#1037

Easy

LeetCode Problem Workspace

Valid Boomerang

Determine if three points on a 2D plane form a boomerang, based on distinctness and non-collinearity.

Problem Statement

You are given an array of three points, where each point is represented as [xi, yi] in a 2D plane. Your task is to determine if these three points form a valid boomerang.

A boomerang is defined as a set of three distinct points that do not lie on a straight line. The points must form a triangle with a non-zero area, which ensures they are not collinear.

Examples

Example 1

Input: points = [[1,1],[2,3],[3,2]]

Output: true

Example details omitted.

Example 2

Input: points = [[1,1],[2,2],[3,3]]

Output: false

Example details omitted.

Constraints

points.length == 3
points[i].length == 2
0 <= xi, yi <= 100

Solution Approach

Check Collinearity Using Area

To check if three points form a valid boomerang, calculate the area of the triangle formed by the points using the determinant method. If the area is non-zero, the points are not collinear and form a boomerang.

Verify Distinct Points

Ensure that all three points are distinct. If any two points are identical, they cannot form a boomerang.

Efficient Calculation

Given the fixed size of the input (three points), the solution will have constant time complexity for both space and time, making the approach efficient and easy to implement.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Since the problem always involves exactly three points, both the time and space complexities are constant, O(1).

What Interviewers Usually Probe

Look for efficient use of mathematical properties to solve the problem.
Expect the candidate to explain collinearity and how the area of a triangle can help determine this.
Candidates should confirm that all points are distinct before proceeding with further checks.

Common Pitfalls or Variants

Common pitfalls

Confusing collinearity check with simple distance calculations, which do not guarantee non-collinearity.
Overlooking the distinctness of the points, leading to incorrect conclusions about the boomerang formation.
Not understanding how the area formula directly applies to this problem for determining valid boomerangs.

Follow-up variants

Handling higher dimensions if the problem was extended to more than 2D.
Checking for a set of points beyond three, such as determining if any subset forms a boomerang.
Adapting the solution for larger inputs or multiple sets of points.

FAQ

What defines a boomerang in the context of this problem?

A boomerang consists of three distinct points that do not lie on a straight line, i.e., they form a triangle with a non-zero area.

Why do we calculate the area to check for collinearity?

The area of the triangle formed by the points will be zero if they are collinear. A non-zero area confirms they are not in a straight line.

What happens if two points are the same?

If any two points are identical, the points do not form a valid boomerang, as the set must consist of distinct points.

How is this problem related to geometry?

The problem involves basic geometry concepts, particularly the area of a triangle and collinearity of points in a 2D plane.

How does GhostInterview assist with problems like this?

GhostInterview helps by offering a detailed solution approach, focusing on essential steps like checking for collinearity and distinctness, which are key to solving geometric problems efficiently.

terminal

Solution

Solution 1: Slope Comparison

Let the three points be $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$. The formula for calculating the slope between two points is $\frac{y_2 - y_1}{x_2 - x_1}$.

1

2

3

4

class Solution:
    def isBoomerang(self, points: List[List[int]]) -> bool:
        (x1, y1), (x2, y2), (x3, y3) = points
        return (y2 - y1) * (x3 - x2) != (y3 - y2) * (x2 - x1)

Continue Practicing

#1030 matrix cells in distance order #973 k closest points to origin #963 minimum area rectangle ii