Find the point where maximum intervals overlap

Consider a big party where a log register for guest’s entry and exit times is maintained. Find the time at which there are maximum guests in the party. Note that entries in register are not in any order.

Example :

Input: arrl[] = {1, 2, 9, 5, 5}
       exit[] = {4, 5, 12, 9, 12}
First guest in array arrives at 1 and leaves at 4, 
second guest arrives at 2 and leaves at 5, and so on.

Output: 5
There are maximum 3 guests at time 5.  

Below is a Simple Method to solve this problem.



1) Traverse all intervals and find min and max time (time at which first guest arrives and time at which last guest leaves)

2) Create a count array of size ‘max – min + 1’. Let the array be count[].

3) For each interval [x, y], run a loop for i = x to y and do following in loop.
     count[i – min]++;

4) Find the index of maximum element in count array. Let this index be ‘max_index’, return max_index + min.

Above solution requires O(max-min+1) extra space. Also time complexity of above solution depends on lengths of intervals. In worst case, if all intervals are from ‘min’ to ‘max’, then time complexity becomes O((max-min+1)*n) where n is number of intervals.

An Efficient Solution is to use sorting n O(nLogn) time. The idea is to consider all events (all arrivals and exits) in sorted order. Once we have all events in sorted order, we can trace the number of guests at any time keeping track of guests that have arrived, but not exited.

Consider the above example.

    arr[]  = {1, 2, 10, 5, 5}
    dep[]  = {4, 5, 12, 9, 12}

Below are all events sorted by time.  Note that in sorting, if two
events have same time, then arrival is preferred over exit.
 Time     Event Type         Total Number of Guests Present
------------------------------------------------------------
   1        Arrival                  1
   2        Arrival                  2
   4        Exit                     1
   5        Arrival                  2
   5        Arrival                  3    // Max Guests
   5        Exit                     2
   9        Exit                     1
   10       Arrival                  2 
   12       Exit                     1
   12       Exit                     0 

Total number of guests at any time can be obtained by subtracting
total exits from total arrivals by that time.

So maximum guests are three at time 5.

Following is the implementation of above approach. Note that the implementation doesn’t create a single sorted list of all events, rather it individually sorts arr[] and dep[] arrays, and then uses merge process of merge sort to process them together as a single sorted array.

C++


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// Program to find maximum guest at any time in a party
#include<iostream>
#include<algorithm>
using namespace std;
  
void findMaxGuests(int arrl[], int exit[], int n)
{
   // Sort arrival and exit arrays
   sort(arrl, arrl+n);
   sort(exit, exit+n);
  
   // guests_in indicates number of guests at a time
   int guests_in = 1, max_guests = 1, time = arrl[0];
   int i = 1, j = 0;
  
   // Similar to merge in merge sort to process
   // all events in sorted order
   while (i < n && j < n)
   {
      // If next event in sorted order is arrival,
      // increment count of guests
      if (arrl[i] <= exit[j])
      {
          guests_in++;
  
          // Update max_guests if needed
          if (guests_in > max_guests)
          {
              max_guests = guests_in;
              time = arrl[i];
          }
          i++;  //increment index of arrival array
      }
      else // If event is exit, decrement count
      {    // of guests.
          guests_in--;
          j++;
      }
   }
  
   cout << "Maximum Number of Guests = " << max_guests
        << " at time " << time;
}
  
// Driver program to test above function
int main()
{
    int arrl[] = {1, 2, 10, 5, 5};
    int exit[] = {4, 5, 12, 9, 12};
    int n = sizeof(arrl)/sizeof(arrl[0]);
    findMaxGuests(arrl, exit, n);
    return 0;
}

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Java

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// Java Program to find maximum guest 
// at any time in a party
import java.util.*;
  
class GFG {
  
    static void findMaxGuests(int arrl[], int exit[],
                                          int n)    
    {   
    // Sort arrival and exit arrays
    Arrays.sort(arrl);
    Arrays.sort(exit);
  
