Abstract
One of the important tasks in the analysis of spatio-temporal data collected from moving entities is to find a group: a set of entities that travel together for a sufficiently long period of time. Buchin et al.2 introduce a formal definition of groups, analyze its mathematical structure, and present efficient algorithms for computing all maximal groups in a given set of trajectories. In this paper, we refine their definition and argue that our proposed definition corresponds better to human intuition in certain cases, particularly in dense environments. We present algorithms to compute all maximal groups from a set of moving entities according to the new definition. For a set of [Formula: see text] moving entities in [Formula: see text], specified by linear interpolation in a sequence of [Formula: see text] time stamps, we show that all maximal groups can be computed in [Formula: see text] time. A similar approach applies if the time stamps of entities are not the same, at the cost of a small extra factor of [Formula: see text] in the running time, where [Formula: see text] denotes the inverse Ackermann function. In higher dimensions, we can compute all maximal groups in [Formula: see text] time (for any constant number of dimensions), regardless of whether the time stamps of entities are the same or not. We also show that one [Formula: see text] factor can be traded for a much higher dependence on [Formula: see text] by giving a [Formula: see text] algorithm for the same problem. Consequently, we give a linear-time algorithm when the number of entities is constant and the input size relates to the number of time stamps of each entity. Finally, we provide a construction to show that it might be difficult to develop an algorithm with polynomial dependence on [Formula: see text] and linear dependence on [Formula: see text].
Highlights
Nowadays, inexpensive modern devices with advanced tracking technologies make it easy to track movements of an entity
We show that for a set X of n moving entities in R1 with τ time stamps each, the number of maximal groups by the refined definition is O(τ n3), which is tight in the worst case. 48:4 A Refined Definition for Groups of Moving Entities and its Computation In Section 3, we present algorithms to compute all maximal groups in R1
When the time stamps of trajectories are not the same, we show that our algorithm runs in O(τ 2n4α(n)) time
Summary
Inexpensive modern devices with advanced tracking technologies make it easy to track movements of an entity. To analyze moving object data, a number of methods have been developed in recent times These methods perform similarity analysis or compute a clustering, outliers, a segmentation, or various patterns that may emerge from the movement of the entities (for surveys see [3, 15]). In [2], a group for an entity inter-distance ε, a minimum required duration δ, and a minimum required size m, is defined as a subset G ⊆ X and corresponding time interval I for which three conditions hold: (i) G contains at least m entities. These entities in between are not part of the maximal group, but they do cause x and y to be ε-connected by the previous definition This can have counter-intuitive effects especially in dense crowds. We do not consider {a, h} a group in the interval I, and not a maximal group
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