Min difference between maximum and minimum element in all Y size subarrays

Given an array arr[] of size N and integer Y, the task is to find a minimum of all the differences between the maximum and minimum elements in all the sub-arrays of size Y.

Examples:

Input: arr[] = { 3, 2, 4, 5, 6, 1, 9 }  Y = 3
Output: 2
Explanation:
All subarrays of length = 3 are:
{3, 2, 4} where maximum element = 4 and  minimum element = 2  difference = 2
{2, 4, 5} where maximum element = 5 and  minimum element = 2  difference = 3
{4, 5, 6} where maximum element = 6 and  minimum element = 4  difference = 2
{5, 6, 1} where maximum element = 6 and  minimum element = 1  difference = 5
{6, 1, 9} where maximum element = 9 and  minimum element = 6  difference = 3 
Out of these minimum is 2. 
 

Input: arr[] = { 1, 2, 3, 3, 2, 2  } Y = 4
Output: 1
Explanation:
All subarrays of length = 4 are:
{1, 2, 3, 3} maximum element = 3 and  minimum element = 1  difference = 2
{2, 3, 3, 2} maximum element = 3 and  minimum element = 2  difference = 1
{3, 3, 2, 2} maximum element = 3 and  minimum element = 2  difference = 1 
Out of these minimum is 1.                          

 

Naive Approach: The naive idea is to traverse for every index i in the range [0, N – Y] use another loop to traverse from ith index to (i + Y – 1)th index and then calculate the minimum and maximum element in a subarray of size Y and hence calculate the difference of maximum and minimum element for that ith iteration. Finally, by keeping a check on differences, evaluate the minimum difference.



Time Complexity: O(N*Y)
Auxiliary Space: O(1)

Efficient Approach: The idea is to use the concept of the approach discussed in the Next Greater Element article. Below are the steps:

  1. Build two arrays maxarr[] and minarr[], where maxarr[] will store the index of the element which is next greater to the element at ith index and minarr[] will store the index of next elements which is less than the element at the ith index.
  2. Initialize a stack with 0 to store the indices in both the above cases.
  3. For each index, i in the range [0, N – Y], using a nested loop and sliding window approach form two arrays submax and submin. These arrays will store maximum and minimum elements in subarray in ith iteration.
  4. Finally, calculate the minimum difference.

Below is the implementation of the above approach:

C++

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// C++ program for the above approach 
#include <bits/stdc++.h> 
using namespace std; 
  
// Function to get the maximum of all 
// the subarrays of size Y 
vector<int> get_submaxarr(int* arr, 
                        int n, int y) 
{ 
    int j = 0; 
    stack<int> stk; 
  
    // ith index of maxarr array 
    // will be the index upto which 
    // Arr[i] is maximum 
    vector<int> maxarr(n); 
    stk.push(0); 
  
    for (int i = 1; i < n; i++) { 
  
        // Stack is used to find the 
        // next larger element and 
        // keeps track of index of 
        // current iteration 
        while (stk.empty() == false
            and arr[i] > arr[stk.top()]) { 
  
            maxarr[stk.top()] = i - 1; 
            stk.pop(); 
        } 
        stk.push(i); 
    } 
  
    // Loop for remaining indexes 
    while (!stk.empty()) { 
  
        maxarr[stk.top()] = n - 1; 
        stk.pop(); 
    } 
    vector<int> submax; 
  
    for (int i = 0; i <= n - y; i++) { 
  
        // j < i used to keep track 
        // whether jth element is 
        // inside or outside the window 
        while (maxarr[j] < i + y - 1 
            or j < i) { 
            j++; 
        } 
  
        submax.push_back(arr[j]); 
    } 
  
    // Return submax 
    return submax; 
} 
  
// Function to get the minimum for 
// all subarrays of size Y 
vector<int> get_subminarr(int* arr, 
                        int n, int y) 
{ 
    int j = 0; 
  
    stack<int> stk; 
  
    // ith index of minarr array 
    // will be the index upto which 
    // Arr[i] is minimum 
    vector<int> minarr(n); 
    stk.push(0); 
  
    for (int i = 1; i < n; i++) { 
  
        // Stack is used to find the 
        // next smaller element and 
        // keeping track of index of 
        // current iteration 
        while (stk.empty() == false
            and arr[i] < arr[stk.top()]) { 
  
            minarr[stk.top()] = i; 
            stk.pop(); 
        } 
        stk.push(i); 
    } 
  
