1. Spatio-temporal Databases Time Parameterized Queries

2. Intro The results of conventional spatial queries are not very useful in dynamic environments because they may be invalidated very soon due to the movements of objects and queries.

3. The Time Parameterized (TP) Window Query Returns: The current query result R The validity period T of R The change of result C at the end of T R={b} Result:

4. The TP Window Query Returns: The current query result R The validity period T of R The change of result C at the end of T R={b}, T=1, C={-b} Result:

5. The TP Nearest Neighbor Query Returns: The current query result R The validity period T of R The change of result C at the end of T R={d} Result:

6. The TP Nearest Neighbor Query Returns: The current query result R The validity period T of R The change of result C at the end of T R={d}, T=1.5, C={f} Result:

7. The TP Spatial Join Query R={(A1, B1), (B3, A4)} Result:

8. The TP Spatial Join Query R={(A1, B1), (B3, A4)} T={1} C={A3, B2} Result:

9. Point Nearest Neighbor (NN) Queries [Roussopoulos et al SIGMOD95, Hjaltason and Samet TODS 99] Branch and bound algorithms use mindist between the query point q and an R-tree entry E, to prune the search space: mindist(E, q) = The minimum distance between E and q

10. Nearest Neighbor Search (NN) with R-Trees Depth-first (DF) and Best-first (BF) algorihms:

11. Reducing TP Window Queries to NN Search Definition: The influence time T INF (o, q) of a data object o indicates the time when o will change the current result of q. The object (C component) invalidating the current query result is the one with the smallest influence time (T component), i.e., a NN query using T INF as the distance metric.

12. Reducing TP Window Queries to NN Search Definition: The influence time T INF (o, q) of a data object o indicates the time when o will change the current result of q. The object (C component) invalidating the current query result is the one with the smallest influence time (T component), i.e., a NN query using T INF as the distance metric.

13. o7o7 Nearest Neighbor (NN) Search with R- trees NN query The algorithm is based on the Branched and Bound framework. p1p1 p2p2 p3p3 o1o1 o2o2 o3o3 o4o4 o5o5 o6o6 o7o7 o1o1 p1p1 o2o2 p2p2 o3o3 o4o4 o5o5 o6o6 p3p3 2 3 3 2.5 q We need 2 metrics: (i) dist(o, q), and (ii) mindist(E, q). mindist(p 3, q) mindist(p 2, q) mindist(p 1, q)

14. Processing TP Queries Treating T INF as the distance function, we may apply the branch and bound paradigm to answer TP queries. Of course we must derive T INF for specific query types. Similar to mindist(E, q) for the NN search, we also need T MININF (E, q), which is the minimum influence time T INF (o, q) among all objects o that can be in the subtree of E.

15. T INF for TP Window Query If an object intersects query q now, its T INF equals the earliest time it stops intersecting q in the future. If an object does not intersect query q now, its T INF equals the earliest time it starts intersecting q in the future. current time 0 T INF (u, q)=2, T INF (v, q)=1

16. T MININF for TP Window Query T MININF (E, q) equals the earliest future time E starts to intersect q if, at the current time E does not intersect query q, or E is contained query q T MININF (E, q)=0 if E intersects (but is not contained in) q TMININF(E, q)=0TMININF(E, q)=1

17. T INF for TP NN Query Assume P NN be the current nearest neighbor of query q; T INF of a data point o equals the time q crosses the perpendicular bisector of line segment P NN o Note that T INF for a TP NN query relies on the current result (i.e., the current NN), while T INF for a TP window query does not. T INF (o, q)=1.5

18. T MININF for TP NN Query T MININF (E, q) of a non-leaf entry E equals the time mindist(E, q)=dist(q, P NN ). As with T INF, it depends on the current query result.

19. BaB Algorithms for TP Queries For those queries (e.g., TP window) where T MIN and T MININF do not depend on the current query result, the T and C components can be retrieved together with the R component in a single traversal of the index structure. For other queries (e.g., TP K-NN) where T MIN and T MININF depend on the current query result, the T and C components can be retrieved together with the R component in separate traversals of the index structure.

20. TP Spatial Join A1A1 A2A2 A3A3 A4A4 A5A5 B1B1 2  B2B2  1  B3B3  42 The TP spatial join is reduced to a closest pair query following the similar idea.

21. Processing Complex Queries The proposed algorithms apply to other mobility combination of objects and queries as well (i.e., mobile objects and static queries, mobile objects and mobile queries). TP queries are preliminary components for Continuous queries Earliest event queries

22. Continuous Nearest Neighbor Data: A set of points Query: A line segment q=[s, e] Result: The nearest neighbor (NN) of every point on q. Result representation: {s(.NN=a), s 1 (.NN=c), s 2 (.NN=f), s 3 (.NN=h), e}

23. One Approach – Time Parameterized Queries Step 1: Find the NN of the start point s, i.e., point a. Step 2: Use the TP technique to find: The first point on the line segment (s c ) where there is a change in the NN (i.e., point c) will become the next NN.

24. TP NN (cont) From Step 2 we have decided the next NN change is point c at s 1 Step 3: Perform another TP NN to find: Starting from s 1, how far we need to travel for the current NN (i.e., c) to change. Repeat this until we finish the entire segment.

25. Another Method: Using Split points Find all split points s 1, s 2, s 3 (as well as the corresponding NN for each partition) with a single traversal of the dataset. Term1: The set of split points (including s and e) constitute the split list. Term2: The circle that centers at split point s i with radius dist(s i, s i.NN) is the vicinity circle of s i. Term3: We say a data point u covers a point s if u=s.NN. E.g., points a, c, f, h cover segments [s, s 1 ], [s 1, s 2 ], [s 2, s 3 ], [s 3, e].

26. Conclusions The time-parameterized query can be integrated with any spatial query type to retrieve predictive information. Processing of TP queries can be reduced to NN search by defining appropriate distance functions. TP queries are preliminary building blocks for more complex queries.