Overlapplengten

Overlapplengten is a term used to describe the lengths of overlap between two or more line segments (intervals) on a common axis. In mathematics, the overlap length between two closed intervals [a, b] and [c, d] on the real line is defined as max(0, min(b, d) − max(a, c)). If the intervals do not intersect, the overlap length is zero. If they merely touch at a point, the overlap length is also zero under this convention.

For a collection of intervals, the concept can be applied in two related ways. The length of the common overlap is the length of the intersection of all intervals, given by max(0, min_i b_i − max_i a_i) for intervals [a_i, b_i]. Alternatively, one may consider pairwise overlaps, where the sum or distribution of all pairwise overlap lengths is analyzed. In practice, the appropriate definition depends on the application.

Computational aspects include simple evaluation for two intervals and more efficient calculations for many intervals using sweep-line techniques or interval trees. In general, the common overlap across all intervals can be computed by identifying the largest left endpoint and the smallest right endpoint and taking their difference.

Applications of overlapplengten appear in scheduling, genomics (to quantify read or feature overlaps), computer graphics, and time-series analysis. The concept also extends to higher dimensions, where overlap length becomes overlap area or volume for rectangles or hyperrectangles.

Example: intervals [1,5] and [3,7] have an overlap length of 2. Intervals [0,10], [5,15], and [7,8] share a common overlap length of 1.