#3143

Medium

auto_awesomeArray scanning plus hash lookup

LeetCode Problem Workspace

Maximum Points Inside the Square

Find the maximum number of points inside a valid square centered at the origin using a scanning and hashing approach.

array hash table string binary search sorting

Problem Statement

You are given a list of points, each with 2D coordinates, and a string where each character represents a tag for the corresponding point. A valid square is defined as a square centered at the origin with edges parallel to the axes. Your goal is to find the maximum number of points that can be enclosed within such a square, ensuring no square contains points with the same tag.

The challenge is to compute this efficiently, especially given the constraints. The smallest possible square to contain a point at position (x, y) has an edge length of max(abs(x), abs(y)) * 2. Utilize array scanning to assess the potential maximum, and hash lookups to ensure no tags overlap within each square.

Examples

Example 1

Input: points = [[2,2],[-1,-2],[-4,4],[-3,1],[3,-3]], s = "abdca"

Output: 2

The square of side length 4 covers two points points[0] and points[1] .

Example 2

Input: points = [[1,1],[-2,-2],[-2,2]], s = "abb"

Output: 1

The square of side length 2 covers one point, which is points[0] .

Example 3

Input: points = [[1,1],[-1,-1],[2,-2]], s = "ccd"

Output: 0

It's impossible to make any valid squares centered at the origin such that it covers only one point among points[0] and points[1] .

Constraints

1 <= s.length, points.length <= 105
points[i].length == 2
-109 <= points[i][0], points[i][1] <= 109
s.length == points.length
points consists of distinct coordinates.
s consists only of lowercase English letters.

Solution Approach

Array Scanning

First, determine the smallest square that can contain each point. For each point, compute the square’s edge length as max(abs(x), abs(y)) * 2. Then, scan through the points to find the largest square possible for the given set of points.

Hash Lookup for Tags

While scanning, maintain a hash table of the tags seen within the current square. This allows you to quickly check whether any point in the square has a duplicate tag, ensuring the square remains valid.

Optimization with Sorting

By sorting the points based on their coordinates or tags, you can reduce the need for repeated comparisons, making the scanning and hash lookup process more efficient. Sorting helps in narrowing down the search for potential square boundaries.

Complexity Analysis

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

The time and space complexities depend on the approach taken. A brute force solution could involve iterating through all points for each possible square, leading to higher time complexity. Optimized solutions using sorting and hash tables can reduce the complexity significantly, particularly in managing the tag uniqueness within squares.

What Interviewers Usually Probe

Look for a clear understanding of how to calculate the minimal square edge length for each point.
Evaluate the candidate's ability to optimize the brute force approach using sorting or hashing.
Check for knowledge of how to efficiently track tag uniqueness within squares.

Common Pitfalls or Variants

Common pitfalls

Overlooking the tag uniqueness constraint, which can lead to invalid square selections.
Using inefficient algorithms that do not scale well with large inputs.
Failing to properly handle edge cases such as when points are too close to each other or when no valid squares can be formed.

Follow-up variants

Allow squares to be rotated and test how that changes the computation.
Use a different geometric shape for the boundary instead of a square.
Extend the problem to higher dimensions or add more constraints on the square's location.

FAQ

What is the key pattern in solving Maximum Points Inside the Square?

The key pattern is array scanning combined with hash lookup. This helps efficiently compute the maximum number of points inside a valid square while ensuring no duplicate tags.

How does sorting improve the solution for this problem?

Sorting helps by reducing unnecessary comparisons during the scanning phase. It optimizes the search for potential square boundaries and tag uniqueness.

What is the time complexity of the brute force solution?

The brute force solution could have a time complexity of O(n^2), where n is the number of points, as it involves checking each point against every other point.

How do you ensure tag uniqueness within the square?

By using a hash table, you can efficiently track the tags within the current square and check for duplicates in constant time.

What happens if no valid square can be formed?

If no valid square can be formed, the result will be 0, as no points can be enclosed within a square that satisfies the tag uniqueness constraint.

terminal

Solution

Solution 1: Hash Table + Sorting

For a point $(x, y)$, we can map it to the first quadrant with the origin as the center, i.e., $(\max(|x|, |y|), \max(|x|, |y|))$. In this way, we can map all points to the first quadrant and then sort them according to the distance from the point to the origin.

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class Solution:
    def maxPointsInsideSquare(self, points: List[List[int]], s: str) -> int:
        g = defaultdict(list)
        for i, (x, y) in enumerate(points):
            g[max(abs(x), abs(y))].append(i)
        vis = set()
        ans = 0
        for d in sorted(g):
            idx = g[d]
            for i in idx:
                if s[i] in vis:
                    return ans
                vis.add(s[i])
            ans += len(idx)
        return ans

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