Clustering moving entities in euclidean space
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Date
2020-08
Authors
Hassan, Md Yeakub
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Abstract
Clustering is a fundamental problem of spatio-temporal data analysis. Given a set X of n moving entities, each of which corresponds to a sequence of tau time-stamped points in R^d, a k-clustering of X is a partition of X into k disjoint subsets that optimizes a given objective function. In this thesis, we consider two clustering problems, k-Center and k-MM, where the goal is to minimize the maximum value of the objective function over the duration of motion for the worst-case input X. We show that both problems are NP-hard when k is an arbitrary input parameter, even when the motion is restricted to R. We provide an exact algorithm for the 2-MM clustering problem in R^d that runs in O(tau dn^2) time. The running time can be improved to O(tau n log n) when the motion is restricted to R. We show that the 2-Center clustering problem is NP-hard in R^2. Our 2-MM clustering algorithm provides a 1.15-approximate solution to the 2-Center clustering problem in R2. Moreover, finding a (1.15 - epsilon)-approximate solution remains NP-hard for any epsilon > 0. For both the k-MM and k-Center clustering problems in R^d, we provide a 2-approximation algorithm that runs in O(tau dnk) time. We show that the worst case of the k-MM clustering over a continuous time interval is realized at the interval's endpoints. Furthermore, we apply our k-MM algorithm for clustering sets of trajectories to real-world data recorded from motion capture on humans, GPS cycling trajectories, and GPS trajectories from a fleet of taxis. We examine what the resulting clusters “look like” when the algorithm is applied to various common real-word scenarios for different values of k.
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Keywords
Trajectories, Clustering, Moving entities, K-Center, Algorithms
Citation
Durocher, S., & Hassan, M. Y. (2020). Clustering Moving Entities in Euclidean Space. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik.