Time-Space Trade-Off for Finding the k-Visibility Region of a Point in a Polygon

Abstract

We study the problem of computing the k-visibility region in the memory-constrained model. In this model, the input resides in a randomly accessible read-only memory of O(n) words, with $O(\log {n})$ bits each. An algorithm can read and write O(s) additional words of workspace during its execution, and it writes its output to write-only memory. In a given polygon P and for a given point $q \in P$, we say that a point p is inside the k-visibility region of q, if and only if the line segment pq intersects the boundary of P at most k times. Given a simple n-vertex polygon P stored in a read-only input array and a point $q \in P$, we give a time-space trade-off algorithm which reports the k-visibility region of q in P in $O(cn/s+n\log {s}+ \min \{{\lceil k/s \rceil n,n \log {\log _s{n}}}\})$ expected time using O(s) words of workspace. Here $c\le n$ is the number of critical vertices for q, i.e., the vertices of P where the visibility region may change. We also show how to generalize this result for polygons with holes and for sets of non-crossing line segments.

This work was partially supported by DFG project MU/3501-2 and by the Natural Sciences and Engineering Research Council of Canada (NSERC).