Longest subsequence from an array of pairs having first element increasing and second element decreasing.

Given an array of pairs A[][] of size N, the task is to find the longest subsequences where the first element is increasing and the second element is decreasing.

Examples:

Input: A[]={{1, 2}, {2, 2}, {3, 1}}, N = 3
Output: 2
Explanation: The longest subsequence satisfying the conditions is of length 2 and consists of {1, 2} and {3, 1};

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

Naive Approach: The simplest approach is to use Recursion. For every pair in the array, there are two possible choices, i.e. either to include the current pair in the subsequence or not. Therefore, iterate over the array recursively and find the required longest subsequence.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Recursive function to find the length of
// the longest subsequence of pairs whose first
// element is increasing and second is decreasing
int longestSubSequence(pair<int, int> A[], int N,
                       int ind = 0,
                       int lastf = INT_MIN,
                       int lasts = INT_MAX)
{

    // Base case
    if (ind == N)
        return 0;

    // Not include the current pair
    // in the longest subsequence
    int ans = longestSubSequence(A, N, ind + 1,
                                 lastf, lasts);

    // Including the current pair
    // in the longest subsequence
    if (A[ind].first > lastf
        && A[ind].second < lasts)

        ans = max(ans, longestSubSequence(A, N, ind + 1,
                                          A[ind].first,
                                          A[ind].second)
                           + 1);

    return ans;
}

// Driver Code
int main()
{
    // Given Input
    pair<int, int> A[] = { { 1, 2 },
                           { 2, 2 },
                           { 3, 1 } };

    int N = sizeof(A) / sizeof(A[0]);

    // Function Call
    cout << longestSubSequence(A, N) << "\n";

    return 0;
}

Java

// Java program for the above approach
import java.io.*;

class GFG{

// Recursive function to find the length of
// the longest subsequence of pairs whose first
// element is increasing and second is decreasing
public static Integer longestSubSequence(int[][] A, int N, int ind,
                                         int lastf, int lasts)
{
    ind = (ind > 0 ? ind : 0);
    lastf = (lastf > 0 ? lastf: Integer.MIN_VALUE);
    lasts = (lasts > 0 ? lasts: Integer.MAX_VALUE);

    // Base case
    if (ind == N)
        return 0;

    // Not include the current pair
    // in the longest subsequence
    int ans = longestSubSequence(A, N, ind + 1,
                                 lastf, lasts);

    // Including the current pair
    // in the longest subsequence
    if (A[ind][0] > lastf && A[ind][1] < lasts)

        ans = Math.max(ans, longestSubSequence(A, N, ind + 1,
                                   A[ind][0], A[ind][1]) + 1);

    return ans;
}

public static int longestSubSequence(int[][] A, int N)
{
    return longestSubSequence(A, N, 0, 0, 0);
}

// Driver Code
public static void main(String args[])
{

    // Given Input
    int[][] A = { { 1, 2 }, { 2, 2 }, { 3, 1 } };
    int N = A.length;

    // Function Call
    System.out.println(longestSubSequence(A, N));
}
}

// This code is contributed by _saurabh_jaiswal

Python3

# Python 3 program for the above approach
import sys

# Recursive function to find the length of
# the longest subsequence of pairs whose first
# element is increasing and second is decreasing
def longestSubSequence(A,  N,
                       ind=0,
                       lastf=-sys.maxsize-1,
                       lasts=sys.maxsize):

    # Base case
    if (ind == N):
        return 0

    # Not include the current pair
    # in the longest subsequence
    ans = longestSubSequence(A, N, ind + 1,
                             lastf, lasts)

    # Including the current pair
    # in the longest subsequence
    if (A[ind][0] > lastf
            and A[ind][1] < lasts):

        ans = max(ans, longestSubSequence(A, N, ind + 1,
                                          A[ind][0],
                                          A[ind][1])
                  + 1)

    return ans


# Driver Code
if __name__ == "__main__":

    # Given Input
    A = [[1, 2],
         [2, 2],
         [3, 1]]

    N = len(A)

