2013 Detect Squares

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 3000 calls in total will be made to add and count.

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class DetectSquares:

    def __init__(self):
        self.points = defaultdict(int)  # (x,y): count

    def add(self, point: List[int]) -> None:
        self.points[tuple(point)] += 1

    def count(self, point: List[int]) -> int:
        res = 0
        x1, y1 = point
        for (x2, y2), n in self.points.items():
            dx, dy = abs(x1-x2), abs(y1-y2)
            if dx == dy and dx > 0:
                # point at diagonal, c1, c2 are points at anti-diagonal
                c1, c2 = (x1, y2), (x2, y1)
                if c1 in self.points and c2 in self.points:
                    res += n * self.points[c1] * self.points[c2]
        return res

# Your DetectSquares object will be instantiated and called as such:
# obj = DetectSquares()
# obj.add(point)
# param_2 = obj.count(point)