Minimize segments required to be removed such that at least one segment intersects with all remaining segments

Given an array arr[] consisting of N pairs [L, R], where L and R denotes the start and end indices of a segment, the task is to find the minimum number of segments that must be deleted from the array such that the remaining array contains at least one segment which intersects with all other segments present in the array.

Examples:

Input: arr[] = {{1, 2}, {5, 6}, {6, 7}, {7, 10}, {8, 9}}
Output: 2
Explanation: Delete the segments {1, 2} and {5, 6}. Therefore, the remaining array contains the segment {7, 10} which intersects with all other segments.

Input: a[] = {{1, 2}, {2, 3}, {1, 5}, {4, 5}}
Output: 0
Explanation: The segment {1, 5} already intersects with all other remaining segments. Hence, no need to delete any segment.

Approach: The maximum possible answer is (N – 1), since after deleting (N – 1) segments from arr[], only one segment will be left. This segment intersects with itself. To achieve the minimum answer, the idea is to iterate through all the segments, and for each segment, check the number of segments which do not intersect with it.

Two segments [f₁, s₁] and [f₂, s₂] intersect only when max(f₁, f₂) ? min(s₁, s₂). 
Therefore, if [f₁, s₁] does not intersect with [f₂, s₂], then there are only two possibilities:

1. s₁ < f₂ i.e segment 1 ends before the start of segment 2
2. f₁ > s₂  i.e segment 1 starts after the end of segment 2.

Follow the steps below to solve the problem:

• Traverse the array arr[] and store the starting point and ending point of each segment in startPoints[], and endPoints[] respectively.
• Sort both the arrays, startPoints[] and endPoints[] in increasing order.
• Initialize ans as (N – 1) to store the number of minimum deletions required.
• Again traverse the array, arr[] and for each segment:
  • Store the number of segments satisfying the first and the second condition of non-intersection in leftDelete and rightDelete respectively.
  • If leftDelete + rightDelete is less than ans, then set ans to leftDelete + rightDelete.
• After the above steps, print the value of ans as the result.

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 find the minimum number
// of segments required to be deleted
void minSegments(pair<int, int> segments[],
                 int n)
{
    // Stores the start and end points
    int startPoints[n], endPoints[n];

    // Traverse segments and fill the
    // startPoints and endPoints
    for (int i = 0; i < n; i++) {
        startPoints[i] = segments[i].first;
        endPoints[i] = segments[i].second;
    }

    // Sort the startPoints
    sort(startPoints, startPoints + n);

    // Sort the startPoints
    sort(endPoints, endPoints + n);

    // Store the minimum number of
    // deletions required and
    // initialize with (N - 1)
    int ans = n - 1;

    // Traverse the array segments[]
    for (int i = 0; i < n; i++) {

        // Store the starting point
        int f = segments[i].first;

        // Store the ending point
        int s = segments[i].second;

        // Store the number of segments
        // satisfying the first condition
        // of non-intersection
        int leftDelete
            = lower_bound(endPoints,
                          endPoints + n, f)
              - endPoints;

        // Store the number of segments
        // satisfying the second condition
        // of non-intersection
        int rightDelete = max(
            0, n - (int)(upper_bound(startPoints,
                                     startPoints + n, s)
                         - startPoints));

        // Update answer
        ans = min(ans,
                  leftDelete
                      + rightDelete);
    }

    // Print the answer
    cout << ans;
}

// Driver Code
int main()
{
    pair<int, int> arr[] = {
        { 1, 2 }, { 5, 6 },
        { 6, 7 }, { 7, 10 }, { 8, 9 }
    };
    int N = sizeof(arr) / sizeof(arr[0]);

    // Function Call
    minSegments(arr, N);

    return 0;
}

Java

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

class GFG{

// Pair class
static class Pair
{
    int first;
    int second;

