CSIS7101 – Advanced Database Technologies

1. CSIS7101 – Advanced Database Technologies Spatio-Temporal Data (Part 1) On Indexing Mobile Objects Kwong Chi Ho Leo Wong Chi Kwong Simon Lui, Tak Sing Arthur

2. Content • 1 Introduction • 2 Indexing Mobile Objects • 3 Indexing in 2 Dimensions • 4 Summary • 5 Reference Spatio-Temporal Data (Part 1)

3. 1. Introduction • Traditional Database Model assume that data stored in the database remain constant. • Not appropriate for applications with continuously changing data. • Better approach: Object location as a function of time f(t), and update the database only when the parameters of f change. • This approach introduces a novel problem since the database is not directly storing data values but function to compute these values. Spatio-Temporal Data (Part 1)

4. 2. Indexing Mobile Objects • 2.1 Partition the mobile objects into two categories: Let’s say objects are moving on an 1-dimensional plane. • Object with low speed v  0. • Objects with speed between v min and v max. (i.e. Moving Objects) • 2.2 Space-Range Representation • 2.3 Dual Space-Time Representation • 2.4 Simplex Range Searching • 2.5 Achieve Logarithmic Query Time Spatio-Temporal Data (Part 1)

5. 2. Indexing Mobile Objects • 2.2 Space-Range Representation • Plot the trajectories y(t) = v t + a of the mobile objects as line in time-location (t,y) plane. • The query is expressed as 2-dimensional interval [(y1q, y2q), (t1q, t2q)]. Spatio-Temporal Data (Part 1)

6. 2.2 Space-Range Representation • Shortcomings: • Minimum Bounding Rectangle (MBR) much larger than a Line. • Object retains trajectory until updated, all lines extend to “infinity”. • Mapping a line segment as a point in 4 dimensions, will not work. • SAM can only access queries until end of the current session. Spatio-Temporal Data (Part 1)

7. 2.3 Dual Space-Time Representation • Map a line from primal plane (t, y) to a point in the dual plane. • Hough-X plane: • y(t) = v t + ais represented by a point (v, a) in dual space. • Hough-Y plane: • y(t) = vt + ais transformed to t = 1/v y(t) – a/v. In dual space, the coordinates are n = 1/v and b = -a/v. Spatio-Temporal Data (Part 1)

8. 2.3 Dual Space-Time Representation • Hough-X Dual Space • Query is transformed in a polygon query in the dual space. This polygon uses a linear constraint polygon. Spatio-Temporal Data (Part 1)

9. 2.3 Dual Space-Time Representation • The 1-dimensional MOR query in dual Hough-X is expressed as For v > 0 the query is : Q = C1 C2 C3 C4, where C1 = v  v min, C2 = v  v max, C3= a + t2qv > y1q, and C4 = a + t1qv <= y2q. For v < 0. the query is : Q = D1 D2 D3 D4, where D1 = v  -v min, D2 = v  -v max, D3 = a + t1qv  y1q, and D4 = a + t2qv  y2q. Spatio-Temporal Data (Part 1)

10. 2.3 Dual Space-Time Representation Problem: range of a = [-v maxx t now , y max-v minx t now ] • Time is increasing, values of the intercepts are not bounded. • V max is significant, values of intercept become very large, and potentially a problem (unbounded range of real numbers) To solve: • Assuming that when an object crosses a border it issues an update (deletion or reflection). • With minimal speed, all objects updated their motion information at least once during a T period time instant, where T period = y max / v min. Spatio-Temporal Data (Part 1)

11. 2.3 Dual Space-Time Representation • Two distinct index structures. Each object stores only once. [0, T period] and[T period, 2T period] 1st index: The intercept is computed using the line t = 0. 2nd index: The intercept is computed using the line t = T period. • With this, the intercept will always have the values between 0 and v max × T period. • Both indices are used in database query. Spatio-Temporal Data (Part 1)

