Find the sum of medians of all odd length subarrays

Given an array arr[] of size N, the task is to find the sum of medians of all sub-array of odd-length.

Examples:

Input: arr[] = {4, 2, 5, 1}
Output: 18
Explanation : Sub-Arrays of odd length and their medians are :

• [4]  -> Median is 4
• [4, 2, 5]  -> Median is 4
• [2]  -> Median is 2
• [2, 5, 1]  -> Median is 2
• [5]  -> Median is 5
• [1]  -> Median is 1

Their sum = 4 + 4+ 2 + 2 + 5 +1 = 18

Input: arr[] = {1, 2}
Output: 3

Pre-requisites: Median of Stream of Running Integers using STL

Naive Approach: Generate each and every sub-array. If the length of the sub-array is odd, then sort the sub-array and return the middle element.

Below is the implementation of the above approach: 

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find sum of medians
// of all odd-length subarrays
int solve(vector<int> arr)
{
    int ans = 0;
    int n = arr.size();

    // Loop to calculate the sum
    for(int i = 0; i < n; i++)
    {
        vector<int> new_arr;
        for(int j = i; j < n; j++)
        {
            new_arr.push_back(arr[j]);

            // Odd length subarray
            if ((new_arr.size() % 2) == 1)
            {
                sort(new_arr.begin(), new_arr.end());
                int mid = new_arr.size() / 2;
                ans += new_arr[mid];
            }
        }
    }
    return ans;
}

// Driver Code
int main()
{
    vector<int> arr = { 4, 2, 5, 1 };
    cout << solve(arr);
}

// This code is contributed by Samim Hossain Mondal.

Java

// Java program for the above approach
import java.util.*;

class GFG {
    // Function to find sum of medians
    // of all odd-length subarrays
    static int solve(int[] arr) {
        int ans = 0;
        int n = arr.length;

        // Loop to calculate the sum
        for (int i = 0; i < n; i++) {
            List<Integer> new_arr = new LinkedList<Integer>();
            for (int j = i; j < n; j++) {
                new_arr.add(arr[j]);

                // Odd length subarray
                if ((new_arr.size() % 2) == 1) {
                    Collections.sort(new_arr);
                    int mid = new_arr.size() / 2;
                    ans += new_arr.get(mid);
                }
            }
        }
        return ans;
    }

    // Driver Code
    public static void main(String[] args) {
        int[] arr = { 4, 2, 5, 1 };
        System.out.println(solve(arr));
    }
}
// This code is contributed by 29AjayKumar

Python3

# Python program for the above approach

# Function to find sum of medians
# of all odd-length subarrays
def solve(arr):
        ans = 0
        n = len(arr)

        # Loop to calculate the sum
        for i in range(n):
            new_arr = []
            for j in range(i, n, 1):
                new_arr.append(arr[j])

                # Odd length subarray
                if (len(new_arr)) % 2 == 1:
                    new_arr.sort()
                    mid = len(new_arr)//2
                    ans += new_arr[mid]
        return (ans)

# Driver Code
if __name__ == "__main__":
    arr = [4, 2, 5, 1]
    print(solve(arr))

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG
{

  // Function to find sum of medians
  // of all odd-length subarrays
  static int solve(int[] arr) {
    int ans = 0;
    int n = arr.Length;

    // Loop to calculate the sum
    for (int i = 0; i < n; i++) {
      List<int> new_arr = new List<int>();
      for (int j = i; j < n; j++) {
        new_arr.Add(arr[j]);

        // Odd length subarray
        if ((new_arr.Count % 2) == 1) {
          new_arr.Sort();
          int mid = new_arr.Count / 2;
          ans += new_arr[mid];
        }
      }
    }
    return ans;
  }

  // Driver Code
  public static void Main() {
    int[] arr = { 4, 2, 5, 1 };
    Console.Write(solve(arr));
  }
}

