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Count of subarrays of size K which is a permutation of numbers from 1 to K

  • Last Updated : 31 May, 2021

Given an array arr of distinct integers, the task is to find the count of sub-arrays of size i having all elements from 1 to i, in other words, the sub-array is any permutation of elements from 1 to i, with 1 < = i <= N.

Examples:

Input: arr[] = {2, 3, 1, 5, 4} 
Output:
Explanation: 
we have {1}, {2, 3, 1} and {2, 3, 1, 5, 4} subarray for i=1, i=3, i=5 respectively. 
Permutation of size 4 and size 2 can’t be made because 5 and 3 are in the way respectively.

Input: arr[] = {1, 3, 5, 4, 2} 
Output:
Explanation: 
we have {1} and {1, 3, 5, 4, 2} subarray for i=1 and i=5 respectively.

A Naive approach is to start from each index and try to find the subarray of every size(i) and check whether all elements from 1 to i are present. 
Time complexity: O(N2)

An Efficient approach can be given by checking if it is possible to create a subarray of size i for every value of i from 1 to N.
As we know, every subarray of size K must be a permutation of all elements from 1 to K, knowing that we can look at the index of the numbers from 1 to N in order and calculate the index of the minimum and maximum values at every step.



  • If maximum_ind – minimum_ind + 1 = K, then we have a permutation of size K, else not.
  • Update the value of minimum_ind and maximum_ind at every step.

Time complexity: O(n)
Illustration:

Given Arr = {2, 3, 1, 5, 4}, let’s start with min_ind = INF and max_ind = -1

  1. index of 1 is 2, so min_ind = min(min_ind, 2) = 2 and max_ind = max(max_ind, 2) = 2, 
    2-2+1 = 1 so we have a permutation of size 1
  2. index of 2 is 0, so min_ind = min(min_ind, 0) = 0 and max_ind = max(max_ind, 0) = 2, 
    2-0+1 = 3 so we don’t have a permutation of size 2
  3. index of 3 is 1, so min_ind = min(min_ind, 1) = 0 and max_ind = max(max_ind, 1) = 2, 
    2-0+1 = 3 so we have a permutation of size 3
  4. index of 4 is 4, so min_ind = min(min_ind, 4) = 0 and max_ind = max(max_ind, 4) = 4, 
    4-0+1 = 5 so we don’t have a permutation of size 4
  5. index of 5 is 3, so min_ind = min(min_ind, 3) = 0 and max_ind = max(max_ind, 4) = 4, 
    4-0+1 = 5 so we have a permutation of size 5

So answer is 3

Below is the implementation of the above approach:

C++




// C++ program to implement
// the above approach
 
#include <iostream>
#include <unordered_map>
#include <vector>
using namespace std;
 
int find_permutations(vector<int>& arr)
{
    int cnt = 0;
    int max_ind = -1, min_ind = 10000000;
    int n = arr.size();
    unordered_map<int, int> index_of;
 
    // Save index of numbers of the array
    for (int i = 0; i < n; i++) {
        index_of[arr[i]] = i + 1;
    }
 
    for (int i = 1; i <= n; i++) {
 
        // Update min and max index
        // with the current index
        // and check if it's a valid permutation
        max_ind = max(max_ind, index_of[i]);
        min_ind = min(min_ind, index_of[i]);
        if (max_ind - min_ind + 1 == i)
            cnt++;
    }
 
    return cnt;
}
 
// Driver code
int main()
{
    vector<int> nums;
    nums.push_back(2);
    nums.push_back(3);
    nums.push_back(1);
    nums.push_back(5);
    nums.push_back(4);
 
    cout << find_permutations(nums);
 
    return 0;
}

Java




// Java program to implement
// the above approach
import java.util.*;
 
class GFG{
     
public static int find_permutations(
    Vector<Integer> arr)
{
    int cnt = 0;
    int max_ind = -1, min_ind = 10000000;
    int n = arr.size();
     
