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Minimum number of pairs required to be selected to cover a given range

  • Difficulty Level : Hard
  • Last Updated : 30 Jun, 2021

Given an array arr[] consisting of N pairs and a positive integer K, the task is to select the minimum number of pairs such that it covers the entire range [0, K]. If it is not possible to cover the range [0, K], then print -1.

Examples :

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Input: arr[] = {{0, 3}, {2, 5}, {5, 6}, {6, 8}, {6, 10}}, K = 10
Output: 4
Explanation: One of the optimal ways is to select the ranges {{0, 3}, {2, 5}, {5, 6}, {6, 10}} which covers the entire range [0, 10].



Input: arr[] = {{0, 4}, {1, 5}, {5, 6}, {8, 10}}, N = 10
Output : -1

Naive Approach: The simplest approach to solve the problem is to generate all possible subsets and check for each subset, if it covers the entire range or not. 

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

Efficient Approach: The above approach can be optimized by sorting the array in increasing order on the basis of the left element and for pairs with equal left elements, sort the elements in decreasing order of their right element. Now, choose the pairs optimally. 

Follow the steps below to solve the problem:

  • Sort the vector of pairs arr[] in increasing order of left element and if left elements are equal, then sort the array in decreasing order of right element of pair.
  • Check if arr[0][0] != 0 then print -1.
  • Initialize a variable, say R as arr[0][1] which stores the right bound of the range and ans as 1 which stores number of ranges needed to cover the range [0,  K].
  • Iterate in the range [0, N-1] using the variable i and perform the following steps:
    • If R == K, terminate the loop.
    • Initialize a variable, say maxr as R which stores the maximum right bound where left bound ≤ R.
    • Iterate in the range [i+1, N-1] using the variable j and perform the following steps:
      • If arr[j][0] ≤ R, then modify the value of maxr as max(maxr, arr[j][1]).
    • Modify the value of i as j-1 and the value of R as maxr and increment the value of ans by 1.
  • If R is not equal to K, then print -1.
  • Otherwise, print the value of ans.

Below is the implementation of the above approach:

C++14




// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to sort the elements in
// increasing order of left element and
// sort pairs with equal left element in
// decreasing order of right element
static bool cmp(pair<int, int> a, pair<int, int> b)
{
    if (a.first == b.first) {
        return a.second > b.second;
    }
    else {
        return b.first > a.first;
    }
}
 
// Function to select minimum number of pairs
// such that it covers the entire range [0, K]
int MinimumPairs(vector<pair<int, int> > arr, int K)
{
    int N = arr.size();
 
    // Sort the array using comparator
    sort(arr.begin(), arr.end(), cmp);
 
    // If the start element
    // is not equal to 0
    if (arr[0].first != 0) {
        return -1;
    }
 
    // Stores the minimum
    // number of pairs required
    int ans = 1;
 
    // Stores the right bound of the range
    int R = arr[0].second;
 
    // Iterate in the range[0, N-1]
    for (int i = 0; i < N; i++) {
        if (R == K) {
            break;
        }
 
        // Stores the maximum right bound
        // where left bound ≤ R
        int maxr = R;
 
        int j;
 
        // Iterate in the range [i+1, N-1]
        for (j = i + 1; j < N; j++) {
 
            // If the starting point of j-th
            // element is less than equal to R
            if (arr[j].first <= R) {
 
                maxr = max(maxr, arr[j].second);
            }
 
            // Otherwise
            else {
                break;
            }
        }
 
        i = j - 1;
        R = maxr;
        ans++;
    }
 
    // If the right bound
    // is not equal to K
    if (R != K) {
        return -1;
    }
 
    // Otherwise
    else {
        return ans;
    }
}
 
// Driver Code
int main()
{
    // Given Input
    int K = 4;
    vector<pair<int, int> > arr{ { 0, 6 } };
 
    // Function Call
    cout << MinimumPairs(arr, K);
}

Python3




# Python3 program for the above approach
 
# Function to select minimum number of pairs
# such that it covers the entire range [0, K]
def MinimumPairs(arr, K):
     
    N = len(arr)
 
    # Sort the array using comparator
    arr.sort()
 
    # If the start element
    # is not equal to 0
    if (arr[0][0] != 0):
        return -1
 
    # Stores the minimum
    # number of pairs required
    ans = 1
 
    # Stores the right bound of the range
    R = arr[0][1]
 
    # Iterate in the range[0, N-1]
    for i in range(N):
        if (R == K):
            break
 
        # Stores the maximum right bound
        # where left bound ≤ R
        maxr = R
 
        j = 0
 
        # Iterate in the range [i+1, N-1]
        for j in range(i + 1, N, 1):
             
            # If the starting point of j-th
            # element is less than equal to R
            if (arr[j][0] <= R):
                maxr = max(maxr, arr[j][1])
 
            # Otherwise
            else:
                break
 
        i = j - 1
        R = maxr
        ans += 1
 
    # If the right bound
    # is not equal to K
    if (R != K):
        return -1
 
    # Otherwise
    else:
        return ans
 
# Driver Code
if __name__ == '__main__':
     
    # Given Input
    K = 4
    arr = [ [ 0, 6 ] ]
 
    # Function Call
    print(MinimumPairs(arr, K))
 
# This code is contributed by bgangwar59
Output
-1

Time Complexity: O(NlogN)
Auxiliary Space: O(1)

 




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