Dynamic Geodesic Nearest Neighbor Searching in a Simple Polygon
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We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set of point sites S in a static simple polygon P. Our data structure allows us to insert a new site in S, delete a site from S, and ask for the site in S closest to an arbitrary query point q ∈ P. All distances are measured using the geodesic distance, that is, the length of the shortest path that is completely contained in P. Our data structure supports queries in O(√(n) n^2 m) time, where n is the number of sites currently in S, and m is the number of vertices of P, and updates in O(√(n)^3 m) time. The space usage is O(n m + m). If only insertions are allowed, we can support queries in worst-case O(^2 n^2 m) time, while allowing for O( n^3 m) amortized time insertions. We can achieve the same running times in case there are both insertions and deletions, but the order of these operations is known in advance.
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