Detect Squares Leetcode Problem 2013 [Python Solution]

Are you ready to dive into the world of coding and algorithmic problem-solving?

Today, we have an exciting challenge that's part of LeetCode's contest â€“ the Detect Squares problem.

In this blog post, we'll explore the problem, break it down step by step, and present a Python solution that will help you understand the underlying concepts.

Problem Overview

Detect Squares is a medium-level problem that falls under the Math & Geometry category.

It's an interesting problem that requires you to design an algorithm to do the following:

Add new points from a stream to a data structure.

These points are in the form of [x, y] coordinates.

Duplicate points are allowed and should be treated as different points.

Given a query point, count the number of ways to choose three points from the data structure such that these three points, along with the query point, form an axis-aligned square with a positive area.

An axis-aligned square is defined as a square whose edges are all the same length and are either parallel or perpendicular to the x-axis and y-axis.

Now, let's structure our approach to tackle this problem effectively.

Understanding the Problem

Before we dive into the solution, let's take a moment to understand the problem's key points:

We need to maintain a data structure that can store points in the form of [x, y] coordinates.
We should handle duplicate points properly, treating them as separate points.
For each query point, we need to count the number of ways to choose three other points to form an axis-aligned square with the query point.
The squares must have a positive area, meaning that the four points should not be collinear or overlap.

With these points in mind, we can proceed to implement our solution.

Constraints

It's important to be aware of the problem's constraints, as they guide our approach:

The coordinates of the points are in the range [0, 1000].
You'll need to handle at most 3000 calls in total for adding points and counting squares.

Detect Squares LeetCode Problem Solution

Let's get into the solution.

We'll break it down into key sections to make it easier to follow.

1. Bruteforce Approach

A brute-force approach to solving this problem involves iterating through all possible combinations of points to check if they can form a square with the query point.

However, this approach has a time complexity of O(n^3), where n is the number of points in the data structure, and it's not very efficient.

2. An Efficient Approach with Explanation

To optimize our solution, we'll use a more efficient approach.

Here's a Python solution that leverages a hash map:

View on GitHub ➹

from collections import defaultdict

class DetectSquares:
    def __init__(self):
        self.ptsCount = defaultdict(int)
        self.pts = []

    def add(self, point: List[int]) -&gt; None:
        self.ptsCount[tuple(point)] += 1
        self.pts.append(point)

    def count(self, point: List[int]) -&gt; int:
        res = 0
        px, py = point
        for x, y in self.pts:
            if (abs(py - y) != abs(px - x)) or x == px or y == py:
                continue
            res += self.ptsCount[(x, py)] * self.ptsCount[(px, y)]
        return res

Now, let's break down this efficient approach:

We use a hash map, ptsCount, to store the counts of points.

The keys are tuples (x, y) representing the coordinates, and the values are the counts of each point.

We also maintain a list, pts, to store the actual points.

This list helps us iterate through the points efficiently.

In the add method, we increment the count of the given point (in tuple form) in the ptsCount map and append the point to the pts list.
In the count method, we initialize a variable res to keep track of the count of squares that can be formed.
We extract the x and y coordinates of the query point.
We iterate through each point in the pts list.

For each point, we check if it can potentially form a square with the query point.

We do this by comparing the differences in x and y coordinates.

If the differences are not equal, or if the point coincides with the query point along either the x or y axis, we continue to the next iteration.

If the point is a potential diagonal point for forming a square, we calculate the count of the top left point and the count of the bottom right point by looking them up in the ptsCount map.
We multiply these counts and add the result to res.

This accounts for all the ways we can create squares with the given query point.

Finally, we return the value of res, which represents the count of valid squares.

This efficient approach significantly reduces the time complexity, making it a practical solution for the problem.

Time and Space Complexity

Let's analyze the time and space complexity of our solution:

Time Complexity: The add operation has a time complexity of O(1) because it involves adding a point to the hash map and the list.

The count operation, which iterates through the pts list, has a time complexity of O(n), where n is the number of points in the data structure.

Space Complexity: The space complexity is O(n) because we store the points in the pts list, and the hash map ptsCount also grows with the number of unique points.

Reasoning Behind Our Approach

The key insight behind our efficient approach is to avoid the brute-force approach, which would involve checking all possible combinations of points to form a square.

Instead, we focus on diagonal points, as these are the crucial elements for creating squares.

By efficiently counting diagonal points and looking up their counts in the hash map, we can quickly determine how many squares can be formed with the given query point.

Conclusion

In this blog post, we explored the Detect Squares problem, a medium-level challenge on LeetCode.

We discussed the problem statement, constraints, and an efficient Python solution using a hash map and a list to store points.

Our approach allows us to count the number of ways to create squares with a query point while avoiding the need to check all combinations exhaustively.

Understanding and solving algorithmic problems like this one is a great way to improve your coding skills and deepen your understanding of data structures and algorithms.

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For further practice, consider exploring additional problems on LeetCode and other coding platforms.

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If you'd like to dive deeper into the problem or explore more coding challenges, you can find the original problem on LeetCode here: Detect Squares LeetCode Problem.

Happy coding, and best of luck with your coding journey!