach: From the above approach, it can be observed that the task is to find the longest subsequence common in both the arrays. Therefore, the above approach can be optimized by finding the Longest Common Subsequence between the two arrays using Dynamic Programming.

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to count maximum number
// of uncrossed lines between the
// two given arrays
int uncrossedLines(int* a, int* b,
                   int n, int m)
{
    // Stores the length of lcs
    // obtained upto every index
    int dp[n + 1][m + 1];
  
    // Iterate over first array
    for (int i = 0; i <= n; i++) {
  
        // Iterate over second array
        for (int j = 0; j <= m; j++) {
  
            if (i == 0 || j == 0)
  
                // Update value in dp table
                dp[i][j] = 0;
  
            // If both characters
            // are equal
            else if (a[i - 1] == b[j - 1])
  
                // Update the length of lcs
                dp[i][j] = 1 + dp[i - 1][j - 1];
  
            // If both characters
            // are not equal
            else
  
                // Update the table
                dp[i][j] = max(dp[i - 1][j],
                               dp[i][j - 1]);
        }
    }
  
    // Return the answer
    return dp[n][m];
}
  
// Driver Code
int main()
{
    // Given array A[] and B[]
    int A[] = { 3, 9, 2 };
    int B[] = { 3, 2, 9 };
  
    int N = sizeof(A) / sizeof(A[0]);
    int M = sizeof(B) / sizeof(B[0]);
  
    // Function Call
    cout << uncrossedLines(A, B, N, M);
    return 0;
}

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Java

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// Java program for the above approach
import java.io.*;
  
class GFG{
  
// Function to count maximum number
// of uncrossed lines between the
// two given arrays
static int uncrossedLines(int[] a, int[] b,
                          int n, int m)
{
      
    // Stores the length of lcs
    // obtained upto every index
    int[][] dp = new int[n + 1][m + 1];
  
    // Iterate over first array
    for(int i = 0; i <= n; i++) 
    {
          
        // Iterate over second array
        for(int j = 0; j <= m; j++) 
        {
            if (i == 0 || j == 0)
              
                // Update value in dp table
                dp[i][j] = 0;
  
            // If both characters
            // are equal
            else if (a[i - 1] == b[j - 1])
  
                // Update the length of lcs
                dp[i][j] = 1 + dp[i - 1][j - 1];
  
            // If both characters
            // are not equal
            else
  
                // Update the table
                dp[i][j] = Math.max(dp[i - 1][j],
                                    dp[i][j - 1]);
        }
    }
  
    // Return the answer
    return dp[n][m];
}
  
// Driver Code
public static void main (String[] args)
{
      
    // Given array A[] and B[]
    int A[] = { 3, 9, 2 };
    int B[] = { 3, 2, 9 };
  
    int N = A.length;
    int M = B.length;
  
    // Function call
    System.out.print(uncrossedLines(A, B, N, M));
}
}
  
// This code is contributed by code_hunt

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Python3

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# Python3 program for 
# the above approach
  
# Function to count maximum number
# of uncrossed lines between the
# two given arrays
def uncrossedLines(a, b,
                   n, m):
  
    # Stores the length of lcs
    # obtained upto every index
    dp = [[0 for x in range(m + 1)]
             for y in range(n + 1)]
   
    # Iterate over first array
    for i in range (n + 1):
   
        # Iterate over second array
        for j in range (m + 1):
   
            if (i == 0 or j == 0):
   
                # Update value in dp table
                dp[i][j] = 0
   
            # If both characters
            # are equal
            elif (a[i - 1] == b[j - 1]):
   
                # Update the length of lcs
                dp[i][j] = 1 + dp[i - 1][j - 1]
   
            # If both characters
            # are not equal
            else:
   
                # Update the table
                dp[i][j] = max(dp[i - 1][j],
                               dp[i][j - 1])
   
    # Return the answer
    return dp[n][m]
   
# Driver Code
if __name__ == "__main__":
    
    # Given array A[] and B[]
    A = [3, 9, 2]
    B = [3, 2, 9]
   
    N = len(A)
    M = len(B)
   
    # Function Call
    print (uncrossedLines(A, B, N, M))
  
# This code is contributed by Chitranayal

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C#

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// C# program for the above approach
using System;
  
class GFG{
  
// Function to count maximum number
// of uncrossed lines between the
// two given arrays
static int uncrossedLines(int[] a, int[] b,
                          int n, int m)
{
      
    // Stores the length of lcs
    // obtained upto every index
    int[,] dp = new int[n + 1, m + 1];
  
    // Iterate over first array
    for(int i = 0; i <= n; i++)
    {
  
        // Iterate over second array
        for(int j = 0; j <= m; j++) 
        {
            if (i == 0 || j == 0)
  
                // Update value in dp table
                dp[i, j] = 0;
  
            // If both characters
            // are equal
            else if (a[i - 1] == b[j - 1])
  
                // Update the length of lcs
                dp[i, j] = 1 + dp[i - 1, j - 1];
  
            // If both characters
            // are not equal
            else
  
                // Update the table
                dp[i, j] = Math.Max(dp[i - 1, j],
                                    dp[i, j - 1]);
        }
    }
  
    // Return the answer
    return dp[n, m];
}
  
// Driver Code
public static void Main (String[] args)
{
      
    // Given array A[] and B[]
    int[] A = { 3, 9, 2 };
    int[] B = { 3, 2, 9 };
  
    int N = A.Length;
    int M = B.Length;
  
    // Function call
    Console.Write(uncrossedLines(A, B, N, M));
}
}
  
// This code is contributed by code_hunt
}

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Output: 

2

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

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Improved By : chitranayal, code_hunt