Segment Visibility Counting Queries in Polygons
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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(nm^2) space. In case A also contains only points, we present a smaller, O(n + m^2 + εlog 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 nlog nm) query time using only O(nm^2 + ε + n^2) space. Finally, we show that we can even handle the case where the objects in A are segments with the same bounds.
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