Detect Squares

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

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

An axis-aligned square has all four sides parallel to the x- and y-axes.

Implement the DetectSquares class:

• DetectSquares() - initializes the object.
• add(point) - adds a new point [x, y] to the data structure.
• count(point) - counts the number of axis-aligned squares that include point as a corner.

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Example 1

(3,10)      (11,10)

(3,2)       (11,2)   <- query point

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: After adding (3,10), (11,2), (3,2): count([11,10]) finds 1 square. count([14,8]) finds 0. After adding a duplicate (11,2): count([11,10]) finds 2 squares.

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Constraints

• 0 <= x, y <= 1000
• At most 3000 calls will be made to add and count.