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# 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++

 // Program to find maximum guest at any time in a party#include#includeusing 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 functionint 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;}

## Java

 // Java Program to find maximum guest// at any time in a partyimport 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

## Python3

 # Program to find maximum guest# at any time in a partydef 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 Codearrl = [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

## C#

 // C# Program to find maximum guest// at any time in a partyusing 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.

## PHP

 \$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.?>

## Javascript



Output :

Maximum Number of Guests = 3 at time 5

Time Complexity: O(nLogn).
Auxiliary Space: O(1) as no extra space has been taken.

Another Efficient Solution :

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++

 #includeusing namespace std; void maxOverlap(vector& start, vector& 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 "< start = {13, 28, 29, 14, 40, 17, 3},                    end = {107, 95, 111, 105, 70, 127, 74};                             maxOverlap(start,end);    return 0;}

## Java

 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;     // Creating 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 codepublic 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);}}

## Python3

 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

## C#

 // C# implementation of above approachusing 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;     // Creating 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 codepublic 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



## Javascript



Output

Maximum value is 7 at position 40