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

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.