    Pair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}

public static int lower_bound(int arr[], int key)
{
    int l = -1, r = arr.length;
    while (l + 1 < r)
    {
        int m = (l + r) >>> 1;
        if (arr[m] >= key)
            r = m;
        else
            l = m;
    }
    return r;
}

public static int upper_bound(int arr[], int key)
{
    int l = -1, r = arr.length;
    while (l + 1 < r)
    {
        int m = (l + r) >>> 1;
        if (arr[m] <= key)
            l = m;
        else
            r = m;
    }
    return l + 1;
}

// Function to find the minimum number
// of segments required to be deleted
static void minSegments(Pair segments[], int n)
{

    // Stores the start and end points
    int startPoints[] = new int[n];
    int endPoints[] = new int[n];

    // Traverse segments and fill the
    // startPoints and endPoints
    for(int i = 0; i < n; i++)
    {
        startPoints[i] = segments[i].first;
        endPoints[i] = segments[i].second;
    }

    // Sort the startPoints
    Arrays.sort(startPoints);

    // Sort the startPoints
    Arrays.sort(endPoints);

    // Store the minimum number of
    // deletions required and
    // initialize with (N - 1)
    int ans = n - 1;

    // Traverse the array segments[]
    for(int i = 0; i < n; i++)
    {

        // Store the starting point
        int f = segments[i].first;

        // Store the ending point
        int s = segments[i].second;

        // Store the number of segments
        // satisfying the first condition
        // of non-intersection
        int leftDelete = lower_bound(endPoints, f);

        // Store the number of segments
        // satisfying the second condition
        // of non-intersection
        int rightDelete = Math.max(
            0, n - (int)(upper_bound(startPoints, s)));

        // Update answer
        ans = Math.min(ans, leftDelete + rightDelete);
    }

    // Print the answer
    System.out.println(ans);
}

// Driver Code
public static void main(String[] args)
{
    Pair arr[] = { new Pair(1, 2), new Pair(5, 6),
                   new Pair(6, 7), new Pair(7, 10),
                   new Pair(8, 9) };
    int N = arr.length;

    // Function Call
    minSegments(arr, N);
}
}

// This code is contributed by Kingash

Python3

# Python3 program for the above approach

from bisect import bisect_left,bisect_right
# Function to find the minimum number
# of segments required to be deleted
def minSegments(segments, n):
    # Stores the start and end points
    startPoints = [0 for i in range(n)]
    endPoints = [0 for i in range(n)]

    # Traverse segments and fill the
    # startPoints and endPoints
    for i in range(n):
        startPoints[i] = segments[i][0]
        endPoints[i] = segments[i][1]

    # Sort the startPoints
    startPoints.sort(reverse = False)

    # Sort the startPoints
    endPoints.sort(reverse= False)

    # Store the minimum number of
    # deletions required and
    # initialize with (N - 1)
    ans = n - 1

    # Traverse the array segments[]
    for i in range(n):
        # Store the starting point
        f = segments[i][0]

        # Store the ending point
        s = segments[i][1]

        # Store the number of segments
        # satisfying the first condition
        # of non-intersection
        leftDelete = bisect_left(endPoints, f)

        # Store the number of segments
        # satisfying the second condition
        # of non-intersection
        rightDelete = max(0, n - bisect_right(startPoints,s))

        # Update answer
        ans = min(ans, leftDelete + rightDelete)

    # Print the answer
    print(ans)

# Driver Code
if __name__ == '__main__':
    arr = [[1, 2],[5, 6], [6, 7],[7, 10],[8, 9]]
    N = len(arr)

    # Function Call
    minSegments(arr, N)

    # This code is contributed by ipg2016107.

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

public class GFG {

    // Pair class
    class Pair {
        public int first;
        public int second;
        public Pair(int first, int second)
        {
            this.first = first;
            this.second = second;
        }
    }

    public static int lower_bound(int[] arr, int key)
    {
        int l = -1, r = arr.Length;
        while (l + 1 < r) {
            int m = (l + r) >> 1;
            if (arr[m] >= key)
                r = m;
            else
                l = m;
        }
        return r;
    }

    public static int upper_bound(int[] arr, int key)
    {
        int l = -1, r = arr.Length;
        while (l + 1 < r) {
            int m = (l + r) >> 1;
            if (arr[m] <= key)
                l = m;
            else
                r = m;
        }
        return l + 1;
    }

    static void minSegments(Pair[] segments, int n)
    {
        // Stores the start and end points
        int[] startPoints = new int[n];
        int[] endPoints = new int[n];