    // Loop for remaining indexes 
    while (!stk.empty()) { 
  
        minarr[stk.top()] = n; 
        stk.pop(); 
    } 
  
    vector<int> submin; 
  
    for (int i = 0; i <= n - y; i++) { 
  
        // j < i used to keep track 
        // whether jth element is inside 
        // or outside the window 
  
        while (minarr[j] <= i + y - 1 
            or j < i) { 
            j++; 
        } 
  
        submin.push_back(arr[j]); 
    } 
  
    // Return submin 
    return submin; 
} 
  
// Function to get minimum difference 
void getMinDifference(int Arr[], int N, 
                    int Y) 
{ 
    // Create submin and submax arrays 
    vector<int> submin 
        = get_subminarr(Arr, N, Y); 
  
    vector<int> submax 
        = get_submaxarr(Arr, N, Y); 
  
    // Store initial difference 
    int minn = submax[0] - submin[0]; 
    int b = submax.size(); 
  
    for (int i = 1; i < b; i++) { 
  
        // Calculate temporary difference 
        int dif = submax[i] - submin[i]; 
        minn = min(minn, dif); 
    } 
  
    // Final minimum difference 
    cout << minn << "\n"; 
} 
  
// Driver Code 
int main() 
{ 
    // Given array arr[] 
    int arr[] = { 1, 2, 3, 3, 2, 2 }; 
    int N = sizeof arr / sizeof arr[0]; 
  
    // Given subarray size 
    int Y = 4; 
  
    // Function Call 
    getMinDifference(arr, N, Y); 
    return 0; 
}

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Python3














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# Python3 program for the above approach  


  


# Function to get the maximum of all  


# the subarrays of size Y  


def get_submaxarr(arr, n, y):  


      


    j = 0


    stk = [] 


      


    # ith index of maxarr array  


    # will be the index upto which  


    # Arr[i] is maximum  


    maxarr = [0] * n  


    stk.append(0)  


  


    for i in range(1, n): 


  


        # Stack is used to find the  


        # next larger element and  


        # keeps track of index of  


        # current iteration  


        while (len(stk) > 0 and 


                 arr[i] > arr[stk[-1]]): 


            maxarr[stk[-1]] = i - 1


            stk.pop()  


              


        stk.append(i) 


  


    # Loop for remaining indexes  


    while (stk): 


        maxarr[stk[-1]] = n - 1


        stk.pop() 


          


    submax = []  


      


    for i in range(n - y + 1): 


  


        # j < i used to keep track  


        # whether jth element is  


        # inside or outside the window  


        while (maxarr[j] < i + y - 1 or


                       j < i): 


            j += 1


              


        submax.append(arr[j]) 


  


    # Return submax  


    return submax 


  


# Function to get the minimum for  


# all subarrays of size Y  


def get_subminarr(arr, n, y): 


      


    j = 0


    stk = []  


      


    # ith index of minarr array  


    # will be the index upto which  


    # Arr[i] is minimum  


    minarr = [0] * n 


    stk.append(0) 


      


    for i in range(1 , n): 


          


        # Stack is used to find the  


        # next smaller element and  


        # keeping track of index of  


        # current iteration  


        while (stk and arr[i] < arr[stk[-1]]): 


            minarr[stk[-1]] = i 


            stk.pop()  


              


        stk.append(i)  


          


    # Loop for remaining indexes  


    while (stk): 


        minarr[stk[-1]] = n 


        stk.pop() 


          


    submin = [] 


      


    for i in range(n - y + 1): 


          


        # j < i used to keep track  


        # whether jth element is inside  


        # or outside the window  


        while (minarr[j] <= i + y - 1 or


                      j < i): 


            j += 1


              


        submin.append(arr[j]) 


          


    # Return submin  


    return submin  


  


# Function to get minimum difference  


def getMinDifference(Arr, N, Y): 


      


    # Create submin and submax arrays 


    submin = get_subminarr(Arr, N, Y) 


    submax = get_submaxarr(Arr, N, Y) 


      


    # Store initial difference  


    minn = submax[0] - submin[0] 


    b = len(submax) 


      


    for i in range(1, b): 


          


        # Calculate temporary difference  


        diff = submax[i] - submin[i] 


        minn = min(minn, diff) 


      


    # Final minimum difference  


    print(minn) 


      


# Driver code 


  


# Given array arr[] 


arr = [ 1, 2, 3, 3, 2, 2 ] 


N = len(arr) 


  


# Given subarray size 


Y = 4


  


# Function call 


getMinDifference(arr, N, Y) 


      


# This code is contributed by Stuti Pathak 



















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Output: 


1



 


Time Complexity: O(N)
Auxiliary Space: O(N)


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