Print Maximum Length Chain of Pairs

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.

Examples:

Input:  (5, 24), (39, 60), (15, 28), (27, 40), (50, 90)
Output: (5, 24), (27, 40), (50, 90)

Input:  (11, 20), {10, 40), (45, 60), (39, 40)
Output: (11, 20), (39, 40), (45, 60)

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

In previous post, we have discussed about Maximum Length Chain of Pairs problem. However, the post only covered code related to finding the length of maximum size chain, but not to the construction of maximum size chain. In this post, we will discuss how to construct Maximum Length Chain of Pairs itself.

The idea is to first sort given pairs in increasing order of their first element. Let arr[0..n-1] be the input array of pairs after sorting. We define vector L such that L[i] is itself is a vector that stores Maximum Length Chain of Pairs of arr[0..i] that ends with arr[i]. Therefore for an index i, L[i] can be recursively written as –

L[0] = {arr[0]}
L[i] = {Max(L[j])} + arr[i] where j < i and arr[j].b < arr[i].a 
	 = arr[i], if there is no such j

For example, for (5, 24), (39, 60), (15, 28), (27, 40), (50, 90)

L[0]: (5, 24)
L[1]: (5, 24) (39, 60)
L[2]: (15, 28)
L[3]: (5, 24) (27, 40)
L[4]: (5, 24) (27, 40) (50, 90)

Please note sorting of pairs is done as we need to find the maximum pair length and ordering doesn’t matter here. If we don’t sort, we will get pairs in increasing order but they won’t be maximum possible pairs.

Below is C++ implementation of above idea –

/* Dynamic Programming solution to construct 
   Maximum Length Chain of Pairs */
#include <bits/stdc++.h> 
using namespace std; 
  
struct Pair 
{ 
    int a; 
    int b; 
}; 
  
// comparator function for sort function 
int compare(Pair x, Pair y) 
{ 
    return x.a < y.a; 
} 
  
// Function to construct Maximum Length Chain 
// of Pairs 
void maxChainLength(vector<Pair> arr) 
{ 
    // Sort by start time 
    sort(arr.begin(), arr.end(), compare); 
  
    // L[i] stores maximum length of chain of 
    // arr[0..i] that ends with arr[i]. 
    vector<vector<Pair> > L(arr.size()); 
  
    // L[0] is equal to arr[0] 
    L[0].push_back(arr[0]); 
  
    // start from index 1 
    for (int i = 1; i < arr.size(); i++) 
    { 
        // for every j less than i 
        for (int j = 0; j < i; j++) 
        { 
            // L[i] = {Max(L[j])} + arr[i] 
            // where j < i and arr[j].b < arr[i].a 
            if ((arr[j].b < arr[i].a) && 
                (L[j].size() > L[i].size())) 
                L[i] = L[j]; 
        } 
        L[i].push_back(arr[i]); 
    } 
  
    // print max length vector 
    vector<Pair> maxChain; 
    for (vector<Pair> x : L) 
        if (x.size() > maxChain.size()) 
            maxChain = x; 
  
    for (Pair pair : maxChain) 
        cout << "(" << pair.a << ", "
             << pair.b << ") "; 
} 
  
// Driver Function 
int main() 
{ 
    Pair a[] = {{5, 29}, {39, 40}, {15, 28}, 
                {27, 40}, {50, 90}}; 
    int n = sizeof(a)/sizeof(a[0]); 
  
    vector<Pair> arr(a, a + n); 
  
    maxChainLength(arr); 
  
    return 0; 
} 

Output:

(5, 29) (39, 40) (50, 90)

Time complexity of above Dynamic Programming solution is O(n²) where n is the number of pairs.
Auxiliary space used by the program is O(n²).

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