Solution Explanation: Minimum Area Rectangle

This problem asks to find the minimum area of a rectangle formed by given points in a 2D plane, with sides parallel to the axes. The solution employs a hash table (dictionary in Python) to efficiently identify pairs of points that form the sides of a rectangle.

Approach

1. Data Preprocessing: The input points is processed to organize points by their x-coordinates. We use a dictionary (or TreeMap in Java, map in C++, or map in Go) where keys are x-coordinates, and values are lists of corresponding y-coordinates. This structure allows quick access to y-coordinates for a given x.

2. Iterating and Finding Rectangles: The algorithm iterates through the sorted x-coordinates. For each x-coordinate, it considers all pairs of y-coordinates. If a pair (y1, y2) has been encountered before with a different x-coordinate (this signifies a rectangle), the area is calculated and compared with the current minimum area (ans).

3. Using a Hash Table for Efficiency: A hash table (pos in the code) stores pairs of (y1, y2) along with the x-coordinate where they were previously encountered. Checking if a pair exists in pos takes O(1) average time, making the overall search significantly faster than a brute-force approach that compares all point pairs.

4. Minimum Area Tracking: The algorithm maintains a variable ans to store the minimum area found so far. Initially, ans is set to a large value (infinity or a very large integer). Whenever a smaller rectangle area is found, ans is updated.

5. Handling No Rectangles: If no rectangles are found, ans remains at its initial large value, and the function returns 0.

Time and Space Complexity Analysis

• Time Complexity: O(N log N), where N is the number of points. The sorting of y-coordinates for each x-coordinate contributes to the log N factor. The nested loops iterate through pairs of y-coordinates for each x, which is roughly O(N^2) in the worst case. However, the hash table lookups are O(1) on average, which significantly improves the overall time complexity. Dominating factor becomes the sorting, thus O(N log N).

• Space Complexity: O(N) to store the dictionary/map. The space used by pos is also proportional to the number of unique (y1,y2) pairs, but it's bounded by N in the worst case (all points have different y-values).

Code Explanation (Python)

The Python code uses a defaultdict(list) to efficiently manage the points. The inf constant represents infinity. The code elegantly handles the case where no rectangles are found by returning 0 if ans remains unchanged (infinity).

from collections import defaultdict
import math
 
class Solution:
    def minAreaRect(self, points: List[List[int]]) -> int:
        d = defaultdict(list)
        for x, y in points:
            d[x].append(y)
        pos = {}
        ans = math.inf  # Use math.inf for infinity
        for x in sorted(d):
            ys = d[x]
            ys.sort()
            n = len(ys)
            for i, y1 in enumerate(ys):
                for y2 in ys[i + 1:]:
                    if (y1, y2) in pos:
                        ans = min(ans, (x - pos[(y1, y2)]) * (y2 - y1))
                    pos[(y1, y2)] = x
        return 0 if ans == math.inf else ans
 

The Java, C++, and Go code follow a similar logic, adapting data structures to their respective language conventions. The key idea of using a hash table for efficient rectangle detection remains consistent across all implementations.