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You are given a 0-indexed integer array nums of length n. A triplet (i, j, k) is considered a mountain triplet if the following conditions hold:

Your goal is to find the minimum sum of a mountain triplet or return -1 if no mountain triplet exists.

Example

Input: nums = [1, 3, 1, 2, 4, 1]
Output: 5
Explanation: The triplet (0, 1, 2) is a mountain triplet with the sum 1 + 3 + 1 = 5.

Clarifying Questions

  1. Input Size: What is the expected range of n (length of nums)?
  2. Element Range: What is the range of values for elements in nums? Are they positive, negative, or both?
  3. Edge Cases: Should we consider cases where n is less than 3 and return -1?

Once these clarifications are made, we can proceed with the solution.

Solution Code

Strategy

  1. Initialization: We’ll initialize two arrays, left_min and right_min, where:
    • left_min[j] will store the minimum value to the left of j for all valid i.
    • right_min[j] will store the minimum value to the right of j for all valid k.
  2. Fill Arrays:
    • Traverse the nums array to fill up the left_min and right_min arrays.
  3. Finding Minimum Triplet Sum:
    • Iterate through the array with j as the middle element of the triplet.
    • For each valid j, look at the potential values from left_min and right_min to calculate the triplet sum.
    • Track the minimum sum found.
  4. Edge Cases:
    • If no mountain triplets are found, return -1.

Time Complexity

Implementation

def minimum_sum_mountain_triplet(nums):
    n = len(nums)
    
    if n < 3:
        return -1
    
    left_min = [None] * n
    right_min = [None] * n
    
    left_min[0] = float('inf')
    for j in range(1, n):
        left_min[j] = min(left_min[j - 1], nums[j - 1])

    right_min[n - 1] = float('inf')
    for j in range(n - 2, -1, -1):
        right_min[j] = min(right_min[j + 1], nums[j + 1])
    
    min_sum = float('inf')
    found = False
    
    for j in range(1, n - 1):
        if left_min[j] < nums[j] and nums[j] > right_min[j]:
            found = True
            min_sum = min(min_sum, left_min[j] + nums[j] + right_min[j])
    
    return min_sum if found else -1

# Example usage
nums = [1, 3, 1, 2, 4, 1]
print(minimum_sum_mountain_triplet(nums))  # Output: 5

Steps for Clarification

  1. Confirm the max bounds of n.
  2. Validate expected ranges for nums elements.
  3. Establish that the return value for no valid triplet is indeed -1.

After the initial confirmation and understanding, the approach is optimized for both run-time and space complexity.

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