    # Function Call
    print(longestSubSequence(A, N))

    # This code is contributed by ukasp.

C#

// C# program for the above approach
using System;

class GFG{

// Recursive function to find the length of
// the longest subsequence of pairs whose first
// element is increasing and second is decreasing
public static int longestSubSequence(int[,] A, int N, int ind,
                                         int lastf, int lasts)
{
    ind = (ind > 0 ? ind : 0);
    lastf = (lastf > 0 ? lastf: Int32.MinValue);
    lasts = (lasts > 0 ? lasts: Int32.MaxValue);

    // Base case
    if (ind == N)
        return 0;

    // Not include the current pair
    // in the longest subsequence
    int ans = longestSubSequence(A, N, ind + 1,
                                 lastf, lasts);

    // Including the current pair
    // in the longest subsequence
    if (A[ind, 0] > lastf && A[ind, 1] < lasts)

        ans = Math.Max(ans, longestSubSequence(A, N, ind + 1,
                                   A[ind, 0], A[ind, 1]) + 1);

    return ans;
}

public static int longestSubSequence(int[,] A, int N)
{
    return longestSubSequence(A, N, 0, 0, 0);
}

// Driver Code
public static void Main()
{
    // Given Input
    int[,] A = { { 1, 2 }, { 2, 2 }, { 3, 1 } };
    int N = A.GetLength(0);

    // Function Call
    Console.Write(longestSubSequence(A, N));
}
}

// This code is contributed by target_2.

Javascript

<script>

// JavaScript program for the above approach


// Function to find the length of the
// longest subsequence of pairs whose first
// element is increasing and second is decreasing
function longestSubSequence(A, N)
{
    // dp[i]: Stores the longest
    // subsequence upto i
    let dp = new Array(N);
    for (let i = 0; i < N; i++) {

        // Base case
        dp[i] = 1;

        for (let j = 0; j < i; j++) {

            // When the conditions hold
            if (A[j][0] < A[i][0]
                && A[j][1] > A[i][1]) {

                dp[i] = Math.max(dp[i], dp[j] + 1);
            }
        }
    }

    // Finally, print the required answer
    document.write(dp[N - 1] + "<br>");
}

// Driver Code

    // Given Input
    let A = [ [ 1, 2 ],
            [ 2, 2 ],
            [ 3, 1 ] ];

    let N = A.length;

    // Function Call
    longestSubSequence(A, N);

</script>

Output

2

Time Complexity: O(2^N)
Auxiliary Space: O(1)

Efficient Approach: This problem has Overlapping Subproblems property and Optimal Substructure property. Therefore, this problem can be solved using Dynamic Programming. Like other typical Dynamic Programming (DP) problems, recomputation of same subproblems can be avoided by constructing a temporary array that stores the results of the subproblems. 

Follow the steps below to solve this problem: 

• Initialize a dp[] array, where dp[i] stores the length of the longest subsequence that can be formed using elements up to index i.
• Iterate over the range [0, N-1] using variable i:
  • Base case: Update dp[i] as 1.
  • Iterate over the range [0, i – 1] using a variable j:
    • If A[j].first is less than A[i].first and A[j].second is greater than A[i].second, then update dp[i] as maximum of dp[i] and dp[j] + 1.
  • Finally, print dp[N-1].

Below is the implementation of the above approach: 

C++

// C++ program for the above approach

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

// Function to find the length of the
// longest subsequence of pairs whose first
// element is increasing and second is decreasing
void longestSubSequence(pair<int, int> A[], int N)
{
    // dp[i]: Stores the longest
    // subsequence upto i
    int dp[N];
    for (int i = 0; i < N; i++) {

        // Base case
        dp[i] = 1;

        for (int j = 0; j < i; j++) {

            // When the conditions hold
            if (A[j].first < A[i].first
                && A[j].second > A[i].second) {

                dp[i] = max(dp[i], dp[j] + 1);
            }
        }
    }

    // Finally, print the required answer
    cout << dp[N - 1] << endl;
}

// Driver Code
int main()
{
    // Given Input
    pair<int, int> A[] = { { 1, 2 },
                           { 2, 2 },
                           { 3, 1 } };

    int N = sizeof(A) / sizeof(A[0]);

