1. Indexing of Network Constrained Moving Objects Dieter Pfoser Christian S. Jensen Chia-Yu Chang

2. 2 Outline Introduction The Trajectory Case Reducing Dimensionality Performance Studies Conclusions

3. 3 Introduction (1/2) Concern with the indexing of the movements of mobile objects for post-processing (e.g. data mining) purpose. The movement of an object may be represented by a trajectory, or polyline, in the three dimensional (x, y, t) space. time x y

4. 4 Introduction (2/2) Three movement scenarios: 1. Unconstrained movement (vessels at sea) 2. Constrained movement (pedestrians) 3. Movement in transportation networks (trains, cars)

5. 5 The Trajectory Case First approach: simply store the position. → we couldn’t answer queries about the object’s movements at time s in-between those of the sampled positions. → use linear interpolation

6. 6 Indexing Trajectory Trajectory are 3D spatial entities, and they can be indexed using spatial methods. The R-tree approximates the data objects by Minimum Bounding Boxes (MBBs). Large amounts of “dead space”.

7. 7 Reducing Dimensionality Translate 2D (network) into one Dimension. Translate 3D into two Dimensions. e.g., cars move on roads. Overall, we have to devise mappings for 1. the Network 2. the Trajectories 3. the Queries

8. 8 Network Mapping (1/2) Algorithm NetworkMapping (network) LOCALSrange //highest coordinate low //lower coordinate of edge in 1D space up //upper coordinate of edge in 1D space NM1 sort edges by their FOR ALL edges NM2 compute length of edge NM3 low = range+ 1 NM4 up = range+ 1+ length NM5 write edge(low, up) NM6 range = up END FOR Hilbert value

9. 9 Network Mapping (2/2) Algorithm NetworkMapping (network) FOR ALL edges NM2 compute length of edge NM3 low = range+ 1 NM4 up = range+ 1+ length NM5 write edge (low, up) NM6 range = up END FOR

10. 10 Trajectory Mapping (1/2) Algorithm TrajectoryMapping (trajectory, 2Dnetwork, 1Dnetwork) FOR ALL segments of the trajectory TM1 find traversed network edge in 2Dnetwork TM2 det. traversed portion of edge in 2Dnetwork TM3 x0, x1 = respective 1Dnetwork coordinates TM4 write segment(x0, t0, x1, t1) END FOR

11. 11 Trajectory Mapping (2/2) Algorithm TrajectoryMapping (trajectory, 2Dnetwork, 1Dnetwork) FOR ALL segments of the trajectory TM1 find traversed network edge in 2Dnetwork TM2 det. traversed portion of edge in 2Dnetwork TM3 x0, x1 = respective 1Dnetwork coordinates TM4 write segment(x0, t0, x1, t1) END FOR

12. 12 Query Mapping Algorithm QueryMapping(query, 2Dnetwork) //2Dnetwork access using an R-tree structure QM1 given a query window, take the spatial extent and retrieve the portion contained in it QM2 lift the retrieved edges by the temporal extent of the query window

13. 13 Performance Studies (1/9) Three synthetic networks: 1. Hilbert network, “h”, 1023 2. Raster network, “r2”, 544 3. Parallel network, “p”, 33

14. 14 Performance Studies (2/9) Two real networks: 1. San Jose, CA, 24123 2. Oldenburg, Germany, 7035

15. 15 Performance Studies (3/9) Index structure for 3D and 2D Trajectory: R-Tree implementation in C. Page size of each node is 1024 bytes which results in maximum fanouts of 36 for 3D and 51 for 2D indexes. Different types of networks. The impact of varying number of edges. → r1 (144), r2 (544), r2 (2112), r4 (8320)

16. 16 Performance Studies (4/9) 500 moving objects which positions are sampled 250 times each. → 125k trajectory segments each. Sizes of 2D and 3D indexes are 2.5MB and 3.35MB. 500 quadratic query windows, each with spatial extents of 0.25%, 0.5%, 1%, 2%, 4%, and 8% of the quadratic 2D space. Temporal extent of the query was kept constant at 10%.

17. 17 Performance Studies (5/9) Different types of synthetic networks:

18. 18 Performance Studies (6/9) Networks of the same type but varying lengths and numbers of edges:

19. 19 Performance Studies (7/9) Varying temporal extent for the Raster network:

20. 20 Performance Studies (8/9) Different types of real networks:

21. 21 Performance Studies (9/9) Different types of real networks:

22. 22 Conclusion The dimensionality of trajectories can be reduced from three to two. The number of 2D queries that result from the mapping of a 3D query is critical. The larger it is, the less likely it is that the mapping approach outperforms querying data in the original space. The lower complexity of a network, the more likely the mapping approach proves to be beneficial over indexing the data in 3D space.