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Clarifying Questions

  1. Input Constraints:
    • What is the maximum number of points provided?
    • Are all points unique, or can there be duplicates?
  2. Output Format:
    • Should the area be returned with a specific precision, or is any floating-point number acceptable?

Strategy

To find the largest area of a triangle from a set of points, we need to consider every combination of three points and calculate the area of the triangle those points form. The maximum area from these calculations will be our answer.

The area of a triangle given vertices ( (x1, y1), (x2, y2), (x3, y3) ) can be calculated using the determinant formula: [ \text{Area} = 0.5 \times | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) | ]

Code

Below is the Python code to find the largest triangle area:

from itertools import combinations

def largestTriangleArea(points):
    def triangle_area(p1, p2, p3):
        x1, y1 = p1
        x2, y2 = p2
        x3, y3 = p3
        return 0.5 * abs(x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2))
    
    max_area = 0
    for p1, p2, p3 in combinations(points, 3):
        max_area = max(max_area, triangle_area(p1, p2, p3))
    
    return max_area

# Test case
points = [[0,0],[0,1],[1,0],[0,2],[2,0]]
print(largestTriangleArea(points))  # Example output: 2.0

Time Complexity

The time complexity for this solution is ( O(n^3) ) because we are checking all combinations of three points from the list of points. Given that the maximum number of points (according to LeetCode constraints) is 50, this approach is computationally feasible.

If the number of points were significantly larger, we would need to consider a more efficient method, but for ( n \leq 50 ), this approach is effective and straightforward. Additionally, the space complexity is ( O(1) ), aside from the input storage itself and the combinations generator’s overhead.

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