Dynamic Programming | Set 20 (Maximum Length Chain of Pairs)

2.5

You are given n pairs of numbers. In every pair, the first number is always smaller than the second number. A pair (c, d) can follow another pair (a, b) if b < c. Chain of pairs can be formed in this fashion. Find the longest chain which can be formed from a given set of pairs. Source: Amazon Interview | Set 2

For example, if the given pairs are {{5, 24}, {39, 60}, {15, 28}, {27, 40}, {50, 90} }, then the longest chain that can be formed is of length 3, and the chain is {{5, 24}, {27, 40}, {50, 90}}

This problem is a variation of standard Longest Increasing Subsequence problem. Following is a simple two step process.
1) Sort given pairs in increasing order of first (or smaller) element.
2) Now run a modified LIS process where we compare the second element of already finalized LIS with the first element of new LIS being constructed.

The following code is a slight modification of method 2 of this post.

C

#include<stdio.h>
#include<stdlib.h>

// Structure for a pair
struct pair
{
  int a;
  int b;
};

// This function assumes that arr[] is sorted in increasing order
// according the first (or smaller) values in pairs.
int maxChainLength( struct pair arr[], int n)
{
   int i, j, max = 0;
   int *mcl = (int*) malloc ( sizeof( int ) * n );

   /* Initialize MCL (max chain length) values for all indexes */
   for ( i = 0; i < n; i++ )
      mcl[i] = 1;

   /* Compute optimized chain length values in bottom up manner */
   for ( i = 1; i < n; i++ )
      for ( j = 0; j < i; j++ )
         if ( arr[i].a > arr[j].b && mcl[i] < mcl[j] + 1)
            mcl[i] = mcl[j] + 1;

   // mcl[i] now stores the maximum chain length ending with pair i

   /* Pick maximum of all MCL values */
   for ( i = 0; i < n; i++ )
      if ( max < mcl[i] )
         max = mcl[i];

   /* Free memory to avoid memory leak */
   free( mcl );

   return max;
}


/* Driver program to test above function */
int main()
{
    struct pair arr[] = { {5, 24}, {15, 25},
                          {27, 40}, {50, 60} };
    int n = sizeof(arr)/sizeof(arr[0]);
    printf("Length of maximum size chain is %d\n",
           maxChainLength( arr, n ));
    return 0;
}

Java

class Pair{
    int a;
    int b;
    
    public Pair(int a, int b) {
        this.a = a;
        this.b = b;
    }
    
    // This function assumes that arr[] is sorted in increasing order
    // according the first (or smaller) values in pairs.
    static int maxChainLength(Pair arr[], int n)
    {
       int i, j, max = 0;
       int mcl[] = new int[n];
     
       /* Initialize MCL (max chain length) values for all indexes */
       for ( i = 0; i < n; i++ )
          mcl[i] = 1;
     
       /* Compute optimized chain length values in bottom up manner */
       for ( i = 1; i < n; i++ )
          for ( j = 0; j < i; j++ )
             if ( arr[i].a > arr[j].b && mcl[i] < mcl[j] + 1)
                mcl[i] = mcl[j] + 1;
     
       // mcl[i] now stores the maximum chain length ending with pair i
     
       /* Pick maximum of all MCL values */
       for ( i = 0; i < n; i++ )
          if ( max < mcl[i] )
             max = mcl[i];
     
       return max;
    }

    /* Driver program to test above function */
    public static void main(String[] args) 
    {
        Pair arr[] = new Pair[] {new Pair(5,24), new Pair(15, 25),
                                  new Pair (27, 40), new Pair(50, 60)};
        System.out.println("Length of maximum size chain is " + 
                                  maxChainLength(arr, arr.length));
    }
}


Output:
Length of maximum size chain is 3

Time Complexity: O(n^2) where n is the number of pairs.

Asked in: Amazon

The given problem is also a variation of Activity Selection problem and can be solved in (nLogn) time. To solve it as a activity selection problem, consider the first element of a pair as start time in activity selection problem, and the second element of pair as end time. Thanks to Palash for suggesting this approach.

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