    // guests_in indicates number of guests at a time
    int guests_in = 1, max_guests = 1, time = arrl[0];
    int i = 1, j = 0;
  
    // Similar to merge in merge sort to process
    // all events in sorted order
    while (i < n && j < n)
    {
        // If next event in sorted order is arrival,
        // increment count of guests
        if (arrl[i] <= exit[j])
        {
            guests_in++;
  
            // Update max_guests if needed
            if (guests_in > max_guests)
            {
                max_guests = guests_in;
                time = arrl[i];
            }
            i++; //increment index of arrival array
        }
        else // If event is exit, decrement count
        { // of guests.
            guests_in--;
            j++;
        }
    }
  
    System.out.println("Maximum Number of Guests = "+
                    max_guests + " at time " + time);
    }
  
    // Driver program to test above function
    public static void main(String[] args)
    {
        int arrl[] = {1, 2, 10, 5, 5};
        int exit[] = {4, 5, 12, 9, 12};
        int n = arrl.length;
        findMaxGuests(arrl, exit, n);
    }
}
// This code is contributed by Prerna Saini

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Python3

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# Program to find maximum guest
# at any time in a party
def findMaxGuests(arrl, exit, n):
  
    # Sort arrival and exit arrays
    arrl.sort();
    exit.sort();
  
    # guests_in indicates number of 
    # guests at a time
    guests_in = 1;
    max_guests = 1;
    time = arrl[0];
    i = 1;
    j = 0;
  
    # Similar to merge in merge sort to 
    # process all events in sorted order
    while (i < n and j < n):
          
        # If next event in sorted order is 
        # arrival, increment count of guests
        if (arrl[i] <= exit[j]):
      
            guests_in = guests_in + 1;
  
        # Update max_guests if needed
            if(guests_in > max_guests):
          
                max_guests = guests_in;
                time = arrl[i];
                  
            # increment index of arrival array
            i = i + 1
      
        else:
            guests_in = guests_in - 1;
            j = j + 1;
      
    print("Maximum Number of Guests =",
           max_guests, "at time", time)
  
# Driver Code
arrl = [1, 2, 10, 5, 5];
exit = [4, 5, 12, 9, 12];
n = len(arrl);
findMaxGuests(arrl, exit, n);
  
# This code is contributed 
# by Shivi_Aggarwal 

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C#

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// C# Program to find maximum guest 
// at any time in a party
using System;
class GFG 
{
    static void findMaxGuests(int []arrl, 
                              int []exit, 
                              int n) 
    
    // Sort arrival and exit arrays
    Array.Sort(arrl);
    Array.Sort(exit);
  
    // guests_in indicates number 
    // of guests at a time
    int guests_in = 1, 
        max_guests = 1, 
        time = arrl[0];
    int i = 1, j = 0;
  
    // Similar to merge in merge sort 
    // to process all events in sorted order
    while (i < n && j < n)
    {
        // If next event in sorted 
        // order is arrival, 
        // increment count of guests
        if (arrl[i] <= exit[j])
        {
            guests_in++;
  
            // Update max_guests if needed
            if (guests_in > max_guests)
            {
                max_guests = guests_in;
                time = arrl[i];
            }
              
            //increment index of arrival array
            i++; 
        }
          
         // If event is exit, decrement 
         // count of guests.
        else
        
            guests_in--;
            j++;
        }
    }
  
    Console.Write("Maximum Number of Guests = "+
                                    max_guests + 
                            " at time " + time);
    }
  
    // Driver Code
    public static void Main()
    {
        int []arrl = {1, 2, 10, 5, 5};
        int []exit = {4, 5, 12, 9, 12};
        int n = arrl.Length;
        findMaxGuests(arrl, exit, n);
    }
}
  
// This code is contributed by nitin mittal.

