1. Indexing the Positions of Continuously Moving Objects Saltenis, Jensen, Leutenegger and Lopez

2. April 3rd 2002Indexing Moving Objects2 Applications Mobile phones -> wireless internet terminals Location aware service can be provided -> improvement in QoS Vehicle navigation Monitoring positions of air,sea or land based equipments –Airplanes, boats,trucks,…

3. April 3rd 2002Indexing Moving Objects3 Conventional Approach Data assumed constant - modification explicit Capture Continuous movement –Very frequent updates –Outdated, inaccurate data Requirement –Capture movement directly –Advancement of time ~-> necessary explicit updates Solution –Object’s position –function of time and store –Updates(explicit) – function parameters changes

4. April 3rd 2002Indexing Moving Objects4 Indexing Indexing of history Indexing current and anticipated future positions (focus) – Time-Parameterized R-tree(TPR-tree) Approach –Index functional indefinitely – optimized for specific time horizon- deteriorates as time progresses TPR, Bounding rectangles and the moving points are functions of time Intuitively – bounding rectangles follow moving points or other rectangles as they move

5. April 3rd 2002Indexing Moving Objects5 Problem Setting Object’s position at time t :X(t) =(x 1 (t),x 2 (t),…..x d (t)) –Position modeled as a linear fn of time – 2 params –1 st : position of object at t ref i.e. X(t ref ) –2 nd : Velocity of the object V = (v 1,v 2,….v d ) –Modeling as fn of time enables future prediction and solves frequent update problem –Objects report positions and velocity vectors when they deviate from the current value (db value) by certain threshold X(t)=X(t ref ) + V(t ref )

6. April 3rd 2002Indexing Moving Objects6 Moving objects position time 0

7. April 3rd 2002Indexing Moving Objects7 R-Tree at time 0

8. April 3rd 2002Indexing Moving Objects8 R-Tree at time t

9. April 3rd 2002Indexing Moving Objects9 Better arrangement at time t

10. April 3rd 2002Indexing Moving Objects10 Assignment From perspective of queries at time t, the last assignment of objects to MBRs is better –This yields worst performance @ time 0 Assignment of objects to MBRs must take into a/c when most queries will arrive

11. April 3rd 2002Indexing Moving Objects11 Query types Retrieve all points with positions within specified regions. d-dimensional rectangle R spec d projections –[a 1 |-,a 1 -| ],….[a d |-,a d -| ] R, R 1,R 2 all d-dimensional rectangles t, t |- < t -| 3 time values not less than current time

12. April 3rd 2002Indexing Moving Objects12 Queries(contd)  Time Slice query : Q = (R,t) specifies a hyper rectangle R located at time point t  Window query : Q=(R,t|-,t-|) specifies a hyper rectangle R that covers [t |-,t -| ]  Retrieves all points with trajectories crossing (d+1) hyper rectangle (a 1 |-,a 1 -| ),…, (a d |-,a d -| ), [t |-,t -| ]  Moving query : Q=(R 1,R 2,t |-,t -| ) specifies the (d+1) dimensional trapezoid obtained by connecting R1 at time t |- to R2 at time t -| Window query generalises timeslice query Moving query generalises window query

13. April 3rd 2002Indexing Moving Objects13 12345time Value403020100-10-20-30-40o1 o2 o4 o3 Q0 Q1 Q2 Q3

14. April 3rd 2002Indexing Moving Objects14 Query examples Q0 and Q1 are timeslice queries Q2 is window query and Q3 moving query Iss(Q) – time when query issued –Ref position and velocity depend on issue(Q) because objects update their parameters as time goes O1 : movement desc by 1 trajectory for iss(Q) < 1 –Another for 1 = 3 Answer to query Q1 is o1 if iss(Q1) 1 Queries in far future little value because positions predicted less accurate Real World – expect queries concentrated in some limited time window extending from current time

15. April 3rd 2002Indexing Moving Objects15 Problem Parameters 3 params affect indexing problem and qualities of TPR-tree Querying window(W) : how far queries can look into the future –Iss(Q) <= t <= Iss(Q) + W for timeslice queries –Iss(Q) <= t |- <= t -| <= Iss(Q) + W for other queries Index Usage Time(U) : time interval during which an index will be used for querying –t l <= Iss(Q) <= t l + U t l index creation time Time Horizon(H) : length of the time interval from which t,t |-, t -| are drawn –Time horizon for an index is index usage time plus the querying window

16. April 3rd 2002Indexing Moving Objects16 Newly created index must support queries that reach H units in future H=U+W tltltltl Iss(Q) U t |- t -| W

17. April 3rd 2002Indexing Moving Objects17 Index Structure TPR tree is a balanced, multi way tree with structure of an R- tree Leaf nodes – pairs of positions of a moving point and a pointer to the moving point Internal nodes – pairs of pointer to subtree and a rectangle that bounds the positions of all moving points or other bounding rectangles in subtree Time parameterized d-dimensional bounding rectangles bound d-dimensional moving points or rectangles at all time not earlier than current time

18. April 3rd 2002Indexing Moving Objects18 Conservative bounding rectangles Min at some time but possibly not at later times In 1d case lower bound of a conservative interval is set to move with the minimum speed of the enclosed points, upper bounds move with max speed of enclosed points –Speeds +ve or –ve dep on direction Conservative bounding intervals never shrink Constant size when all points have same velocity vector –It may move

19. April 3rd 2002Indexing Moving Objects19 Querying A bounding interval (x |-,x -|,v |-,v -| ) satisfies a query (([a |-,a -| ]),t q ) iff –a |- = x |- + v |- (t q -t l ) To answer window queries and moving queries we need to check if in (X,t) space the trapezoid of a query intersects with the trapezoid of a bounding rectangle that is b/n the start and end time times of the query –Generic polyhedron-polyhedron intersection tests may be used