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# Given an array and two integers l and r, find the kth largest element in the range [l, r]

Given an unsorted array arr[] of n integers and an integer k, the task is to find the kth largest element in the given index range [l, r]
Examples:

Input: arr[] = {5, 3, 2, 4, 1}, k = 4, l = 1, r = 5
Output:
4 will be the 4th element when arr[0…4] is sorted.
Input: arr[] = {1, 4, 2, 3, 5, 7, 6}, k = 3, l = 3, r = 6
Output:

Approach: A naive solution will be to sort the elements in the range and get the kth largest element, the time complexity of that solution will be nlog(n) for every query. We can solve each query in log(n) by using prefix array and binary search. All we have to do is maintain a 2d prefix array in which the ith row will contain number of elements less than equal to i in the same range as in the given array. After the prefix array is done all we need to do is a simple binary search over the prefix array. Hence the time complexity is drastically reduced.
Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;``#define MAX 1001``static` `int` `prefix[MAX][MAX];``int` `ar[MAX];` `// Function to calculate the prefix``void` `cal_prefix(``int` `n, ``int` `arr[])``{``    ``int` `i, j;` `    ``// Creating one based indexing``    ``for` `(i = 0; i < n; i++)``        ``ar[i + 1] = arr[i];` `    ``// Initializing and creating prefix array``    ``for` `(i = 1; i <= 1000; i++) {``        ``for` `(j = 0; j <= n; j++)``            ``prefix[i][j] = 0;` `        ``for` `(j = 1; j <= n; j++) {` `            ``// Creating a prefix array for every``            ``// possible value in a given range``            ``prefix[i][j] = prefix[i][j - 1]``                           ``+ (``int``)(ar[j] <= i ? 1 : 0);``        ``}``    ``}``}` `// Function to return the kth largest element``// in the index range [l, r]``int` `ksub(``int` `l, ``int` `r, ``int` `n, ``int` `k)``{``    ``int` `lo, hi, mid;` `    ``lo = 1;``    ``hi = 1000;` `    ``// Binary searching through the 2d array``    ``// and only checking the range in which``    ``// the sub array is a part``    ``while` `(lo + 1 < hi) {``        ``mid = (lo + hi) / 2;``        ``if` `(prefix[mid][r] - prefix[mid][l - 1] >= k)``            ``hi = mid;``        ``else``            ``lo = mid + 1;``    ``}` `    ``if` `(prefix[lo][r] - prefix[lo][l - 1] >= k)``        ``hi = lo;` `    ``return` `hi;``}` `// Driver code``int` `main()``{``    ``int` `arr[] = { 1, 4, 2, 3, 5, 7, 6 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);``    ``int` `k = 4;` `    ``// Creating the prefix array``    ``// for the given array``    ``cal_prefix(n, arr);` `    ``// Queries``    ``int` `queries[][3] = { { 1, n, 1 },``                         ``{ 2, n - 2, 2 },``                         ``{ 3, n - 1, 3 } };``    ``int` `q = ``sizeof``(queries) / ``sizeof``(queries[0]);` `    ``// Perform queries``    ``for` `(``int` `i = 0; i < q; i++)``        ``cout << ksub(queries[i][0], queries[i][1],``                     ``n, queries[i][2])``             ``<< endl;` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;` `class` `GFG``{``    ` `static` `int` `MAX = ``1001``;``static` `int` `prefix[][] = ``new` `int``[MAX][MAX];``static` `int` `ar[] = ``new` `int``[MAX];` `// Function to calculate the prefix``static` `void` `cal_prefix(``int` `n, ``int` `arr[])``{``    ``int` `i, j;` `    ``// Creating one based indexing``    ``for` `(i = ``0``; i < n; i++)``        ``ar[i + ``1``] = arr[i];` `    ``// Initializing and creating prefix array``    ``for` `(i = ``1``; i <= ``1000``; i++)``    ``{``        ``for` `(j = ``0``; j <= n; j++)``            ``prefix[i][j] = ``0``;` `        ``for` `(j = ``1``; j <= n; j++)``        ``{` `            ``// Creating a prefix array for every``            ``// possible value in a given range``            ``prefix[i][j] = prefix[i][j - ``1``]``                        ``+ (``int``)(ar[j] <= i ? ``1` `: ``0``);``        ``}``    ``}``}` `// Function to return the kth largest element``// in the index range [l, r]``static` `int` `ksub(``int` `l, ``int` `r, ``int` `n, ``int` `k)``{``    ``int` `lo, hi, mid;` `    ``lo = ``1``;``    ``hi = ``1000``;` `    ``// Binary searching through the 2d array``    ``// and only checking the range in which``    ``// the sub array is a part``    ``while` `(lo + ``1` `< hi)``    ``{``        ``mid = (lo + hi) / ``2``;``        ``if` `(prefix[mid][r] - prefix[mid][l - ``1``] >= k)``            ``hi = mid;``        ``else``            ``lo = mid + ``1``;``    ``}` `    ``if` `(prefix[lo][r] - prefix[lo][l - ``1``] >= k)``        ``hi = lo;` `    ``return` `hi;``}` `// Driver code``public` `static` `void` `main(String args[])``{``    ``int` `arr[] = { ``1``, ``4``, ``2``, ``3``, ``5``, ``7``, ``6` `};``    ``int` `n = arr.length;``    ``int` `k = ``4``;` `    ``// Creating the prefix array``    ``// for the given array``    ``cal_prefix(n, arr);` `    ``// Queries``    ``int` `queries[][] = { { ``1``, n, ``1` `},``                        ``{ ``2``, n - ``2``, ``2` `},``                        ``{ ``3``, n - ``1``, ``3` `} };``    ``int` `q = queries.length;` `    ``// Perform queries``    ``for` `(``int` `i = ``0``; i < q; i++)``        ``System.out.println( ksub(queries[i][``0``], queries[i][``1``],``                    ``n, queries[i][``2``]));``}``}` `// This code is contributed by Arnab Kundu`