    HashMap<Integer,
            Integer> index_of = new HashMap<>();
             
    // Save index of numbers of the array
    for(int i = 0; i < n; i++)
    {
        index_of.put(arr.get(i), i + 1);
    }
 
    for(int i = 1; i <= n; i++)
    {
         
        // Update min and max index with
        // the current index and check
        // if it's a valid permutation
        max_ind = Math.max(max_ind, index_of.get(i));
        min_ind = Math.min(min_ind, index_of.get(i));
         
        if (max_ind - min_ind + 1 == i)
            cnt++;
    }
    return cnt;
}
 
// Driver Code
public static void main(String[] args)
{
    Vector<Integer> nums = new Vector<Integer>();
    nums.add(2);
    nums.add(3);
    nums.add(1);
    nums.add(5);
    nums.add(4);
     
    System.out.print(find_permutations(nums));
}
}
 
// This code is contributed by divyeshrabadiya07

Python3




# Python3 program to implement
# the above approach
def find_permutations(arr):
     
    cnt = 0
    max_ind = -1
    min_ind = 10000000;
     
    n = len(arr)
    index_of = {}
 
    # Save index of numbers of the array
    for i in range(n):
        index_of[arr[i]] = i + 1
 
    for i in range(1, n + 1):
 
        # Update min and max index with the
        # current index and check if it's a
        # valid permutation
        max_ind = max(max_ind, index_of[i])
        min_ind = min(min_ind, index_of[i])
         
        if (max_ind - min_ind + 1 == i):
            cnt += 1
             
    return cnt
 
# Driver code
if __name__ == "__main__":
 
    nums = []
    nums.append(2)
    nums.append(3)
    nums.append(1)
    nums.append(5)
    nums.append(4)
 
    print(find_permutations(nums))
 
# This code is contributed by chitranayal

C#




// C# program to implement
// the above approach
using System;
using System.Collections;
using System.Collections.Generic;
 
class GFG{
     
static int find_permutations(ArrayList arr)
{
    int cnt = 0;
    int max_ind = -1, min_ind = 10000000;
    int n = arr.Count;
     
    Dictionary<int,
               int> index_of = new Dictionary<int,
                                              int>();
             
    // Save index of numbers of the array
    for(int i = 0; i < n; i++)
    {
        index_of[(int)arr[i]] = i + 1;
    }
 
    for(int i = 1; i <= n; i++)
    {
         
        // Update min and max index with
        // the current index and check
        // if it's a valid permutation
        max_ind = Math.Max(max_ind, index_of[i]);
        min_ind = Math.Min(min_ind, index_of[i]);
         
        if (max_ind - min_ind + 1 == i)
            cnt++;
    }
    return cnt;
}
 
// Driver Code
public static void Main(string[] args)
{
    ArrayList nums = new ArrayList();
 
    nums.Add(2);
    nums.Add(3);
    nums.Add(1);
    nums.Add(5);
    nums.Add(4);
 
    Console.Write(find_permutations(nums));
}
}
 
// This code is contributed by rutvik_56

Javascript




<script>
// Javascript implementation
function find_permutations(arr)
{
    var cnt = 0;
    var max_ind = -1, min_ind = 10000000;
    var n = arr.length;
    var index_of = new Map();
   
    // Save index of numbers of the array
    for (var i = 0; i < n; i++) {
        index_of.set(arr[i], i + 1);
    }
   
    for (var i = 1; i <= n; i++) {
   
        // Update min and max index
        // with the current index
        // and check if it's a valid permutation
        max_ind = Math.max(max_ind, index_of.get(i));
        min_ind = Math.min(min_ind, index_of.get(i));
        if (max_ind - min_ind + 1 == i)
            cnt++;
    }
   
    return cnt;
}
  
var nums = [];
nums.push(2);
nums.push(3);
nums.push(1);
nums.push(5);
nums.push(4);
 
document.write(find_permutations(nums));
// This code contributed by shubhamsingh10
</script>
Output: 
3




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