Largest Triangle Area

This problem requires a solution for calculating the maximum area of a triangle formed by any three points on a 2D plane.

The first step is to iterate over every possible combination of three points. Python’s itertools.combinations() can be used for this.

Then, for each combination of three points, we can use the formula for calculating the area of a triangle given its vertices. In 2D geometry, given three points (x1, y1), (x2, y2), and (x3, y3), the area of the triangle is given by:

Area = 1/2 * abs( x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2) )

Now, we just need to find the maximum area from all the possible triangles.

Python solution:

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from typing import List
from itertools import combinations
import math

class Solution:
    def largestTriangleArea(self, points: List[List[int]]) -> float:
        max_area = 0
        for combination in combinations(points, 3):
            area = self.calcArea(combination)
            max_area = max(area, max_area)
        return max_area

    def calcArea(self, points):
        x1, y1 = points[0]
        x2, y2 = points[1]
        x3, y3 = points[2]
        return 0.5 * abs(x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2))

This code iterates over all combinations of points, which gives it a time complexity of O(n^3), where n is the number of points. This complexity is acceptable because the problem constraints state there will be at most 50 points. The space complexity is O(1), as no additional space is used that scales with the input size.

Identifying Problem Isomorphism

“Largest Triangle Area” can be approximately mapped to “Valid Triangle Number”.

The “Largest Triangle Area” problem involves selecting three points in a plane and calculating the area of the triangle formed by these points, which is a combinatorial task. The “Valid Triangle Number” problem also has a similar combinatorial nature where it involves selecting three sides from an array to form a triangle.

The reason for approximate mapping is that both problems fundamentally involve the concept of forming a triangle. However, it’s not an exact mapping because “Valid Triangle Number” doesn’t involve calculating the area of triangles, as in “Largest Triangle Area”.

“Valid Triangle Number” is simpler as it only involves checking the triangle inequality condition. “Largest Triangle Area” is more complex as it involves both selecting points to form a triangle and calculating its area.

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class Solution:
    def largestTriangleArea(self, points):
        max_area = 0.0
        for i in range(len(points) - 2):
            for j in range(i + 1, len(points) - 1):
                for k in range(j + 1, len(points)):
                    max_area = max(max_area, self.areaCal(points[i], points[j], points[k]))
        return max_area

    def areaCal(self, pt1, pt2, pt3):
        return abs(pt1[0] * (pt2[1] - pt3[1]) + pt2[0] * (pt3[1] - pt1[1]) + pt3[0] * (pt1[1] - pt2[1])) / 2.0

Problem Classification

The problem is from the Computational Geometry, a branch of computer science devoted to the study of algorithms that can be stated in terms of geometry.

“What” Components:

  1. Input: An array of points in a 2D plane. Each point is represented as an array containing two integers representing its x and y coordinates.

  2. Output: The area of the largest triangle that can be formed by any three different points.

Problem Classification:

  1. Computation Problem: This problem requires calculation - more specifically, the calculation of areas of various possible triangles.

  2. Optimization Problem: We are not simply looking for any triangle, but the one with the largest possible area. Hence, this is an optimization problem where the goal is to maximize the area.

This problem falls into the domain of Computational Geometry as it deals with points in a geometric plane and requires the calculation of the area of triangles formed by these points.

It’s a Computation Problem because the solution involves calculating the areas of triangles formed by the points given as input.

It’s an Optimization Problem as we have to find the triangle with the largest area, not just any triangle. So, the goal is to find an optimal solution (i.e., the maximum area), not just any valid solution.

Language Agnostic Coding Drills

  1. Identifying Distinct Concepts:

    a. Class Definition: The Solution class is a fundamental aspect of Object-Oriented Programming (OOP). It encapsulates related functions that solve the problem.

    b. Function Definition: Two functions, largestTriangleArea and areaCal, are defined. This is the basis of procedural programming, where related pieces of code are encapsulated in reusable functions.

    c. Iteration: The use of nested for loops to iterate over the array of points.

    d. Mathematical Calculation: The areaCal function implements the formula for the area of a triangle given three points in 2D space.

    e. Using built-in functions: The use of Python’s built-in max and abs functions for getting maximum value and absolute value, respectively.

    f. Array Indexing: Accessing the elements of a list (array) using their indices.

  2. Ordered List of Concepts:

    a. Class Definition (Difficulty: 1) - This is the basic concept of creating a class in Python, foundational to OOP in Python.

    b. Function Definition (Difficulty: 2) - Function definition is slightly more difficult as it involves understanding the idea of encapsulating a piece of functionality in a function.

    c. Array Indexing (Difficulty: 3) - You need to understand how Python arrays (lists) are structured and how you can access each element.

    d. Iteration (Difficulty: 4) - This involves understanding the concept of loops, nested loops, and how they can be used to iterate over data structures.

    e. Using built-in functions (Difficulty: 5) - Requires familiarity with Python’s built-in functions and how they can be used to perform common tasks.

    f. Mathematical Calculation (Difficulty: 6) - This is the most complex concept, as it requires understanding the formula for calculating the area of a triangle in 2D space and how to implement it in Python.

  3. Problem-solving Approach:

    a. Define a class and two functions, one for calculating the area and one for finding the largest area.

    b. Implement the mathematical formula for calculating the area of a triangle given three points in the areaCal function.

    c. In the largestTriangleArea function, use nested loops to iterate over the array of points and for each unique set of three points, calculate the area of the triangle they form.

    d. Keep track of the largest area calculated so far and update it whenever a larger area is found.

    e. After going through all sets of points, the largest area calculated is the largest area that can be formed by any three points. Return this value.

Targeted Drills in Python

  1. Python Coding Drills:

    a. Class Definition:

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    class MyClass:
    pass

    b. Function Definition:

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    def my_function():
    pass

    c. Array Indexing:

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    my_list = [1, 2, 3, 4, 5]
first_item = my_list[0]

    d. Iteration:

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    for i in range(5):
    print(i)

    e. Using built-in functions:

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    maximum = max([1, 2, 3, 4, 5])
absolute = abs(-5)

    f. Mathematical Calculation:

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    a = 2
b = 3
area = 0.5 * a * b

  2. Problem-specific Coding Drills:

    a. Area Calculation: This is a critical aspect of our problem. We need to accurately calculate the area of a triangle given three points. Here’s a simple drill for implementing the formula:

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    def calculate_area(pt1, pt2, pt3):
    return abs(pt1[0] * (pt2[1] - pt3[1]) + pt2[0] * (pt3[1] - pt1[1]) + pt3[0] * (pt1[1] - pt2[1])) / 2.0

  3. Integration of Concepts:

    a. Start by defining a class and add two empty methods to it.

    b. Inside the areaCal method, implement the area calculation formula. This involves mathematical calculation and array indexing drills.

    c. In the largestTriangleArea method, implement nested loops to iterate over all possible combinations of three points. This uses the iteration and array indexing drills.

    d. Inside these loops, calculate the area for each triangle using the areaCal method and update the maximum area found so far. This will involve the use of built-in function (max) drill.

    e. Finally, after the loops, return the maximum area found. This covers the function definition drill.

These drills need to be assembled in the correct order. The mathematical calculation is part of the areaCal method. This method, along with the built-in functions, is used inside the loops in the largestTriangleArea method, which is part of the class definition.