Intersect Motion Threshold Periods Across Cameras

Implement common_motion_periods(streams, threshold, end_time). Each camera stream is a nonempty list of [timestamp, intensity] readings sorted by strictly increasing integer timestamp. Intensities are integers from 0 through 1000000, representing values from 0.000000 through 1.000000; threshold uses the same scale.

A reading at timestamp t remains in effect at every integer timestamp from t through one less than the next reading's timestamp. The final reading remains in effect through end_time, inclusive. A camera has no defined intensity before its first reading, so analysis begins at the latest first timestamp among all streams.

Return every maximal inclusive interval [start, end] during which every camera's effective intensity is at least threshold. Intervals must be sorted by start and must not overlap or touch; touching qualifying spans are one maximal interval. With one stream, the function solves the single-camera part.

The discrete time domain, persistence rule, scaling, and explicit horizon are pedagogical assumptions because the source gives timestamp-intensity pairs and inclusive output endpoints without defining interpolation or the last reading's end.

Function Contract

common_motion_periods(streams: list[list[list[int]]], threshold: int, end_time: int) -> list[list[int]]

Constraints

• 1 <= len(streams) <= 1000
• Every stream has at least one reading, and its timestamps are strictly increasing.
• Every reading timestamp is in [0, end_time].
• 0 <= threshold <= 1000000
• 0 <= end_time <= 10^12
• The total number of readings is at most 200000.

Examples

Example 1

Input:
streams = [[[1, 200000], [3, 700000], [6, 800000], [8, 100000]]]
threshold = 600000
end_time = 10
Output: [[3, 7]]

The single camera is above threshold from timestamp 3 through 7; the change at 6 does not break the qualifying period.

Example 2

Input:
streams = [
 [[0, 800000], [5, 300000], [8, 900000]],
 [[2, 700000], [6, 900000], [9, 200000]]
]
threshold = 600000
end_time = 10
Output: [[2, 4], [8, 8]]

Both cameras qualify from 2 through 4 and again only at timestamp 8.