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PHP

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<?php
// PHP Program to find maximum 
// guest at any time in a party
  
function findMaxGuests($arrl, $exit, $n)
{
      
    // Sort arrival and exit arrays
    sort($arrl);
    sort($exit);
      
    // guests_in indicates number
    // of guests at a time
    $guests_in = 1; 
    $max_guests = 1; 
    $time = $arrl[0]; 
    $i = 1;
    $j = 0;
      
    // Similar to merge in merge
    // sort to process all events 
    // in sorted order
    while ($i < $n and $j < $n)
    {
          
        // If next event in sorted
        // order is arrival,
        // increment count of guests
        if ($arrl[$i] <= $exit[$j])
        {
            $guests_in++;
      
            // Update max_guests if needed
            if ($guests_in > $max_guests)
            {
                $max_guests = $guests_in;
                $time = $arrl[$i];
            }
              
            // increment index of 
            // arrival array
            $i++; 
        }
          
        // If event is exit, decrement
        // count of guests.
        else 
        {                             
            $guests_in--;
            $j++;
        }
    }
      
    echo "Maximum Number of Guests = " , $max_guests
                               , " at time " , $time;
}
  
    // Driver Code
    $arr1 = array(1, 2, 10, 5, 5);
    $exit = array(4, 5, 12, 9, 120);
    $n = count($arr1);
    findMaxGuests($arr1, $exit, $n);
      
// This code is contributed by anuj_67.
?>

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Output :

Maximum Number of Guests = 3 at time 5

Time Complexity of this method is O(nLogn).

Thanks to Gaurav Ahirwar for suggesting this method.

Another Efficient Solution :
Approach :
1). Create an auxiliary array used for storing dynamic data of starting and ending points.

2). Loop through the whole array of elements and increase the value at the starting point by 1 and similarly decrease the value after ending point by 1.
[Here we use the expressions “x[start[i]]-=1” and “x[end[i]+1]-=1”]

3). While looping, after calculating the auxiliary array: permanently add the value at current index and check for the maximum valued index traversing from left to right.

C++

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#include<bits/stdc++.h>
using namespace std;
  
void maxOverlap(vector<int>& start, vector<int>& end )
{
      
    int n= start.size(); 
      
    // Finding maximum starting time O(n)
    int maxa=*max_element(start.begin(), start.end()); 
  
    // Finding maximum ending time O(n)
    int maxb=*max_element(end.begin(), end.end());
  
    int maxc = max(maxa, maxb); 
      
    int x[maxc + 2];
    memset(x, 0, sizeof x);
          
        int cur = 0, idx;
          
        // Creating and auxiliary array O(n)
        for(int i = 0; i < n; i++)
        
            //Lazy addition
            ++x[start[i]];
            --x[end[i]+1];
        }
          
        int maxy = INT_MIN;
          
        //Lazily Calculating value at index i O(n)
        for(int i = 0; i <= maxc; i++)
        {
            cur += x[i];
            if(maxy < cur)
            
                maxy = cur;
                idx = i;
                  
            }         
        }
        cout<<"Maximum value is "<<maxy<<" at position "<<idx<<endl;
}
  
// Driver code
int main() 
{     
        vector<int> start = {13, 28, 29, 14, 40, 17, 3},
                    end = {107, 95, 111, 105, 70, 127, 74};
                      
        maxOverlap(start,end);
    return 0;
}

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Java

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import java.io.*;
import java.util.*;
import java.text.*;
import java.math.*;
import java.util.regex.*;
  
class GFG
{
  
public static void maxOverlap(int []start,int [] end ,int n)
{
    // Finding maximum starting time
    int maxa = Arrays.stream(start).max().getAsInt();
      
    // Finding maximum ending time
    int maxb = Arrays.stream(end).max().getAsInt();
      
    int maxc = Math.max(maxa,maxb); 
          
    int []x = new int[maxc + 2];
    Arrays.fill(x, 0);
          
    int cur=0,idx=0;
  
    // reating an auxiliary array
    for(int i = 0; i < n; i++)
    {
        // Lazy addition
        ++x[start[i]];
        --x[end[i]+1];
    }
          
    int maxy=Integer.MIN_VALUE;
      
    //Lazily Calculating value at index i
    for(int i = 0; i <= maxc; i++) 
    {
        cur+=x[i];
        if(maxy < cur)
        {
            maxy = cur;
            idx = i;
              