// This code is contributed by Saurabh Jaiswal

Javascript

<script>

// javascript program for the above approach
    // Function to find sum of medians
    // of all odd-length subarrays
    function solve(arr) {
        var ans = 0;
        var n = arr.length;

        // Loop to calculate the sum
        for (var i = 0; i < n; i++) {
            var new_arr= new Array();
            for (var j = i; j < n; j++) {
                new_arr.push(arr[j]);

                // Odd length subarray
                if ((new_arr.length % 2) == 1) {
                    new_arr.sort();
                    var mid = Math.floor(new_arr.length / 2);

                    // document.write(mid);
                    ans += new_arr[mid];
                }
            }
        }
        return ans;
    }

    // Driver Code
var arr = [ 4, 2, 5, 1 ];
document.write(solve(arr));

// This code is contributed by shikhasingrajput
</script>

Output

18

Time Complexity: O(N³ * Log(N))
Auxiliary Space: O(N)

Note: Instead of sorting array each time, which costs (N*logN), insertion sort can be applied. But still, overall Time Complexity will be O(N³).

Efficient Approach: The median of the sorted array is the value separating the higher half from the lower half in the array. For finding out the median, we only need the middle element, rather than the entire sorted array. The approach of Median of Stream of Running Integers can be applied over here. Follow the steps mentioned below 

1. Use max and min heaps to calculate the running median.
2. Traverse each and every element in the array.
3. While creating a new subarray, add an element into the heaps and return median if the size is odd else return 0.
4. Max_heap is used to store lower half elements such that the maximum element is at the top and min_heap is used to store higher half elements such that the minimum element is at the top.
5. The difference between both the heaps should not be greater than one, and one extra element is always placed in max_heap.

Note: Here max_heap is implemented using min_heap, by just negating the values so that the maximum negative element can be popped.

Below is the implementation of the above approach:  

C++

// C++ program for the above approach
#include <bits/stdc++.h>

// Class to find median
class find_median {
  public:
  // Constructor to declare two heaps
  find_median()
  {
    // Store lower half elements such that
    // maximum element is at top
    max_heap
      = std::priority_queue<int, std::vector<int>,
    std::less<int> >();

    // Store higher half elements such that
    // minimum element is at top
    min_heap
      = std::priority_queue<int, std::vector<int>,
    std::less<int> >();
  }

  // Function to add element
  int add(int val)
  {

    // If max_heap is empty or current element
    // smaller than top of max_heap
    if (max_heap.empty() || max_heap.top() > val) {
      max_heap.push(-val);
    }

    else {
      min_heap.push(val);
    }

    // If size of max_heap + 1 less
    // than min_heap
    if (max_heap.size() + 1 > min_heap.size()) {
      val = -max_heap.top();
      max_heap.pop();
      min_heap.push(val);
    }

    // If size of min_heap
    // less than max_heap
    if (min_heap.size() > max_heap.size()) {
      val = min_heap.top();
      min_heap.pop();
      max_heap.push(-val);
    }

    // Finally if sum of sizes is odd,
    // return median
    if ((min_heap.size() + max_heap.size()) % 2 == 1) {
      return -max_heap.top();
    }

    // Else return 0
    else {
      return 0;
    }
  }

  std::priority_queue<int, std::vector<int>,
  std::less<int> >
    max_heap;
  std::priority_queue<int, std::vector<int>,
  std::less<int> >
    min_heap;
};

// Function to calculate the sum of all odd length subarrays
int solve(std::vector<int> arr)
{
  int ans = 0;

  // Size of the array
  int n = arr.size();
  for (int i = 0; i < n; i++) {
    // Create an object of class find_median
    find_median obj;

    for (int j = i; j < n; j++)
    {

      // Add value to the heaps using object
      int val = obj.add(arr[j]);
      ans += val;
    }
  }

  return (ans);
}

// Driver Code
int main()
{
  std::vector<int> arr = { 4, 2, 5, 1 };
  std::cout << solve(arr);
  return 0;
}

// This code is contributed by phasing17.

