Klee's Algorithm (Length Of Union Of Segments of a line)

Last Updated : 22 Feb, 2023

Given starting and ending positions of segments on a line, the task is to take the union of all given segments and find length covered by these segments.
Examples:

Input : segments[] = {{2, 5}, {4, 8}, {9, 12}}
Output : 9 
Explanation:
segment 1 = {2, 5}
segment 2 = {4, 8}
segment 3 = {9, 12}
If we take the union of all the line segments,
we cover distances [2, 8] and [9, 12]. Sum of 
these two distances is 9 (6 + 3)

Approach:

The algorithm was proposed by Klee in 1977. The time complexity of the algorithm is O (N log N). It has been proven that this algorithm is the fastest (asymptotically) and this problem can not be solved with a better complexity.

Description :
1) Put all the coordinates of all the segments in an auxiliary array points[].
2) Sort it on the value of the coordinates.
3) An additional condition for sorting - if there are equal coordinates, insert the one which is the left coordinate of any segment instead of a right one.
4) Now go through the entire array, with the counter "count" of overlapping segments.
5) If the count is greater than zero, then the result is added to the difference between the points[i] - points[i-1].
6) If the current element belongs to the left end, we increase "count", otherwise reduce it.
Illustration:

Lets take the example :
segment 1 : (2,5)
segment 2 : (4,8)
segment 3 : (9,12)

Counter = result = 0;
n = number of segments = 3;

for i=0,  points[0] = {2, false}
          points[1] = {5, true}
for i=1,  points[2] = {4, false}
          points[3] = {8, true}
for i=2,  points[4] = {9, false}
          points[5] = {12, true}

Therefore :
points = {2, 5, 4, 8, 9, 12}
         {f, t, f, t, f, t}

after applying sorting :
points = {2, 4, 5, 8, 9, 12}
         {f, f, t, t, f, t}

Now,
for i=0, result = 0;
         Counter = 1;

for i=1, result = 2;
         Counter = 2;

for i=2, result = 3;
         Counter = 1;

for i=3, result = 6;
         Counter = 0;

for i=4, result = 6;
         Counter = 1;

for i=5, result = 9;
         Counter = 0;

Final answer = 9;

C++

// C++ program to implement Klee's algorithm
#include<bits/stdc++.h>
using namespace std;

// Returns sum of lengths covered by union of given
// segments
int segmentUnionLength(const vector< 
                          pair <int,int> > &seg)
{
    int n = seg.size();

    // Create a vector to store starting and ending
    // points
    vector <pair <int, bool> > points(n * 2);
    for (int i = 0; i < n; i++)
    {
        points[i*2]     = make_pair(seg[i].first, false);
        points[i*2 + 1] = make_pair(seg[i].second, true);
    }

    // Sorting all points by point value
    sort(points.begin(), points.end());

    int result = 0; // Initialize result

    // To keep track of counts of 
    // current open segments
    // (Starting point is processed, 
    // but ending point
    // is not)
    int Counter = 0;

    // Traverse through all points
    for (unsigned i=0; i<n*2; i++)
    {
        // If there are open points, then we add the
        // difference between previous and current point.
        // This is interesting as we don't check whether
        // current point is opening or closing,
        if (Counter)
            result += (points[i].first - 
                        points[i-1].first);

        // If this is an ending point, reduce, count of
        // open points.
        (points[i].second)? Counter-- : Counter++;
    }
    return result;
}

// Driver program for the above code
int main()
{
    vector< pair <int,int> > segments;
    segments.push_back(make_pair(2, 5));
    segments.push_back(make_pair(4, 8));
    segments.push_back(make_pair(9, 12));
    cout << segmentUnionLength(segments) << endl;
    return 0;
}

Java

// Java program to implement Klee's algorithm
import java.io.*;
import java.util.*;

class GFG {

  // to use create a pair of segments
  static class SegmentPair
  {
    int x,y;
    SegmentPair(int xx, int yy){
      this.x = xx;
      this.y = yy;
    }
  }

  //to create a pair of points
  static class PointPair{
    int x;
    boolean isEnding;
    PointPair(int xx, boolean end){
      this.x = xx;
      this.isEnding = end;
    }
  }

  // creates the comparator for comparing objects of PointPair class
  static class Comp implements Comparator<PointPair>
  {
    
    // override the compare() method
    public int compare(PointPair p1, PointPair p2)
    {
      if (p1.x < p2.x) {
        return -1;
      }
      else {
        if(p1.x == p2.x){
          return 0;
        }else{
          return 1;
        }
      }
    }
  }

  public static int segmentUnionLength(List<SegmentPair> segments){
    int n = segments.size();

    // Create a list to store 
    // starting and ending points
    List<PointPair> points = new ArrayList<>();
    for(int i = 0; i < n; i++){
      points.add(new PointPair(segments.get(i).x,false));
      points.add(new PointPair(segments.get(i).y,true));
    }
    
    // Sorting all points by point value
    Collections.sort(points, new Comp());

    int result = 0; // Initialize result

    // To keep track of counts of
    // current open segments
    // (Starting point is processed,
    // but ending point
    // is not)
    int Counter = 0;

    // Traverse through all points
    for(int i = 0; i < 2 * n; i++)
    {
      
      // If there are open points, then we add the
      // difference between previous and current point.
      // This is interesting as we don't check whether
      // current point is opening or closing,
      if (Counter != 0)
      {
        result += (points.get(i).x - points.get(i-1).x);
      }

