Formatted question description: https://leetcode.ca/all/2013.html

2013. Detect Squares (Medium)

You are given a stream of points on the X-Y plane. Design an algorithm that:

  • Adds new points from the stream into a data structure. Duplicate points are allowed and should be treated as different points.
  • Given a query point, counts the number of ways to choose three points from the data structure such that the three points and the query point form an axis-aligned square with positive area.

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

Implement the DetectSquares class:

  • DetectSquares() Initializes the object with an empty data structure.
  • void add(int[] point) Adds a new point point = [x, y] to the data structure.
  • int count(int[] point) Counts the number of ways to form axis-aligned squares with point point = [x, y] as described above.

 

Example 1:

Input
["DetectSquares", "add", "add", "add", "count", "count", "add", "count"]
[[], [[3, 10]], [[11, 2]], [[3, 2]], [[11, 10]], [[14, 8]], [[11, 2]], [[11, 10]]]
Output
[null, null, null, null, 1, 0, null, 2]

Explanation
DetectSquares detectSquares = new DetectSquares();
detectSquares.add([3, 10]);
detectSquares.add([11, 2]);
detectSquares.add([3, 2]);
detectSquares.count([11, 10]); // return 1. You can choose:
                               //   - The first, second, and third points
detectSquares.count([14, 8]);  // return 0. The query point cannot form a square with any points in the data structure.
detectSquares.add([11, 2]);    // Adding duplicate points is allowed.
detectSquares.count([11, 10]); // return 2. You can choose:
                               //   - The first, second, and third points
                               //   - The first, third, and fourth points

 

Constraints:

  • point.length == 2
  • 0 <= x, y <= 1000
  • At most 5000 calls in total will be made to add and count.

Solution 1. Enumerate seen x values

Intuition: For query (x, y), we try each x values we’ve seen, say x1, as the x value of the other vertical edge. Because we want to form squares, there are only two square candidates possible.

Algorithm:

Given query (x, y), we try each x1 values we’ve seen. Another point must be (x1, y).

Since the side length d = abs(x - x1), the y value of other horizontal side must be either y1 = y + d or y2 = y - d.

So the answer is count(x1, y) * [ count(x, y1) * count(x1, y1) + count(x, y2) * count(x1, y2) ].

// OJ: https://leetcode.com/problems/detect-squares/
// Time:
//      DetectSquares: O(1)
//      add: O(1)
//      count: O(U) where `U` is the number of unique `x` values
// Space: O(N)
class DetectSquares {
    unordered_map<int, int> cnt;
    unordered_set<int> xs;
    inline int key(int x, int y) {
        return 10000 * x + y;
    }
    inline int count(int x, int y) {
        int k = key(x, y); 
        return cnt.count(k) ? cnt[k] : 0;
    }
public:
    DetectSquares() {}
    void add(vector<int> p) {
        cnt[key(p[0], p[1])]++;
        xs.insert(p[0]);
    }
    int count(vector<int> p) {
        int x = p[0], y = p[1], ans = 0;
        for (int x1 : xs) {
            if (x1 == x) continue;
            int c = count(x1, y), d = abs(x - x1), y1 = y - d, y2 = y + d;
            if (c == 0) continue;
            ans += c * (count(x, y1) * count(x1, y1) + count(x, y2) * count(x1, y2));
        }
        return ans;
    }
};

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