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

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 2 Content  1 Introduction  2 Indexing Mobile Objects  3 Indexing in 2 Dimensions  4 Summary  5 Reference

CSIS Advanced Database Technologies 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.

CSIS Advanced Database Technologies 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. 1. Object with low speed v  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

CSIS Advanced Database Technologies 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 [(y 1q, y 2q ), (t 1q, t 2q )].

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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 + a is represented by a point (v, a) in dual space.  Hough-Y plane: y(t) = vt + a is transformed to t = 1/v y(t) – a/v. In dual space, the coordinates are n = 1/v and b = -a/v.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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 C 1 = v  v min, C 2 = v  v max, C 3 = a + t 2q v > y 1q, and C 4 = a + t 1q v <= y2q. For v < 0. the query is : Q = D1  D2  D3  D4, where D 1 = v  -v min, D 2 = v  -v max, D 3 = a + t 1q v  y 1q, and D 4 = a + t 2q v  y 2q.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Dual Space-Time Representation Problem : range of a = [- v max x t now, y max - v min x 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Dual Space-Time Representation  Two distinct index structures. Each object stores only once. [0, T period ] and [T period, 2T period ] 1 st index: The intercept is computed using the line t = 0. 2 nd 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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, 2T period ] and [2T period, 3T period ]  With this method, intercept is bounded, and Performance of index structure remains asymptotically the same.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Simplex range searching  Two approaches: 1. Point Access Methods 2. Query Approximation Algorithm

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Query Approximation Algorithm  Query Approximation Algorithm  Using Hough-Y dual plane

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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 = E 1 + E 2, and

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Query Approximation Algorithm  E is based on y r (where the b coordinate is computed) and the query interval (y 1q, y 2q ), which is unknown.  If a constant c is use to keep equidistant y r, we can have observation index, y i = y max /c × i

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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 [y 1q, y 2q ], two cases will be considered,  Case 1: y 2a – y 1q ≤ y max / c The area is bounded by

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Query Approximation Algorithm  Case 2: y 2q – y 1q > 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).

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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 = t 1q = t 2q ).

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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, y r ] in O(log 2 N + K) time.  Consider objects {p 1, p 2, ……., p n }, p i has relative position y i and a velocity v i. At time t q, the position p i is p i = y i + v i × t q. The objects stored in the binary tree are sorted.  In O(log 2 N) time we can find the positions of y l and y r relative the objects at time t q, and we report the objects that lie between.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Achieve Logarithmic Query Time  e.g. An object list {p 1, p 2, p 3, p 4, p 5, p 6, p 7 }, and at time t q and say, [y l, y r ] = [p 3, p 6 ]

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Achieve Logarithmic Query Time  Lemma 3 We can find all the crossings of objects in time O (Nlog 2 N + Mlog 2 M), where M is the number of crosses in the time period [0,T].  Consider N Objects {p 1, p 2, ……, p N }, p i has relative position y i and a velocity v i at time 0. Another sorted order list {p t(1), p t(2), ……p t(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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Achieve Logarithmic Query Time  e.g. Ordered list {p 1, p 2, p 3, p 4, p 5, p 6, p 7 } at time t = 0 Ordered list {p 1, p 3, p 2, p 6, p 4, p 5, p 7 } at time t = T {, p 3, p 2, p 6, p 4, p 5, p 7 } {, p 3,, p 6, p 4, p 5, p 7 } crossing {,,, p 6, p 4, p 5, p 7 } {,,, p 6,, p 5, p 7 } crossing {,,, p 6,,, p 7 } crossing {,,,,,, p 7 } {,,,,,, } 3 Crossings !  M crossings define M ordered list of the N objects.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Achieve Logarithmic Query Time  Lemma 4 We store the O(M) order lists of N objects 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(t 1 ), B(t 2 ), …, B(t M ).  Post the entries of s as 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) 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.

CSIS Advanced Database Technologies Spatio-Temporal Data (Part 1) Reference  George Kollios, Dimitrios Gunopulos and Vassilis Tsotras. On Indexing Mobile Objects. ACM PODS, 1999.

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