        }         
    }
        System.out.println("Maximum value is:"+
                        maxy+" at position: "+idx+"");
          
  
// Driver code
public static void main(String[] args) 
{
    int [] start = new int[]{13, 28, 29, 14, 40, 17, 3 };
    int [] end = new int[]{107, 95, 111, 105, 70, 127, 74};
    int n = start.length;
  
    maxOverlap(start,end,n);
}
}

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Python3

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import sys
   
def maxOverlap(start,end):
   
    n= len(start)
    maxa = max(start)# Finding maximum starting time
    maxb = max(end)  # Finding maximum ending time
    maxc=max(maxa,maxb)
    x =(maxc+2)*[0]
    cur=0; idx=0
   
    for i in range(0,n) :# CREATING AN AUXILIARY ARRAY
        x[start[i]]+=1 # Lazy addition
        x[end[i]+1]-=1
        
    maxy=-1
    #Lazily Calculating value at index i
    for i in range(0,maxc+1): 
        cur+=x[i]
        if maxy<cur :
            maxy=cur
            idx=i     
    print("Maximum value is: {0:d}".format(maxy),
                     " at position: {0:d}".format(idx))
if __name__ == "__main__":
       
    start=[13,28,29,14,40,17,3]
    end=[107,95,111,105,70,127,74]
                     
    maxOverlap(start,end)

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C#

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// C# implementation of above approach
using System;
using System.Linq;
  
class GFG
{
  
public static void maxOverlap(int []start,int [] end ,int n)
{
    // Finding maximum starting time
    int maxa = start.Max();
      
    // Finding maximum ending time
    int maxb = end.Max();
      
    int maxc = Math.Max(maxa,maxb); 
          
    int[] x = new int[maxc + 2];
          
    int cur = 0,idx = 0;
  
    // reating an auxiliary array
    for(int i = 0; i < n; i++)
    {
        // Lazy addition
        ++x[start[i]];
        --x[end[i]+1];
    }
          
    int maxy=int.MinValue;
      
    //Lazily Calculating value at index i
    for(int i = 0; i <= maxc; i++) 
    {
        cur+=x[i];
        if(maxy < cur)
        {
            maxy = cur;
            idx = i;
              
        }     
    }
        Console.WriteLine("Maximum value is "+
                        maxy+" at position: "+idx+"");
          
  
// Driver code
public static void Main() 
{
    int []start = {13, 28, 29, 14, 40, 17, 3 };
    int []end = {107, 95, 111, 105, 70, 127, 74};
    int n = start.Length;
  
    maxOverlap(start,end,n);
}
}
  
// This code is contributed by chandan_jnu

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PHP

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<?php
  
function maxOverlap($start, $end )
{
      
    $n= count($start);
      
    // Finding maximum starting time O(n)
    $maxa = max($start); 
  
    //Finding maximum ending time O(n)
    $maxb = max($end);
  
    $maxc = max($maxa, $maxb); 
      
    $x = array_fill(0, $maxc + 2, 0);
          
        $cur = 0;
          
        // Creating and auxiliary array O(n)
        for($i = 0; $i < $n; $i++)
        
            // Lazy addition
            ++$x[$start[$i]];
            --$x[$end[$i]+1];
        }
          
        $maxy=-PHP_INT_MAX;
          
        // Lazily Calculating value at index i O(n)
        for($i = 0; $i <= $maxc; $i++)
        {
            $cur += $x[$i];
            if($maxy < $cur)
            {
                $maxy = $cur;
                $idx = $i;
            }     
        }
        echo "Maximum value is ".$maxy." at position ".$idx."\n";
}
  
    // Driver code
        $start = array(13,28,29,14,40,17,3);
        $end = array(107,95,111,105,70,127,74);
                      
        maxOverlap($start,$end);
      
// This code is contributed by chandan_jnu
?>

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Output:

Maximum value is 7 at position 40

Time Complexity : O(n)
Auxiliary Space : O(n)
Thanks to Harshit Saini for suggesting this method.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



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