Minimum distance to travel to cover all intervals

Given many intervals as ranges and our position. We need to find minimum distance to travel to reach such a point which covers all the intervals at once.
Examples:

Input : Intervals = [(0, 7), (2, 14), (4, 6)]
        Position = 3
Output : 1
We can reach position 4 by traveling
distance 1, at which all intervals will
be covered. So answer will be 1

Input : Intervals = [(1, 2), (2, 3), (3, 4)]
        Position = 2
Output : -1 
It is not possible to cover all intervals
at once at any point

Input : Intervals = [(1, 2), (2, 3), (1, 4)]
        Position = 2
Output : 0
All Intervals are covered at current 
position only so no need travel and
answer will be 0

All above examples are shown in below diagram.


We can solve this problem by concentrating only on endpoints. Since the requirement is to cover all intervals by reaching a point, all intervals must a share a point for answer to exist. Even the interval with leftmost end point must overlap with the interval right most start point.

First, we find right most start point and left most end point from all intervals. Then we can compare our position with these points to get the result which is explained below :

  1. If this right most start point is to the right of left most end point then it is not possible to cover all intervals simultaneously. (as in example 2)
  2. If our position is in mid between to right most start and left most end then there is no need to travel and all intervals will be covered by current position only (as in example 3)
  3. If our position is left to both points then we need to travel up to the rightmost start point and if our position is right to both points then we need to travel up to leftmost end point.

Refer above diagram to understand these cases. As in the first example, right most start is 4 and left most end is 6, so we need to reach 4 from current position 3 to cover all intervals.
Please see below code for better understanding.

// C++ program to find minimum distance to 
// travel to cover all intervals
#include <bits/stdc++.h>
using namespace std;

//  structure to store an interval
struct Interval
{
    int start, end;
    Interval(int start, int end) : start(start), 
                                       end(end)
    {}
};

//  Method returns minimum distance to travel 
// to cover all intervals
int minDistanceToCoverIntervals(Interval intervals[], 
                                       int N, int x)
{
    int rightMostStart = INT_MIN;
    int leftMostEnd = INT_MAX;

    //  looping over all intervals to get right most
    // start and left most end
    for (int i = 0; i < N; i++)
    {
        if (rightMostStart < intervals[i].start)
            rightMostStart = intervals[i].start;

        if (leftMostEnd > intervals[i].end)
            leftMostEnd = intervals[i].end;
    }
    
    int res;

    /*  if rightmost start > leftmost end then all 
        intervals are not aligned and it is not 
        possible to cover all of them  */
    if (rightMostStart > leftMostEnd)
        res = -1;

    //  if x is in between rightmoststart and 
    // leftmostend then no need to travel any distance
    else if (rightMostStart <= x && x <= leftMostEnd)
        res = 0;
    
    //  choose minimum according to current position x 
    else
        res = (x < rightMostStart) ? (rightMostStart - x) :
                                     (x - leftMostEnd);
    
    return res;
}

//  Driver code to test above methods
int main()
{
    int x = 3;
    Interval intervals[] = {{0, 7}, {2, 14}, {4, 6}};
    int N = sizeof(intervals) / sizeof(intervals[0]);

    int res = minDistanceToCoverIntervals(intervals, N, x);
    if (res == -1)
        cout << "Not Possible to cover all intervals\n";
    else
        cout << res << endl;
}

Output:

1

This article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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