Locate Cars with a Boolean Square-Scan API

An external scanner exposes this API: ```python def mock_squaredar_scan(lat_min: float, lon_min: float, side_length: float) -> bool: # True if at least one car is inside the queried square; otherwise False. ``` You are asked to design `get_car_positions` using only this Boolean occupancy API. Explain what contract is needed, how a search could work, and what information the API cannot reveal. ### Constraints and Clarifying Questions - Clarify the search region, coordinate system, required precision, boundary inclusion, and whether cars remain stationary during the search. - Ask whether the desired output is exact coordinates, approximate occupied regions, or one result for every car including cars at the same position. - Treat the coordinates as a plane for this practice discussion; no geographic-distance calculation is specified. - Do not assume that a positive square contains exactly one car or that a negative scan says anything about locations outside that square. ### Part 1 — Identify the Missing Contract Explain why the Boolean API alone does not completely specify an exact car-location operation. Identify the conditions needed for a terminating search and a meaningful output. #### What This Part Should Cover - Search coverage, static or snapshot-consistent occupancy, and precision. - The difference between detecting occupancy and counting cars. - Limits on exact recovery from an unbounded or arbitrary-real coordinate domain. ### Part 2 — Design a Conditional Finite-Grid Search For a concrete practice design, suppose the following additional contract is agreed: - All cars are in a known square with lower corner `(x0, y0)` and side length `L`. - The square contains an `N` by `N` grid, where `N = 2^h` for a nonnegative integer `h`, and `delta = L / N`. - Car coordinates are grid points `(x0 + i * delta, y0 + j * delta)` for integer indices `0 <= i, j < N`. - Scans use half-open squares: each lower boundary is included and each upper boundary is excluded. The scanner is exact and reports one unchanged snapshot throughout the search. - Return each occupied grid coordinate once, regardless of how many cars share it. Output order is not significant. Describe a search using the square API. Explain how it handles several cars in different regions, how it terminates, and why it finds every occupied grid position. #### What This Part Should Cover - A subdivision or bisection strategy compatible with square-shaped queries. - Pruning and coverage without assuming that only one child region is occupied. - Correct handling of boundary points and the smallest resolvable region. ### Part 3 — Analyze Cost and Generalize Carefully Analyze the number of scanner calls and working memory. Then explain what the same search can report if cars instead have arbitrary coordinates and `delta` is only a requested spatial resolution. #### What This Part Should Cover - Query cost in terms of grid size and the number of occupied positions. - The distinction between an occupied cell and an exact coordinate. - What remains unknowable about coincident cars or several cars in one final cell. ```hint Interpret a positive answer carefully Consider what one positive scan proves and what it leaves undecided. Check whether several disjoint subregions could all contain cars. ``` ### What a Strong Answer Covers - An explicit contract separating known scanner behavior from added practice assumptions. - A complete conditional search with coverage, termination, and complexity reasoning. - Honest limits on exactness, multiplicity, and changing observations. ### Follow-up Questions - Why is following only one ordinary binary-search branch insufficient when several cars may exist? - What changes if the scanner uses closed boundaries or approximate floating-point comparisons? - Can repeated Boolean scans distinguish one car from several cars at exactly the same coordinate?