durationcombined

DurationCombined is a metric used in time-based analysis to describe the total span of time covered by a set of intervals, counting overlapping portions only once. Formally, given a collection of intervals [s_i, e_i] with s_i < e_i, durationCombined equals the length of the union of these intervals: length(∪_i [s_i, e_i]). In discrete time, it is the number of time units that belong to at least one interval. This makes durationCombined distinct from the sum of durations, which counts overlaps multiple times.

Computation is commonly performed with a sweep-line approach. Collect all interval endpoints, sort them by time, and traverse them while maintaining the number of currently active intervals. Between consecutive time points, add the elapsed time to the total whenever the active count is greater than zero. The typical complexity is O(n log n) due to the initial sort, with O(n) additional work to accumulate the total.

Examples: Intervals (0,4), (2,6), (7,8) yield durationCombined 7, since the union of the first two is [0,6] of length 6, plus [7,8] of length 1. If intervals are disjoint, durationCombined equals the sum of their individual durations.

Applications include project scheduling, multimedia editing, log analysis, and sensor data processing, where it is useful to quantify total coverage over time. In software, durationCombined is often stored as a metric or feature; it contrasts with durationTotal, which ignores overlaps. Related concepts include union of intervals, coverage, and the sweep-line algorithm.