## Python3

 `# Python3 implementation of the approach` `MAX` `=` `1001``prefix ``=` `[[``0` `for` `i ``in` `range``(``MAX``)]``             ``for` `j ``in` `range``(``MAX``)]``ar ``=` `[``0` `for` `i ``in` `range``(``MAX``)]` `# Function to calculate the prefix``def` `cal_prefix(n, arr):``    ` `    ``# Creating one based indexing``    ``for` `i ``in` `range``(n):``        ``ar[i ``+` `1``] ``=` `arr[i]` `    ``# Initializing and creating prefix array``    ``for` `i ``in` `range``(``1``, ``1001``, ``1``):``        ``for` `j ``in` `range``(n ``+` `1``):``            ``prefix[i][j] ``=` `0` `        ``for` `j ``in` `range``(``1``, n ``+` `1``):``            ` `            ``# Creating a prefix array for every``            ``# possible value in a given range``            ``if` `ar[j] <``=` `i:``                ``k ``=` `1``            ``else``:``                ``k ``=` `0``            ``prefix[i][j] ``=` `prefix[i][j ``-` `1``] ``+` `k` `# Function to return the kth largest element``# in the index range [l, r]``def` `ksub(l, r, n, k):``    ``lo ``=` `1``    ``hi ``=` `1000` `    ``# Binary searching through the 2d array``    ``# and only checking the range in which``    ``# the sub array is a part``    ``while` `(lo ``+` `1` `< hi):``        ``mid ``=` `int``((lo ``+` `hi) ``/` `2``)``        ``if` `(prefix[mid][r] ``-``            ``prefix[mid][l ``-` `1``] >``=` `k):``            ``hi ``=` `mid``        ``else``:``            ``lo ``=` `mid ``+` `1` `    ``if` `(prefix[lo][r] ``-``        ``prefix[lo][l ``-` `1``] >``=` `k):``        ``hi ``=` `lo` `    ``return` `hi` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ``arr ``=` `[``1``, ``4``, ``2``, ``3``, ``5``, ``7``, ``6``]``    ``n ``=` `len``(arr)``    ``k ``=` `4` `    ``# Creating the prefix array``    ``# for the given array``    ``cal_prefix(n, arr)` `    ``# Queries``    ``queries ``=` `[[``1``, n, ``1``],``               ``[``2``, n ``-` `2``, ``2``],``               ``[``3``, n ``-` `1``, ``3``]]``    ``q ``=` `len``(queries)` `    ``# Perform queries``    ``for` `i ``in` `range``(q):``        ``print``(ksub(queries[i][``0``],``                   ``queries[i][``1``], n, queries[i][``2``]))``        ` `# This code is contributed by``# Surendra_Gangwar`

