Point Location in the Continuous-Time Moving Network
We discuss two variations of the moving network Voronoi diagram. The first one addresses the following problem: given a network with n vertices and E edges. Suppose there are m sites (cars, postmen, etc) moving along the network edges and we know their moving trajectories with time information. Which site is the nearest one to a point p located on network edge at time t′? We present an algorithm to answer this query in O(log(mWlogm)) time with O(nmWlog2 m + n 2logn + nE) time and O(nmWlogm + E) space for preprocessing step, where E is the number of edges of the network graph (the definition of W is in section 3). The second variation views query point p as a customer with walking speed v. The question is which site he can catch the first? We can answer this query in O(m + log(mWlogm)) time with same preprocessing time and space as the first case. If the customer is located at some node, then the query can be answered in O(log(mWlogm)) time.
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