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Team-based invasion sports such as football, basketball and hockey are similar in the sense that the players are able to move freely around the playing area; and that player and team performance cannot be fully analysed without considering the movements and interactions of all players as a group. State of the art object tracking systems now produce spatio-temporal traces of player trajectories with high definition and high frequency, and this, in turn, has facilitated a variety of research efforts, across many disciplines, to extract insight from the trajectories. We survey recent research efforts that use spatio-temporal data from team sports as input, and involve non-trivial computation. This article categorises the research efforts in a coherent framework and identifies a number of open research questions.

Given a set V of n points in R d and a real constant t > 1, we present the first O(n log n)-time algorithm to compute a geometric t-spanner on V . A geometric t-spanner on V is a connected graph G = (V, E) with edge weights equal to the Euclidean distances between the endpoints, and with the property that, for all u, v ∈ V , the distance between u and v in G is at most t times the Euclidean distance between u and v. The spanner output by the algorithm has O(n) edges and weight O(1) · wt(MST ), and its degree is bounded by a constant. Introduction.Complete graphs represent ideal communication networks, but they are expensive to build; sparse spanners represent low-cost alternatives. The weight of the spanner network is a measure of its sparseness; other sparseness measures include the number of edges, the maximum degree, and the number of Steiner points. Spanners for complete Euclidean graphs as well as for arbitrary weighted graphs find applications in robotics, network topology design, distributed systems, design of parallel machines, and many other areas and have been a subject of considerable research [1,2,4,8,11].Consider a set V of n points in R d , where the dimension d is a constant. A network on V can be modeled as an undirected graph G with vertex set V and with edges e = (u, v) of weight wt(e). A Euclidean network is a geometric network where the weight of the edge e = (u, v) is equal to the Euclidean distance d(u, v) between its two endpoints u and v. Let t > 1 be a real number. We say that G is a t-spanner for V if, for each pair of points u, v ∈ V , there exists a path in G of weight at most t times the Euclidean distance between u and v. A sparse t-spanner is defined to be a t-spanner of size (number of edges) O(n) and weight (sum of edge weights) O(1) · wt(MST ), where wt(MST ) is the total weight of a minimal spanning tree. Given a geometric network G = (V, E), a (generic) weight function wt defined on its edges, and two vertices u, v ∈ V , we let D {G,wt} (u, v) denote the weight of the shortest path from u to v in G for the weight function wt.The problem of constructing spanners has been investigated by many researchers. Levcopoulos and Lingas [10] presented an O(n log n)-time algorithm that produced a sparse t-spanner for the two-dimensional case. It works by taking any t-spanner which has the form of a (possibly partial) triangulation and achieving almost the same t as that triangulation. However, the problem gets much more difficult in higher *

In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Fréchet distance and the discrete Fréchet distance between subtrajectories, which are invariant under differences in speed. We give several approximation algorithms, and also show that the problem of finding the 'longest' subtrajectory cluster is as hard as MaxClique to compute and approximate.

Data representing moving objects is rapidly getting more available, especially in the area of wildlife GPS tracking. It is a central belief that information is hidden in large data sets in the form of interesting patterns. One of the most common spatio-temporal patterns sought after is flocks. A flock is a large enough subset of objects moving along paths close to each other for a certain pre-defined time. We give a new definition that we argue is more realistic than the previous ones, and by the use of techniques from computational geometry we present fast algorithms to detect and report flocks. The algorithms are analysed both theoretically and experimentally.

Abstract. Widespread availability of location aware devices (such as GPS receivers) promotes capture of detailed movement trajectories of people, animals, vehicles and other moving objects, opening new options for a better understanding of the processes involved. In this paper we investigate spatio-temporal movement patterns in large tracking data sets. We present a natural definition of the pattern 'one object is leading others', which is based on behavioural patterns discussed in the behavioural ecology literature. Such leadership patterns can be characterised by a minimum time length for which they have to exist and by a minimum number of entities involved in the pattern. Furthermore, we distinguish two models (discrete and continuous) of the time axis for which patterns can start and end. For all variants of these leadership patterns, we describe algorithms for their detection, given the trajectories of a group of moving entities. A theoretical analysis as well as experiments show that these algorithms efficiently report leadership patterns.

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