Java

import java.util.*;

// A class to find the median of the array
class FindMedian {

  // Declare two heaps
  // Store lower half elements such that
  // maximum element is at top
  private List<Integer> max_heap;

  // Store higher half elements such that
  // minimum element is at top
  private List<Integer> min_heap;

  // Constructor to initialize the heaps
  public FindMedian() {
    max_heap = new ArrayList<>();
    min_heap = new ArrayList<>();
  }

  public int add(int val) {
    // len(max_heap) == 0 or curr_element
    // smaller than max_heap top
    if (max_heap.size() == 0 || max_heap.get(0) > val) {
      max_heap.add(-val);
    }
    else {
      min_heap.add(val);
    }

    // If size of max_heap + 1 greater
    // than min_heap
    if (max_heap.size() + 1 > min_heap.size()) {
      int v = max_heap.get(max_heap.size() - 1);
      max_heap.remove(max_heap.size() - 1);
      min_heap.add(-v);
    }

    // If size of min_heap
    // greater than max_heap
    if (min_heap.size() > max_heap.size()) {
      int v = min_heap.get(min_heap.size() - 1);
      min_heap.remove(min_heap.size() - 1);
      max_heap.add(-v);
    }

    // Finally if sum of sizes is odd,
    // return median
    if ((min_heap.size() + max_heap.size()) % 2 == 1) {
      return (-max_heap.get(0));
    }

    // Else return 0
    else {
      return 0;
    }
  }
}

class GFG {
  // Function to calculate the sum
  // of all odd length subarrays
  public static int solve(int[] arr) {
    int ans = 0;

    // Size of the array
    int n = arr.length;

    for (int i = 0; i < n; i++) {
      // Create an object
      // of class FindMedian
      FindMedian obj = new FindMedian();
      for (int j = i; j < n; j++) {
        // Add value to the heaps
        // using object
        int val = obj.add(arr[j]);
        ans += val;
      }
    }

    return (ans);
  }

  // Driver Code
  public static void main(String[] args) {
    int[] arr = { 4, 2, 5, 1 };
    System.out.println(solve(arr));
  }
}

Python3

# Python program for the above approach
from heapq import heappush as push, heappop as pop

# Find the sum of medians of odd-length
# subarrays
class find_median():

    # Constructor to declare two heaps
    def __init__(self):

        # Store lower half elements such that
        # maximum element is at top
        self.max_heap = []

        # Store higher half elements such that
        # minimum element is at top
        self.min_heap = []

    def add(self, val):

        # len(max_heap) == 0 or curr_element
        # smaller than max_heap top
        if (len(self.max_heap) == 0 or
            self.max_heap[0] > val):
            push(self.max_heap, -val)

        else:
            push(self.min_heap, val)

        # If size of max_heap + 1 greater
        # than min_heap
        if (len(self.max_heap)+1 >
            len(self.min_heap)):
            val = pop(self.max_heap)
            push(self.min_heap, -val)

        # If size of min_heap
        # greater than max_heap
        if (len(self.min_heap) >
            len(self.max_heap)):
            val = pop(self.min_heap)
            push(self.max_heap, -val)

        # Finally if sum of sizes is odd,
        # return median
        if (len(self.min_heap) +
            len(self.max_heap)) % 2 == 1:
            return (-self.max_heap[0])

        # Else return 0
        else:
            return 0

# Function to calculate the sum
# of all odd length subarrays
def solve(arr):
    ans = 0

    # Size of the array
    n = len(arr)
    for i in range(n):

        # Create an object
        # of class find_median
        obj = find_median()
        for j in range(i, n, 1):

            # Add value to the heaps
            # using object
            val = obj.add(arr[j])
            ans += val

    return (ans)

# Driver Code
if __name__ == "__main__":
    arr = [4, 2, 5, 1]
    print(solve(arr))

C#

// C# Program for the above approach
using System;
using System.Collections.Generic;