    // Function Call
    longestSubSequence(A, N);

    return 0;
}

Java

// Java program for the above approach
import java.io.*;

class GFG{

// Function to find the length of the
// longest subsequence of pairs whose first
// element is increasing and second is decreasing
public static void longestSubSequence(int[][] A, int N)
{

    // dp[i]: Stores the longest
    // subsequence upto i
    int[] dp = new int[N];

    for(int i = 0; i < N; i++)
    {

        // Base case
        dp[i] = 1;

        for(int j = 0; j < i; j++)
        {

            // When the conditions hold
            if (A[j][0] < A[i][0] && A[j][1] > A[i][1])
            {
                dp[i] = Math.max(dp[i], dp[j] + 1);
            }
        }
    }

    // Finally, print the required answer
    System.out.println(dp[N - 1]);
}

// Driver Code
public static void main(String args[])
{

    // Given Input
    int[][] A = { { 1, 2 },
                  { 2, 2 },
                  { 3, 1 } };

    int N = A.length;

    // Function Call
    longestSubSequence(A, N);
}
}

// This code is contributed by gfgking

Python3

# Python3 program for the above approach

# Function to find the length of the
# longest subsequence of pairs whose first
# element is increasing and second is decreasing
def longestSubSequence(A, N):

    # dp[i]: Stores the longest
    # subsequence upto i
    dp = [0]*N
    for i in range(N):

        # Base case
        dp[i] = 1

        for j in range(i):

            # When the conditions hold
            if (A[j][0] < A[i][0] and A[j][1] > A[i][1]):
                dp[i] = max(dp[i], dp[j] + 1)


    # Finally, print the required answer
    print (dp[N - 1])

# Driver Code
if __name__ == '__main__':

    #Given Input
    A = [ [ 1, 2 ],
           [ 2, 2 ],
           [ 3, 1 ] ]

    N = len(A)

    #Function Call
    longestSubSequence(A, N)

# This code is contributed by mohit kumar 29.

C#

// C# program for the above approach
using System;
class GFG {

    // Function to find the length of the
    // longest subsequence of pairs whose first
    // element is increasing and second is decreasing
    static void longestSubSequence(int[,] A, int N)
    {

        // dp[i]: Stores the longest
        // subsequence upto i
        int[] dp = new int[N];

        for(int i = 0; i < N; i++)
        {

            // Base case
            dp[i] = 1;

            for(int j = 0; j < i; j++)
            {

                // When the conditions hold
                if (A[j,0] < A[i,0] && A[j,1] > A[i,1])
                {
                    dp[i] = Math.Max(dp[i], dp[j] + 1);
                }
            }
        }

        // Finally, print the required answer
        Console.Write(dp[N - 1]);
    }

  static void Main()
  {

    // Given Input
    int[,] A = { { 1, 2 },
                  { 2, 2 },
                  { 3, 1 } };

    int N = A.GetLength(0);

    // Function Call
    longestSubSequence(A, N);
  }
}

// This code is contributed by decode2207.

Javascript

<script>

// JavaScript program for the above approach


// Function to find the length of the
// longest subsequence of pairs whose first
// element is increasing and second is decreasing
function longestSubSequence(A, N) {
    // dp[i]: Stores the longest
    // subsequence upto i
    let dp = new Array(N);
    for (let i = 0; i < N; i++) {

        // Base case
        dp[i] = 1;

        for (let j = 0; j < i; j++) {

            // When the conditions hold
            if (A[j][0] < A[i][0]
                && A[j][1] > A[i][1]) {

                dp[i] = Math.max(dp[i], dp[j] + 1);
            }
        }
    }

    // Finally, print the required answer
    document.write(dp[N - 1] + "<br>");
}

// Driver Code

// Given Input
let A = [[1, 2],
[2, 2],
[3, 1]];

let N = A.length;

// Function Call
longestSubSequence(A, N);

</script>

Output

2

Time Complexity: O(N²)
Auxiliary Space: O(N)