Segment Visibility Counting Queries in Polygons

Let P be a simple polygon with n vertices, and let A be a set of m points or line segments inside P . We develop data structures that can efficiently count the number of objects from A that are visible to a query point or a query segment. Our main aim is to obtain fast, O(polylog nm), query times, while using as little space as possible. In case the query is a single point, a simple visibility-polygon-based solution achieves O(log nm) query time using O(nm2) space. In case A also contains only points, we present a smaller, O(n + m2+ε log n)-space, data structure based on a hierarchical decomposition of the polygon. Building on these results, we tackle the case where the query is a line segment and A contains only points. The main complication here is that the segment may intersect multiple regions of the polygon decomposition, and that a point may see multiple such pieces. Despite these issues, we show how to achieve O(log n log nm) query time using only O(nm2+ε + n2) space. Finally, we show that we can even handle the case where the objects in A are segments with the same bounds. 2012 ACM Subject Classification Theory of computation → Computational geometry