#Minimum Length Triangulation

import math

def distance(point1, point2):
    x1, y1 = point1
    x2, y2 = point2
    return math.sqrt((x2 - x1) ** 2 + (y2 - y1) ** 2)

def minimum_length_triangulation(points):
    n = len(points)
    dp = [[0] * n for _ in range(n)]

    for gap in range(2, n):
        for i in range(n - gap):
            j = i + gap
            dp[i][j] = math.inf
            for k in range(i + 1, j):
                cost = dp[i][k] + dp[k][j] + distance(points[i], points[k]) + distance(points[k], points[j])
                dp[i][j] = min(dp[i][j], cost)

    return dp[0][n - 1]

# Example usage
if __name__ == '__main__':
    # Example points
    points = [(0, 0), (1, 0), (0, 1), (1, 1)]

    # Example maximum length
    maximum_length = 3.0

    # Calculate the minimum length triangulation
    minimum_length = minimum_length_triangulation(points)

    # Check if it satisfies the maximum length constraint
    result = minimum_length <= maximum_length

    # Print the result
    if result:
        print("A triangulation with total length", minimum_length, "or less exists.")
    else:
        print("No triangulation with the given maximum length constraint exists.")

