Abstract
Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal et al. (in: Proceedings of the 31st International Symposium on Computational Geometry (SoCG), pp 240–254, 2015). In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al. ’s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for both the time-windowed 2D diameter decision problem and the time-windowed 2D convex hull area decision problem in $$O(n \log n)$$ time, improving Bokal et al. ’s $$O(n \log ^2 n)$$ and $$O(n \log n \log \log n)$$ solutions respectively. Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first $$O(n\, \mathrm{polylog}\, n)$$ algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.
Highlights
Time-windowed geometric problems have been the subject of many recent papers and are motivated by timestamped social network data and Geographic Information System (GIS) data, the latter of which may consist of longitude, latitude, and altitude coordinates and time
We consider problems for which the answer is a Boolean value: given a query interval of time [t1, t2] called a time window, does the subset of S whose time values are within the query window have property P or not? We call these time-windowed decision problems
They achieve the following geometric results: 1. 2D diameter decision: Given a set of n time-labeled points in R2, determine if there exist two points greater than unit distance apart, whose time values are within a query time window
Summary
Time-windowed geometric problems have been the subject of many recent papers and are motivated by timestamped social network data and Geographic Information System (GIS) data, the latter of which may consist of longitude, latitude, and altitude coordinates and time. The query time only increases by O(1) predecessor searches on the query time values In this and the previous paper [5], we focus only on hereditary properties, meaning if a set S has P any superset S ⊇ S has it.. A query for a time window [t1, t2] is answered by looking up the t in the table for start time t1, and checking if t2 ≤ t. For this reason, Bokal et al refer to time-windowed decision problems as the problem of finding maximal contiguous subsequences with hereditary properties. Since answering a query after preprocessing is trivial, for the rest of the paper we focus only on bounding the preprocessing time
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