Count Points in Inclusive Circular Queries

### Problem Each entry of `points` gives integer x and y coordinates; repeated coordinates count as separate points. Each query `[x,y,r]` asks how many points lie inside or on the circle centered at (x,y) with radius r. Return counts in query order. ```cpp class Solution { public: vector<int> countPoints(vector<vector<int>>& points, vector<vector<int>>& queries); }; ``` ### Examples **Example 1:** ```text points = [[0,0],[3,4],[3,4]], queries = [[0,0,5],[3,4,1]] Output: [3,2] Explanation: The duplicate points on the first circle’s boundary both count. ``` **Example 2:** ```text points = [[1,1],[2,2],[4,4]], queries = [[2,2,1],[0,0,2]] Output: [1,1] Explanation: The first query includes only [2,2]; the second includes only [1,1]. ``` ### Constraints - There are `1..500` points, each with two integer coordinates in `[0,500]`. - There are `1..500` queries, each with three integers. - Query centers have coordinates in `[0,500]` and radii in `[1,500]`. ### Clarifying Questions The circle border is included. Duplicate coordinates are not deduplicated, and query ordering must be preserved. ```hint Inspect the structure How should a point on the circular boundary be treated, and do repeated coordinates count separately? ``` ### Reasoning checkpoints - Treat points exactly on each circular boundary as inside. - Count each point occurrence separately. - Discuss the source follow-up about better than linear work per query. ### Optional extensions The source follow-up asks for better than `O(P)` time per query for `P` points. For the fixed point set and the stated integer bounds, give an exact method that meets this goal after preprocessing. State its preprocessing time, storage, and worst-case query time. If you also discuss a spatial index, explain when it prunes work and whether it guarantees the requested bound.