algoadvance

Given an integer array nums, return the largest perimeter of a triangle with a non-zero area, formed from three of these lengths. If it is impossible to form any triangle of a non-zero area, return 0.

Clarifying Questions

  1. Input Size: Are there any constraints on the length of the input array nums?
    • Typically nums.length would range from 3 to 10^4.
  2. Element Constraints: Are the values in nums guaranteed to be positive integers?
    • Yes, nums consists of positive integers.
  3. Edge Cases: Should the function handle small input sizes (less than 3)?
    • If the input array length is less than 3, we can directly return 0 since it’s impossible to form a triangle.

Strategy

To form a valid triangle with sides a, b, and c (where a <= b <= c), the triangle inequality theorem states that a + b > c. The goal is to find the largest possible perimeter under this constraint.

  1. Sorting: Start by sorting the array in descending order to maximize the perimeter. Consider the first three elements and check if they can form a triangle.

  2. Greedy Approach: Iterate over the sorted list, always checking the first three elements that haven’t been ruled out. If they satisfy the triangle condition, their sum is the maximum possible perimeter.

  3. Early Termination: Once a valid triangle is found, return its perimeter immediately since sorting in descending order ensures maximum perimeter is considered first.

Code

def largestPerimeter(nums):
    # Sort the numbers in descending order
    nums.sort(reverse=True)
    
    # Check each triplet of consecutive elements
    for i in range(len(nums) - 2):
        if nums[i] < nums[i + 1] + nums[i + 2]:
            # Found a valid triangle, return its perimeter
            return nums[i] + nums[i + 1] + nums[i + 2]
    
    # No valid triangle found
    return 0

Time Complexity

Overall, the time complexity of the solution is O(n log n) due to the sorting step. The subsequent scan is linear and thus does not affect the leading term.

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