Fastest Rolling One-Kilometer Average Speed from 5-Second Distance Samples

A runner's watch records how far the runner travels during each consecutive 5-second interval. You are given `distances`, where `distances[i]` is the number of meters covered during interval `i`. Find the fastest average speed the runner sustained over a rolling one-kilometer (1000-meter) stretch. ### Function Signature ```python def fastest_rolling_km_speed(distances: list[int]) -> float: ``` ### Rules - A window is a contiguous range of intervals `[l, r]` with `l <= r`. Its distance is `distances[l] + ... + distances[r]` meters and its duration is `5 * (r - l + 1)` seconds. - For each end interval `r`, the rolling one-kilometer window ending at `r` is the shortest window `[l, r]` whose distance is at least `1000` meters. If even `[0, r]` covers less than `1000` meters, interval `r` has no rolling window. - The average speed of a window is its full distance divided by its duration, in meters per second. The whole distance of the window counts even when it exceeds `1000` meters; there is no interpolation inside an interval. - Return the maximum average speed over all rolling windows. - If the total of all distances is less than `1000` meters, no rolling window exists; return `-1.0`. - An answer within an absolute or relative error of `1e-6` of the exact value is accepted. ### Constraints - `1 <= len(distances) <= 100000` - `0 <= distances[i] <= 1000`, and every value is an integer. - The sum of all distances is at most `100000000`. - When a rolling window exists, the answer is greater than `0` and at most `200.0` meters per second. ### Examples **Example 1** - Input: `distances = [300, 400, 300, 200, 500, 400]` - Output: `73.33333333333333` - Explanation: No window ending at interval 0 or 1 reaches 1000 meters. The rolling windows ending at intervals 2, 3, 4 and 5 are `[0, 2]` (1000 m in 15 s), `[0, 3]` (1200 m in 20 s), `[2, 4]` (1000 m in 15 s) and `[3, 5]` (1100 m in 15 s), with average speeds of about 66.667, 60, 66.667 and 73.333 meters per second. The largest is 1100 / 15. **Example 2** - Input: `distances = [250, 250, 250, 240]` - Output: `-1.0` - Explanation: The runner covers only 990 meters in total, so no window reaches one kilometer. **Example 3** - Input: `distances = [0, 600, 500, 0, 600]` - Output: `110.0` - Explanation: The rolling window ending at interval 2 is `[1, 2]`: 1100 meters in 10 seconds, or 110 meters per second. It does not include interval 0, because `[1, 2]` already reaches 1000 meters and is shorter. The rolling windows ending at intervals 3 and 4 are `[1, 3]` and `[2, 4]`, each 1100 meters in 15 seconds.