      // If this is an ending point, reduce, count of
      // open points.
      if(points.get(i).isEnding)
      {
        Counter--;
      }
      else
      {
        Counter++;
      }
    }
    return result;
  }

  // Driver Code
  public static void main (String[] args) {
    List<SegmentPair> segments = new ArrayList<>();
    segments.add(new SegmentPair(2,5));
    segments.add(new SegmentPair(4,8));
    segments.add(new SegmentPair(9,12));
    System.out.println(segmentUnionLength(segments));
  }
}

// This code is contributed by shruti456rawal

Python3

# Python program for the above approach

def segmentUnionLength(segments):
  
    # Size of given segments list
    n = len(segments)
    
    # Initialize empty points container
    points = [None] * (n * 2)
    
    # Create a vector to store starting 
    # and ending points
    for i in range(n):
        points[i * 2] = (segments[i][0], False)
        points[i * 2 + 1] = (segments[i][1], True)
        
    # Sorting all points by point value
    points = sorted(points, key=lambda x: x[0])
    
    # Initialize result as 0
    result = 0
    
    # To keep track of counts of current open segments
    # (Starting point is processed, but ending point
    # is not)
    Counter = 0
    
    # Traverse through all points
    for i in range(0, n * 2):
      
        # If there are open points, then we add the
        # difference between previous and current point.
        if (i > 0) & (points[i][0] > points[i - 1][0]) &  (Counter > 0):
            result += (points[i][0] - points[i - 1][0])
            
        # If this is an ending point, reduce, count of
        # open points.
        if points[i][1]:
            Counter -= 1
        else:
            Counter += 1
    return result


# Driver code
if __name__ == '__main__':
    segments = [(2, 5), (4, 8), (9, 12)]
    print(segmentUnionLength(segments))

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;

// C# program to implement Klee's algorithm
class HelloWorld {


  class GFG : IComparer<KeyValuePair<int, bool>>
  {
    public int Compare(KeyValuePair<int, bool> x,KeyValuePair<int, bool> y)
    {
      // CompareTo() method
      return x.Key.CompareTo(y.Key);
    }
  }


  // Returns sum of lengths covered by union of given
  // segments
  public static int segmentUnionLength(List<KeyValuePair<int,int>> seg)
  {
    int n = seg.Count;
    // Create a vector to store starting and ending
    // points
    List<KeyValuePair<int, bool>> points = new List<KeyValuePair<int, bool>>();
    for(int i = 0; i < 2*n; i++){
      points.Add(new KeyValuePair<int, bool> (0,true));
    }        


    for (int i = 0; i < n; i++)
    {
      points[i*2] = new KeyValuePair<int, bool> (seg[i].Key, false);
      points[i*2 + 1] = new KeyValuePair<int, bool> (seg[i].Value, true);
    }

    // Sorting all points by point value
    GFG gg = new GFG();
    points.Sort(gg);

    int result = 0; // Initialize result

    // To keep track of counts of 
    // current open segments
    // (Starting point is processed, 
    // but ending point
    // is not)
    int Counter = 0;

    // Traverse through all points
    for (int i=0; i<n*2; i++)
    {
      // If there are open points, then we add the
      // difference between previous and current point.
      // This is interesting as we don't check whether
      // current point is opening or closing,
      if (Counter  != 0)
        result += (points[i].Key - points[i-1].Key);

      // If this is an ending point, reduce, count of
      // open points.
      if(points[i].Value != false){
        Counter--;
      }
      else{
        Counter++;
      }
    }
    return result;
  }


  static void Main() {

    List<KeyValuePair<int,int>> segments = new List<KeyValuePair<int,int>> ();
    segments.Add(new KeyValuePair<int,int> (2, 5));
    segments.Add(new KeyValuePair<int,int> (4, 8));
    segments.Add(new KeyValuePair<int,int> (9, 12));

    Console.WriteLine(segmentUnionLength(segments));
  }
}

// The code is contributed by Nidhi goel.

JavaScript

// JavaScript program to implement Klee's algorithm

// Returns sum of lengths covered by union of given
// segments
function segmentUnionLength(seg)
{
    let n = seg.length;

    // Create a vector to store starting and ending
    // points
    let points = new Array(2*n);
    for (let i = 0; i < n; i++)
    {
        points[i*2] = [seg[i][0], false];
        points[i*2 + 1] = [seg[i][1], true];
    }

    // Sorting all points by point value
    points.sort(function(a, b){
        return a[0] - b[0];
    });
    
    let result = 0; // Initialize result

    // To keep track of counts of 
    // current open segments
    // (Starting point is processed, 
    // but ending point
    // is not)
    let Counter = 0;

    // Traverse through all points
    for (let i=0; i<n*2; i++)
    {
        // If there are open points, then we add the
        // difference between previous and current point.
        // This is interesting as we don't check whether
        // current point is opening or closing,
        if (Counter)
            result += (points[i][0] - points[i-1][0]);

        // If this is an ending point, reduce, count of
        // open points.
        if(points[i][1]){
            Counter = Counter - 1;
        }
        else{
            Counter = Counter + 1;
        }
    }
    return result;
}

let segments = new Array();
segments.push([2, 5]);
segments.push([4, 8]);
segments.push([9, 12]);
console.log(segmentUnionLength(segments));

// The code is contributed by Gautam goel (gautamgoel962)

Output

Time Complexity : O(n * log n)
Auxiliary Space: O(n)

Count maximum points on same line

Aarti_Rathi and Abhinandan Mittal

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Mathematical

Klee's Algorithm (Length Of Union Of Segments of a line)

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