        // Traverse segments and fill the
        // startPoints and endPoints
        for (int i = 0; i < n; i++) {
            startPoints[i] = segments[i].first;
            endPoints[i] = segments[i].second;
        }

        // Sort the startPoints
        Array.Sort(startPoints);

        // Sort the startPoints
        Array.Sort(endPoints);

        // Store the minimum number of
        // deletions required and
        // initialize with (N - 1)
        int ans = n - 1;

        // Traverse the array segments[]
        for (int i = 0; i < n; i++) {

            // Store the starting point
            int f = segments[i].first;

            // Store the ending point
            int s = segments[i].second;

            // Store the number of segments
            // satisfying the first condition
            // of non-intersection
            int leftDelete = lower_bound(endPoints, f);

            // Store the number of segments
            // satisfying the second condition
            // of non-intersection
            int rightDelete = Math.Max(
                0, n - (int)(upper_bound(startPoints, s)));

            // Update answer
            ans = Math.Min(ans, leftDelete + rightDelete);
        }

        // Print the answer
        Console.WriteLine(ans);
    }

    static public void Main()
    {

        // Code
        Pair[] arr = { new Pair(1, 2), new Pair(5, 6),
                       new Pair(6, 7), new Pair(7, 10),
                       new Pair(8, 9) };

        int N = arr.Length;
        minSegments(arr, N);
    }
}

// This code is contributed by lokesh (lokeshmvs21).

Javascript

// JS program for the above approach
function lowerBound(arr, key) {
    let l = -1, r = arr.length;
    while (l + 1 < r) {
        let m = (l + r) >>> 1;
        if (arr[m] >= key)
            r = m;
        else
            l = m;
    }
    return r;
}

function upperBound(arr, key) {
    let l = -1, r = arr.length;
    while (l + 1 < r) {
        let m = (l + r) >>> 1;
        if (arr[m] <= key)
            l = m;
        else
            r = m;
    }
    return l + 1;
}

// Function to find the minimum number
// of segments required to be deleted
function minSegments(segments, n) {
    // Stores the start and end points
    let startPoints = [];
    let endPoints = [];

    // Traverse segments and fill the
    // startPoints and endPoints
    for (let i = 0; i < n; i++) {
        startPoints.push(segments[i].first);
        endPoints.push(segments[i].second);
    }

    // Sort the startPoints
    startPoints.sort();

    // Sort the startPoints
    endPoints.sort(function (a, b) { return a - b });

    // Store the minimum number of
    // deletions required and
    // initialize with (N - 1)
    let ans = n - 1;

    // Traverse the array segments[]
    for (let i = 0; i < n; i++) {

        // Store the starting point
        let f = segments[i].first;

        // Store the ending point
        let s = segments[i].second;

        // Store the number of segments
        // satisfying the first condition
        // of non-intersection
        let leftDelete
            = lowerBound(endPoints, f);

        // Store the number of segments
        // satisfying the second condition
        // of non-intersection
        let rightDelete = Math.max(0, n - Math.floor(upperBound(startPoints, s)));

        // Update answer
        ans = Math.min(ans, leftDelete + rightDelete);
    }

    // Print the answer
    console.log(ans);
}

// Driver Code
arr = [{ "first": 1, "second": 2 },
{ "first": 5, "second": 6 },
{ "first": 6, "second": 7 },
{ "first": 7, "second": 10 },
{ "first": 8, "second": 9 }];
let N = arr.length;

// Function Call
minSegments(arr, N);

// This code is contributed by akashish__.

Output:

2

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