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Count of Subarrays

Count of Subarrays: In this article, we are going to see how to find a valid number of subarrays based on some constraints? It's a very common interview problem often asked in interview s of Amazon. Flipkart.
Submitted by Radib Kar, on June 12, 2020

Problem statement:

Given an array of N positive integers a₁, a₂,  ..., a_n. The value of each contiguous subarray of a given array is the maximum element present in that subarray. The task is to return the number of subarrays having value strictly greater than K.

Input:

The first line contains two space-separated positive integers N and K denoting the size of the array and the value of K. The second line contains N space-separated positive integers denoting the elements of the array.

Output:

Output the number of subarrays having value strictly greater than K.

Constraints:

1 <= T <= 50
1 <= N <= 100
1 <= a[i] <= 105

Example:

Test case: 1

Input: 
5 3
3 4 5 2 7

Output:
13

Test case: 2
4 1
1 2 3 4

Output:
9

Explanation:

Test case 1: 
All possible subarrays are listed below 
with their respective value (maximum in the subarray)

3 -> 3
4 -> 4
5 -> 5
2 -> 2
7 -> 7
3, 4 -> 4
4, 5 -> 5
5, 2 -> 5
2, 7 -> 7
3, 4, 5 -> 5
4, 5, 2 -> 5
5, 2, 7 -> 7
3, 4, 5, 2 -> 5
4, 5, 2, 7 -> 7
3, 4, 5, 2, 7 -> 7

So, number of valid subarrays is 13

Solution Approach:

1. Create dp[n][n] to store value (maximum) of subarray;
2. Initialize count = 0 which will be our final result;
3. Base case computation (single length subarrays),

   for i=0 to n
       // since only one element in single length subarray, 
       // just check for that element
       if(a[i]>k) 
           dp[i][i]=a[i];
           count++;
       else
           dp[i][i]=-1; // or arr[i] itslef
   end for
   

4. Computing all length subarray cases,

   for subarray length,len=2 to n
       for start=0 to n-len
           end=start+len-1; // last element of subarray
           if(a[end]>k || dp[start][end-1]>k)
               dp[start][end]=std::max(a[end],dp[start][end-1]);
               count++;
           else
               dp[start][end]=-1;  
       end for
   end for
   

Okay, so the strategy is to compute for subarray a[start..end] with help of already computed one a[start..end-1].

Subarray a[start..end] will only make a count if a[start..end-1] already makes a count ( yes because, it's part of the subarray so, anything max in it would be max in the parent one too) Or if a[end]>k.

In both the cases maximum of the subarray, a[start..end] is greater than K

That's what we have done.

Below is illustration for our test case 1,

Input array: [3, 4, 5, 2, 7] and K=3.

So, let's compute the basic test case first

We need a 5X5 DP array for this

Starting to compute for other values.

Len=2

Start=0, end=1

Start=1, end=2

Start=2, end=3

Start=3, end=4

Len=3

Start=0, end=2

Start=1, end=3

Start=2, end=4

Len=4

Start=0, end=3

Start=1, end=4

Len=5

Start=0, end=3

Done!

Count is 13 (count of positive values in the array).

C++ Implementation:

#include <bits/stdc++.h>
using namespace std;

int maximum(int a, int b)
{
    return (a > b) ? a : b;
}

int subArray(vector<int> a, int n, int k)
{

    int dp[n][n];
    int count = 0;
    for (int i = 0; i < n; i++) {
        if (a[i] > k) {
            dp[i][i] = a[i];
            count++;
        }
        else
            dp[i][i] = -1; //or arr[i] itslef
    }

    for (int len = 2; len <= n; len++) {
        for (int start = 0; start <= n - len; start++) {
            int end = start + len - 1;
            if (a[end] > k || dp[start][end - 1] > k) {
                dp[start][end] = std::max(a[end], dp[start][end - 1]);
                count++;
            }
            else
                dp[start][end] = -1;
        }
    }

    return count;
}

int main()
{
    int n, item, k;

    cout << "Input size of array\n";
    cin >> n;
    
    cout << "Input k\n";
    cin >> k;
    
    cout << "Add the array elements\n";
    vector<int> a;
    
    for (int j = 0; j < n; j++) {
        scanf("%d", &item);
        a.push_back(item);
    }

    cout << "Total count of valid subarray is " << subArray(a, n, k) << endl;

    return 0;
}

Output:

Input size of array
5
Input k
3
Add the array elements
3 4 5 2 7
Total count of valid subarray is 13