// A class to find the median of the array
public class FindMedian
{

  // Declare two heaps
  // Store lower half elements such that
  // maximum element is at top
  private List<int> max_heap;

  // Store higher half elements such that
  // minimum element is at top
  private List<int> min_heap;

  // Constructor to initialize the heaps
  public FindMedian()
  {
    max_heap = new List<int>();
    min_heap = new List<int>();
  }

  public int Add(int val)
  {
    // len(max_heap) == 0 or curr_element
    // smaller than max_heap top
    if (max_heap.Count == 0 || max_heap[0] > val) {
      max_heap.Add(-val);
    }
    else {
      min_heap.Add(val);
    }

    // If size of max_heap + 1 greater
    // than min_heap
    if (max_heap.Count + 1 > min_heap.Count) {
      int v = max_heap[max_heap.Count - 1];
      max_heap.RemoveAt(max_heap.Count - 1);
      min_heap.Add(-v);
    }

    // If size of min_heap
    // greater than max_heap
    if (min_heap.Count > max_heap.Count) {
      int v = min_heap[min_heap.Count - 1];
      min_heap.RemoveAt(min_heap.Count - 1);
      max_heap.Add(-v);
    }

    // Finally if sum of sizes is odd,
    // return median
    if ((min_heap.Count + max_heap.Count) % 2 == 1) {
      return (-max_heap[0]);
    }

    // Else return 0
    else {
      return 0;
    }
  }
}

public class GFG {
  // Function to calculate the sum
  // of all odd length subarrays
  public static int Solve(int[] arr)
  {
    int ans = 0;

    // Size of the array
    int n = arr.Length;

    for (int i = 0; i < n; i++) {
      // Create an object
      // of class find_median
      FindMedian obj = new FindMedian();
      for (int j = i; j < n; j++) {
        // Add value to the heaps
        // using object
        int val = obj.Add(arr[j]);
        ans += val;
      }
    }

    return (ans);
  }

  // Driver Code
  public static void Main()
  {
    int[] arr = { 4, 2, 5, 1 };
    Console.WriteLine(Solve(arr));
  }
}

// This code is contributed by vinayetbi1

Javascript

// JavaScript Program for the above approach

// A class to find the median of the array
class find_median {
    // Constructor to declare two heaps
    constructor() {
        // Store lower half elements such that
        // maximum element is at top
        this.max_heap = [];

        // Store higher half elements such that
        // minimum element is at top
        this.min_heap = [];
    }

    add(val) {
        // len(max_heap) == 0 or curr_element
        // smaller than max_heap top
        if (this.max_heap.length == 0 || this.max_heap[0] > val) {
            this.max_heap.push(-val);
        }
        else {
            this.min_heap.push(val);
        }

        // If size of max_heap + 1 greater
        // than min_heap
        if (this.max_heap.length + 1 > this.min_heap.length) {
            let val = this.max_heap.pop();
            this.min_heap.push(-val);
        }

        // If size of min_heap
        // greater than max_heap
        if (this.min_heap.length > this.max_heap.length) {
            let val = this.min_heap.pop();
            this.max_heap.push(-val);
        }

        // Finally if sum of sizes is odd,
        // return median
        if ((this.min_heap.length + this.max_heap.length) % 2 === 1) {
            return (-this.max_heap[0]);
        }

        // Else return 0
        else {
            return 0;
        }
    }
}

// Function to calculate the sum
// of all odd length subarrays
function solve(arr) {
    let ans = 0;

    // Size of the array
    let n = arr.length;
    for (let i = 0; i < n; i++) {

        // Create an object
        // of class find_median
        let obj = new find_median();
        for (let j = i; j < n; j++) {

            // Add value to the heaps
            // using object
            let val = obj.add(arr[j]);
            ans += val;
        }
    }

    return (ans);
}

// Driver Code
let arr = [4, 2, 5, 1];
console.log(solve(arr));

// This code is contributed by vinayetbi1.

Output

18

Time Complexity: O(N² * Log(N))
Auxiliary Space: O(N)