Distribution-sensitive construction of the greedy spanner (extended abstract)

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it take O(n^2) time, limiting its applicability on large data sets. We observe that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly, and few or no ‘long’ edges that can usually be determined quickly using local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. This characterization gives a O(n log^2 n log^2 log n) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points

Distribution-sensitive construction of the greedy spanner

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take $\varOmega (n^2)$ time, limiting its applicability on large data sets. We propose a novel algorithm design which uses the observation that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly. To find the usually few remaining ‘long’ edges, we use a combination of already determined local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. We give a geometric property that holds with high probability, which in turn implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points. This characterization gives an $O(n \log ^2 n \log ^2 \log n)$ expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give an $O((n + |E|) \log ^2 n \log \log n)$ expected time algorithm on uniformly distributed points that determines whether E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E

Model-based segmentation and classification of trajectories

\u3cp\u3eWe present efficient algorithms for segmenting and classifying trajectories based on a movement model parameterised by a single parameter, like the Brownian bridge movement model. Segmentation is the problem of subdividing a trajectory into interior-disjoint parts such that each part is homogeneous in its movement characteristics. We formalise this using the likelihood of the model parameter, and propose a new algorithm for trajectory segmentation based on this. We consider the case where a discrete set of m parameter values is given and present an algorithm to compute an optimal segmentation with respect to an information criterion in O(nm) time for a trajectory with n sampling points. We also present an algorithm that efficiently computes the optimal segmentation if we allow the parameter values to be drawn from a continuous domain. Classification is the problem of assigning trajectories to classes of similar movement characteristics. The set of trajectories might for instance be the subtrajectories resulting from segmenting a trajectory, thus identifying movement phases. We give an algorithm to compute the optimal classification with respect to an information criterion in O(m\u3csup\u3e2\u3c/sup\u3e+ kmlog m) time for m parameter values and k trajectories, assuming bitonic likelihood functions. We also show that classification is NP-hard if the parameter values are allowed to vary continuously and present an algorithm that solves the problem in polynomial time under mild assumptions on the input.\u3c/p\u3

A framework for trajectory segmentation by stable criteria

We present an algorithmic framework for criteria-based segmentation of trajectories that can efficiently process a large class of criteria. Criteria-based segmentation is the problem of subdividing a trajectory into a small number of parts such that each part satisfies a global criterion. Our framework can handle criteria that are stable, in the sense that these do not change their validity along the trajectory very often. This includes both increasing and decreasing monotone criteria. Our framework takes O(n log n) time for preprocessing and computation, where n is the number of data points. It surpasses the two previous algorithmic frameworks on criteria-based segmentation, which could only handle decreasing monotone criteria, or had a quadratic running time, respectively. Furthermore, we develop an efficient data structure for interactive parameter selection, and provide mechanisms to improve the exact position of break points in the segmentation. We demonstrate and evaluate our framework by performing case studies on real-world data sets

Progressive geometric algorithms

Progressive algorithms are algorithms that, on the way to computing a complete solution to the problem at hand, output intermediate solutions that approximate the complete solution increasingly well. We present a framework for analyzing such algorithms, and develop efficient progressive algorithms for two geometric problems: computing the convex hull of a planar point set, and finding popular places in a set of trajectories