12. 2.3 Dual Space-Time Representation • Keeping on that, after time 2T period, the first index is deleted and a new index is created. We then have [T period, 2Tperiod] and [2Tperiod, 3Tperiod] • With this method, intercept is bounded, and Performance of index structure remains asymptotically the same. Spatio-Temporal Data (Part 1)

13. 2.4 Simplex range searching • The dual space-time representation transforms the indexing mobile objects on a line to simplex range searching in two dimensions. • Theorem 1 (Lower Bound) Simplex reporting in d-dimensions with a query time of O (n δ + k), where N is the number of points, n = N/B, K is the number of the reported points, k = K/B and 0 < δ ≤ 1, requires space Ω (n d(1- δ) – є ) disk blocks, for any fixed є. Spatio-Temporal Data (Part 1)

14. 2.4 Simplex range searching • Two approaches: • Point Access Methods • Query Approximation Algorithm Spatio-Temporal Data (Part 1)

15. 2.4.1 Point Access Method • Point Access Method • Using R-trees to answer simplex range queries, the 1-dimensional MOR query can be answered in the dual Hough-X space. Spatio-Temporal Data (Part 1)

16. 2.4.2 Query Approximation Algorithm • Query Approximation Algorithm • Using Hough-Y dual plane Spatio-Temporal Data (Part 1)

17. 2.4.2 Query Approximation Algorithm • Since rectangular query is an efficient access method, the query is approximated by a rectangular one. • The query rectangle • The query area is enlarged by area E = E1 + E2, and Spatio-Temporal Data (Part 1)

18. 2.4.2 Query Approximation Algorithm • E is based on yr (where the b coordinate is computed) and the query interval (y1q, y2q), which is unknown.   • If a constant c is use to keep equidistant yr, we can have observation index, y i = y max /c × i Spatio-Temporal Data (Part 1)

19. 2.4.2 Query Approximation Algorithm • A given 1-dimensional query MOR query will be forwarded to the index(es) that minimize E. • Rectangle range search = simple range search on the b coordinate axis. • Each of the observation index can be a B+ tree. • To process query interval [y1q, y2q], two cases will be considered, • Case 1: y2a – y1q ≤ y max / c • The area is bounded by Spatio-Temporal Data (Part 1)

20. 2.4.2 Query Approximation Algorithm • Case 2: y2q – y1q > y max / c • The query is decomposed into collection of smaller sub-queries: one sub-query per sub-terrain contained by the original query’s endpoints. The sub-queries can be answered with bounded E using an appropriate “ observation” index. • Lemma 1 The one dimensional MOR query can be answered in time O(log B n + (K + K’) / B), where K’ is the approximation error. The space used is O(c n) where c is a small constant, and the update is O(c log B n). Spatio-Temporal Data (Part 1)

21. 2.5 Achieve Logarithmic Query Time • Restrict our queries to occur before the first time that a point overtakes another. • To index mobile objects in a bounded time interval T in the future. • Consider objects are moving on a line, Find all the objects that lie in the segment [ y l , y r ] at time t q (i.e. t q = t1q = t2q). Spatio-Temporal Data (Part 1)

22. 2.5 Achieve Logarithmic Query Time • Lemma 2 If we have the relative ordering of all the objects at time t q, the position of the objects at time t c that corresponds to the closest crossing event before t q, and the speed of the objects, we can find the objects that are in [y l, yr] in O(log2N + K) time. • Consider objects {p1, p2, ……. , p n}, pihas relative position y i and a velocity vi. At time t q, the position pi is pi = y i + vi × t q. The objects stored in the binary tree are sorted. • In O(log2N) time we can find the positions of yland yr relative the objects at time t q, and we report the objects that lie between. Spatio-Temporal Data (Part 1)

23. 2.5 Achieve Logarithmic Query Time • e.g. An object list {p1, p2, p3, p4, p5, p6, p7}, and at time t q and say, [y l, yr] = [p3, p6] Spatio-Temporal Data (Part 1)

