Find Shared Motion Periods Across Camera Streams

Problem Statement

You are monitoring a fleet of security cameras. Each camera produces a time-series list of sensor readings represented as (timestamp, intensity), sorted in ascending order by timestamp. Here, timestamp is an integer (e.g., in seconds or milliseconds), and intensity is a float between 0.0 and 1.0 indicating motion.

Given a list of streams streams, an intensity threshold threshold, and a maximum allowed gap max_gap between consecutive readings:

1. Extract Active Intervals for each stream:

   • A reading is active if intensity >= threshold.
   • Two consecutive active readings at timestamps t_prev and t_curr belong to the same continuous active interval if t_curr - t_prev max_gap, the current interval closes at t_prev, and a new interval begins at t_curr.
   • An isolated active reading at t forms a closed interval [t, t].

2. Compute Shared Intervals:

   • Find all closed time intervals [start, end] where all cameras in streams are simultaneously active.
   • A single-point overlap [t, t] is valid if all cameras are active at time t.
   • Return the shared intervals in chronological order. Non-overlapping or touching intervals that can be merged should be merged.

Do not mutate the input lists.

Function Signature

def find_shared_motion_periods(
    streams: list[list[tuple[int, float]]],
    threshold: float,
    max_gap: int
) -> list[tuple[int, int]]:
    pass

Constraints

• 0 <= len(streams) <= 100
• Each stream has 0 <= len(stream) <= 10^4 readings.
• Timestamps in each stream are strictly increasing non-negative integers: 0 <= timestamp <= 10^9.
• 0.0 <= intensity <= 1.0
• 0.0 <= threshold <= 1.0
• 0 <= max_gap <= 10^6

Examples

Example 1

streams = [
    [(10, 0.8), (14, 0.9), (20, 0.7), (25, 0.2)],
    [(12, 0.6), (15, 0.8), (18, 0.9), (30, 0.9)]
]
threshold = 0.5
max_gap = 5

# Stream 0 active readings: 10, 14, 20 (gaps: 14-10=4  5)
# Stream 0 intervals: [10, 14], [20, 20]
# Stream 1 active readings: 12, 15, 18, 30 (gaps: 15-12=3  5)
# Stream 1 intervals: [12, 18], [30, 30]
# Intersection of [10, 14] & [20, 20] with [12, 18] & [30, 30]:
# Output: [(12, 14)]

Example 2

streams = [
    [(5, 0.8)],
    [(5, 0.9)]
]
threshold = 0.7
max_gap = 2

# Both have isolated readings at t=5.
# Output: [(5, 5)]