algoadvance

Let’s simplify it: You are given an array of integers. You need to find the minimum possible average of the largest and smallest elements from every possible contiguous subarray of the given array.

Clarifying Questions

Before we move forward, I need to ensure that we have all the necessary details:

  1. Are there any constraints on the size of the array?
  2. Can the array contain negative numbers?
  3. Should the subarrays include the original array itself?

Assuming a general array with typical constraints (1 ≤ size of array ≤ 10^5 and integers can be negative):

Strategy

  1. Brute Force Approach: Loop through all possible subarrays, identify the maximum and minimum of each, compute their average, and track the smallest average. This approach will be computationally expensive as the number of subarrays for an array of size n is n(n+1)/2.

  2. Optimized Approach: Use sliding window or data structures like deques to maintain maximum and minimum in a window. This way, we can manage the maximum and minimum efficiently without recalculating for every subarray.

We will implement the brute force approach first for simplicity, and then discuss an optimized approach if needed.

Code

Here is the brute force approach:

# Import necessary modules
import sys

def min_avg_smallest_largest(arr):
    n = len(arr)
    min_avg = sys.maxsize
    
    # Loop through all possible subarrays
    for i in range(n):
        current_min = arr[i]
        current_max = arr[i]
        for j in range(i, n):
            # Update the minimum and maximum for the current subarray
            current_min = min(current_min, arr[j])
            current_max = max(current_max, arr[j])
            
            # Calculate the current average
            current_avg = (current_min + current_max) / 2
            
            # Update the minimum average found
            min_avg = min(min_avg, current_avg)
    
    return min_avg

# Example usage
arr = [1, 3, 2]
print(min_avg_smallest_largest(arr))  # Output should be 1.0

Optimized Approach (Sliding Window using Deques)

To optimize, we can use two deques to maintain the minimum and maximum of the current subarray efficiently.

from collections import deque

def min_avg_smallest_largest(arr):
    n = len(arr)
    min_avg = float('inf')
    
    for length in range(1, n + 1):
        min_deque = deque()
        max_deque = deque()
        
        for i in range(length):
            while min_deque and arr[min_deque[-1]] >= arr[i]:
                min_deque.pop()
            while max_deque and arr[max_deque[-1]] <= arr[i]:
                max_deque.pop()
                
            min_deque.append(i)
            max_deque.append(i)
        
        for i in range(length, n):
            min_avg = min(min_avg, (arr[min_deque[0]] + arr[max_deque[0]]) / 2)
            
            while min_deque and min_deque[0] <= i - length:
                min_deque.popleft()
            while max_deque and max_deque[0] <= i - length:
                max_deque.popleft()
            
            while min_deque and arr[min_deque[-1]] >= arr[i]:
                min_deque.pop()
            while max_deque and arr[max_deque[-1]] <= arr[i]:
                max_deque.pop()
                
            min_deque.append(i)
            max_deque.append(i)
        
        min_avg = min(min_avg, (arr[min_deque[0]] + arr[max_deque[0]]) / 2)
    
    return min_avg

# Example usage
arr = [1, 3, 2]
print(min_avg_smallest_largest(arr))  # Output should be 1.0

Time Complexity

Implementing and testing the optimized approach will ensure better performance with larger arrays.

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