24. 2.5 Achieve Logarithmic Query Time • Lemma 3 We can find all the crossings of objects in time O (Nlog2N + Mlog2M), where M is the number of crosses in the time period [0,T]. • Consider N Objects {p1, p2, ……, pN}, pihas relative position yi and a velocity vi at time 0. Another sorted order list {pt(1) , pt(2) , ……pt(N)} at time T. Then objects i and j cross if and only if t(j) < t(i). • Keep the objects in a linked list, in the same order they were at time t = 0. Scan the sorted lit of objects at time t = T. All the crossings can be reported in O(N + M). After all crossings are reported we can find when each occurs and sort them on their time attribute. Spatio-Temporal Data (Part 1)

25. 2.5 Achieve Logarithmic Query Time • e.g. Ordered list {p1, p2, p3, p4, p5, p6, p7} at time t = 0 Ordered list {p1, p3, p2, p6, p4, p5, p7} at time t = T { , p3, p2, p6, p4, p5, p7} { , p3, , p6, p4, p5, p7} crossing { , , , p6, p4, p5, p7} { , , , p6, , p5,p7} crossing { , , , p6, , ,p7} crossing { , , , , , ,p7} { , , , , , ,} 3 Crossings ! • M crossings define M ordered list of the N objects. Spatio-Temporal Data (Part 1)

26. 2.5 Achieve Logarithmic Query Time • Lemma 4 We store the O(M) order lists of Nobjects in O(n+m) blocks and perform a search on any list in O(log B(n+m)) I/O’s, where n = N/B and m = M/B. • Embed the binary tree structure of the list of N objects inside a B-tree. Each B tree use O(n) nodes where each node hold B entries. • A s node stores the evolution of trees B(0), B(t1), B(t2), …, B(t M). • Post the entries of sas changes in the history of the parent node p. When a new copy of node s is created, a new record is added on the “log” of p that has the same position l, but a pointer to the new copy of s and the current time. Spatio-Temporal Data (Part 1)

27. 2.5 Achieve Logarithmic Query Time • Overall space O(n+m), and • Query time O(log B(n+m)) • Theorem 2: Given N objects and a time limit T, an one dimensional MOR1 query can be answered in time log B (n +m) using space O(n +m), where m = M/B and M is the number of crosses of objects in the time limit T . Spatio-Temporal Data (Part 1)

28. 3. Indexing in Two Dimensions • The 1.5-dimensional problem • Objects moving in the plane but their movement is restricted on using a given collection of routes (roads) on the finite terrain. • Each predefined route can be represented as a sequence of connected (straight line) segments. • Indexing the line segment by standard SAM. • Indexing the object moving on the given route is an1-dimensional model. Spatio-Temporal Data (Part 1)

29. 3. Indexing in Two Dimensions • The 2-dimensional problem • Decompose the motion of the object into two independent motions, one in the x-axis and the other in the y-axis. • For each axis, use the methods for the 1-dimensional case and answer two 1-dimensional MOR queries. The algorithms for the 1-dimensional case can be used. • Take the intersection of the two answers to find the answer to the initial query. Spatio-Temporal Data (Part 1)

30. 4. Summary • One-dimensional case, We have: • A dynamic, external memory algorithm with guaranteed worst case performance and linear space. • A practical approximation algorithm also in the dynamic, external memory setting, which has linear space and expected logarithmic query time. • An algorithmic query time for a restricted version of the problem. • Two-dimensional case, • Extending the techniques to 2-dimensional networks of 1-dimensional routes as well objects moving on a plane. Spatio-Temporal Data (Part 1)

31. 5. Reference • George Kollios, Dimitrios Gunopulos and Vassilis Tsotras. On Indexing Mobile Objects. ACM PODS, 1999. Spatio-Temporal Data (Part 1)

32. On Indexing Mobile Objects The End Break or Move on to Presentation 2 Spatio-Temporal Data (Part 1)