## C#

 `// C# implementation of the approach``using` `System;` `class` `GFG``{``    ` `static` `int` `MAX = 1001;``static` `int``[,] prefix = ``new` `int``[MAX,MAX];``static` `int``[] ar = ``new` `int``[MAX];` `// Function to calculate the prefix``static` `void` `cal_prefix(``int` `n, ``int``[] arr)``{``    ``int` `i, j;` `    ``// Creating one based indexing``    ``for` `(i = 0; i < n; i++)``        ``ar[i + 1] = arr[i];` `    ``// Initializing and creating prefix array``    ``for` `(i = 1; i <= 1000; i++)``    ``{``        ``for` `(j = 0; j <= n; j++)``            ``prefix[i, j] = 0;` `        ``for` `(j = 1; j <= n; j++)``        ``{` `            ``// Creating a prefix array for every``            ``// possible value in a given range``            ``prefix[i, j] = prefix[i, j - 1]``                        ``+ (``int``)(ar[j] <= i ? 1 : 0);``        ``}``    ``}``}` `// Function to return the kth largest element``// in the index range [l, r]``static` `int` `ksub(``int` `l, ``int` `r, ``int` `n, ``int` `k)``{``    ``int` `lo, hi, mid;` `    ``lo = 1;``    ``hi = 1000;` `    ``// Binary searching through the 2d array``    ``// and only checking the range in which``    ``// the sub array is a part``    ``while` `(lo + 1 < hi)``    ``{``        ``mid = (lo + hi) / 2;``        ``if` `(prefix[mid, r] - prefix[mid, l - 1] >= k)``            ``hi = mid;``        ``else``            ``lo = mid + 1;``    ``}` `    ``if` `(prefix[lo, r] - prefix[lo, l - 1] >= k)``        ``hi = lo;` `    ``return` `hi;``}` `// Driver code``static` `void` `Main()``{``    ``int` `[]arr = { 1, 4, 2, 3, 5, 7, 6 };``    ``int` `n = arr.Length;``    ``//int k = 4;` `    ``// Creating the prefix array``    ``// for the given array``    ``cal_prefix(n, arr);` `    ``// Queries``    ``int` `[,]queries = { { 1, n, 1 },``                        ``{ 2, n - 2, 2 },``                        ``{ 3, n - 1, 3 } };``    ``int` `q = queries.Length/queries.Rank-1;` `    ``// Perform queries``    ``for` `(``int` `i = 0; i < q; i++)``        ``Console.WriteLine( ksub(queries[i,0], queries[i,1],``                    ``n, queries[i, 2]));``}``}` `// This code is contributed by mits`

## PHP

 `= ``\$k``)``            ``\$hi` `= ``\$mid``;``        ``else``            ``\$lo` `= ``\$mid` `+ 1;``    ``}` `    ``if` `(``\$prefix``[``\$lo``][``\$r``] - ``\$prefix``[``\$lo``][``\$l` `- 1] >= ``\$k``)``        ``\$hi` `= ``\$lo``;` `    ``return` `\$hi``;``}` `    ``// Driver code``    ``\$arr` `= ``array``( 1, 4, 2, 3, 5, 7, 6 );``    ``\$n` `= ``count``(``\$arr``);``    ``\$k` `= 4;` `    ``// Creating the prefix array``    ``// for the given array``    ``cal_prefix(``\$n``, ``\$arr``);` `    ``// Queries``    ``\$queries` `= ``array``(``array``( 1, ``\$n``, 1 ),``                        ``array``( 2, ``\$n` `- 2, 2 ),``                        ``array``( 3, ``\$n` `- 1, 3 ));``    ``\$q` `= ``count``(``\$queries``);` `    ``// Perform queries``    ``for` `(``\$i` `= 0; ``\$i` `< ``\$q``; ``\$i``++)``        ``echo` `ksub(``\$queries``[``\$i``][0], ``\$queries``[``\$i``][1],``\$n``, ``\$queries``[``\$i``][2]).``"\n"``;` `    ``// This code is contributed by mits``?>`

## Javascript

 ``

Output:

```1
3
5```

Time Complexity : O(n + q*log(MAX)) ,where n is the size of the array, q is the number of queries and MAX is the number of rows or columns of 2D prefix array.

Space Complexity : O(MAX*MAX) , to store elements in prefix matrix.