Trajectory Compression

Trajectory compression is a very promising field, since the amount of data collected by positioning devices will sooner or later expand with excessive rates. Therefore, research on trajectory compression has to be extended toward many directions.

A first direction is based on the idea of theST Tracealgorithm, where the amount of memory used to store each trajectory is a priori determined. Consider now the following task posed against a trajectory database: compress a given trajectory T with n vertices to produce Q, which must contain only j<n vertices. Although

the ST Tracealgorithm is suitable for such procedures applied on trajectory data streams, it concerns only local optimization. Therefore, in the above case where the trajectory is completely known in advance, it is possible to apply a different algo-rithm concerningglobal optimization. Such algorithms need to be examined in the future, providing also a bounded error.

A second direction also regarding the compression algorithms is the utilization of the insightful idea of amnesic approximation for streaming time series, which was initially introduced in [41], considering that the importance of each measure-ment generally decays with time. However, it is not possible to utilize this idea straightforwardly since the work presented in [41] only handles one-dimensional time series. Moreover, authors in [51] argue that apart from handling multidimen-sional points, the case for trajectories is different because of characteristics inherent in movement: not only spatial locations, but also speed and orientation should not be overlooked when approximating trajectories.

One first solution on this subject is presented with theAmTreeintroduced in [50].

The structure of an AmTreeis illustrated in Fig. 6.20; each leveli, except for the tree root, consists of two nodesRiandLi. NodeR₀at the lowest level accepts data in every timestamp that are reported. Each node at theith level contains informa-tion about twice as many timestamps as a node at the(i−1)th level. Hence, a node at levelicontains information about 2ⁱtimestamps.Amtreeis built in a bottom-up fashion. As new sampled positions are feeding it, they are added into positionR₀, while the contents of nodeR₀are shifted toL₀; with the addition of one more sam-pled position, the contents ofL₀andR₀are combined using a simple function and propagated to the higher tree level. Potamias et al. [50] propose the employment of motion vectors (e.g. tuples of the formx,y,t_start,t_end,dx,dy) inside each tree node; then the combination function produces the summarization of the two vec-tors. A more sophisticated approach would include the treatment of the combination function with a similar algorithm as the one proposed in the previous section. Such

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Fig. 6.20 TheAmTreestructure [50]

182 E. Frentzos et al.

Compressed Trajectory Q

Original Trajectory P B

A

C

Fig. 6.21 Compressing trajectory data under network constraints

a function would compress a given partial trajectoryT constituting ofnvertices to produceQ, which must contain onlyn/2 vertices. This approach would lead to a compressed trajectory with higher compression rates – more abstract information – for aged data and lower compression rates for recent data, with the advantage that it incorporates at the same time the inherited requirements of trajectory data (e.g., considering together the spatial locations, the direction, and the speed of the objects).

A third research direction could consider network constraint compression of tra-jectory data. Consider for example Fig. 6.21 illustrating an uncompressed tratra-jectory P constrained on an underlying road network and the respective compressed tra-jectory Qproduced by one of the previously described algorithms; although the original trajectory is valid under the network constraint, the compressed one is not sinceQdirectly connects nonneighboring network nodes (e.g., line segmentABin Fig. 6.21). A methodology towards the solution of this issue would be to initially compress the trajectory data with one of the existing algorithms and then use the underlying network to produce routes connecting the – under network constraints – unaffected by the compression and trajectory nodes. However, such an approach would be computationally inefficient since it would require performing many – possibly overlapping – routings through the network graph. Moreover, one has to examine whether the produced network route between two remained trajectory nodes still – after the routing – verifies the thresholds utilized by the compression algorithm.

The last issue on the subject of trajectory compression requires the determina-tion of the error introduced in query results when trajectory data are compressed.

For example, rather than determining the mean error of the trajectory approxima-tion, as discussed in [36], it might be more meaningful for a system user to know the influence of the compression in the results returned by the database when execut-ing a range or nearest-neighbor query against the trajectory data. Then, along with the query results, the database would inform the user about an estimation of the per-centage of the false hits introduced in the query results. There are two approaches

that may clarify this issue: either performing experiments under various compressed trajectory data sets or theoretically determining the effect of the compression by assuming uniformity of data and then using spatiotemporal histograms to overpass such an assumption.