Data Structures for Moving Objects Pankaj K. Agarwal Center for - PDF document

Data Structures for Moving Objects Pankaj K. Agarwal Center for Geometric Computing Department of Computer Science Duke University Center for Geometric Computing Geometric Data Structures S : Set of geometric objects Points, segments, polygons ✫ Ask several queries on S • Range searching • Nearest-neighbor searching BSP kd−Tree Quad Tree Data Structures for Moving Objects Center for Geometric Computing 1

Moving Objects: Applications ✫ Traffic management • Location based services • Emergency services • Air traffic control ✫ Digital battlefields ✫ Molecular biology ✫ Deformable objects ✫ Adhoc networks Need data structures for storing, analyzing, querying moving objects. Data Structures for Moving Objects Center for Geometric Computing 2 Modeling Motion p ( t ) = ( x ( t ) , y ( t )) : Position of p at time t . • x ( · ) , y ( · ) : Polynomials • Degree of motion: max degree of x ( · ) , y ( · ) . p(t) • Linear motion: Degree of motion is 1 a , b ∈ R 2 p ( t ) = a t + b , ✫ Mostly assume motion to be linear ✫ Trajectory of points can change ✫ Trajectory can be piecewise linear Issues: ✫ Sampled motion ✫ Hierarchical motion ✫ Uncertainty Data Structures for Moving Objects Center for Geometric Computing 3

Range Searching S : Set of points Preprocess S into a data structure Report all points of S lying inside a query rectangle Data Structure Space Query log n n log n + k Range tree log log n √ n + k n kd-tree External memory data structures also available Example: R-tree Data Structures for Moving Objects Center for Geometric Computing 4 Kinetic Range Searching S : Set of points, each moving with fixed velocity in the plane Preprocess S into a data structure: Q1 Given a rectangle R and a time value t , report all points of S ( t ) ∩ R Q2 Given a rectangle R and time values t 1 , t 2 , report all points that pass through R during the time interval [ t 1 , t 2 ] . t t y y x x Data Structures for Moving Objects Center for Geometric Computing 5

Early Approaches ✫ One-dimensional data structures [Wolfson et al. ’98-’99, Tayeb et al. ’98, Kollios et al. ’99] ✫ Two-dimensional data structures • Map trajectories to higher dimensional points [Kollios et al.] • Build index on trajectories [Pfoser et al.] • Parametric R-trees [Saltenis et al.] Assumes frequent updates on trajectories Data Structures for Moving Objects Center for Geometric Computing 6 Kinetic Range Searching (A., Arge, Erickson, 2001) ✫ Partition-tree based approach • O ( n ) space, ∼ √ n + k query time • log 2 n insertion/deletion/trajectory-change • Time oblivious scheme ✫ Kinetic range trees • n log n/ log log n space, log n + k query • Events: x - or y -coordinates of two points become equal • Θ( n 2 ) events, each requiring log 2 n time • Tradeoff between # events and query time • Queries have to arrive in a chronological order Data Structures for Moving Objects Center for Geometric Computing 7

Partition Tree Based Approach t t yt* xt* t y y x x ✫ Trajectory of a point p i is a line ℓ i in R 3 ✫ p i ( t ) ∈ R ⇐ ⇒ ℓ i intersects ( R, t ) ✫ ℓ x , ℓ y : Projection of ℓ onto the xt - & yt -planes ✫ ℓ intersects ( R, t ) ⇔ ℓ x intersects ( R x , t ) & ℓ y intersects ( R y , t ) ✫ Use duality and partition trees Data Structures for Moving Objects Center for Geometric Computing 8 R-Trees ✫ Bounding box hierarchy, B-tree ✫ Each node v is associated with a subset S v of points and the smallest rectangle R v containing S v ✫ Partition S v into B clusters, each associated with a child of v ✫ Several heuristics are proposed for partitioning S v into B clusters R1 R2 G F R1 R5 E R3 R4 R5 R6 B H A I R3 A B C D E F G H I R2 C D R4 Data Structures for Moving Objects Center for Geometric Computing 9

STAR-Tree ✫ Kinetic R-tree ✫ Maintain the smallest box enclosing the set of moving points • Box is defined by four points • The combinatorial structure can change Ω( n ) times B(t) • Maintain an approximation of the small- est enclosing box ✫ Maintaining the clustering kinetically • Extend the known heuristics • No theoretical nontrivial results known on kinetic clustering Data Structures for Moving Objects Center for Geometric Computing 10 Smallest Enclosing Box ✫ R ( P ( t )) : Smallest box enclosing P at time t ✫ ε -core-set: C ⊆ S ε -coreset if ∀ t (1 − ε ) R ( S ( t )) ⊆ R ( C ( t )) Theorem: ∃ ε -core-set of size 1 / √ ε ; Computation time: n + 1 /ε A more general result on core sets in [A., Har-Peled, Varadarajan] Leads to approximatation algorithms for several problems Data Structures for Moving Objects Center for Geometric Computing 11

Smallest Enclosing Box 1 1 0.99 0.99 0.98 0.98 0.97 Quality Quality 0.97 0.96 0.96 0.95 0.95 0.94 Kernel Size = 16 Kernel Size = 16 Kernel Size = 32 Kernel Size = 28 0.93 0.94 −10 −5 0 5 10 −10 −5 0 5 10 Time Time 18 Linear Motion, Kernel Size = 24 11000 Linear Motion, Input Size = 10,000 16 Quadratic Motion, Kernel Size = 27 10000 Quadratic Motion, Input Size = 10,000 14 9000 12 8000 # Events 7000 10 # Events 6000 8 5000 6 4000 4 3000 2 2000 0 1000 −10 −5 0 5 10 −10 −5 0 5 10 Time Time Data Structures for Moving Objects Center for Geometric Computing 12 Smallest Enclosing Box STAR-tree: Maintain a box enclosing S v at each node v ✫ Compute C v ⊂ S v for each node in a bottom-up manner • Merge the core sets computed at the children of v • Prune the merged set ✫ Maintain the smallest enclosing box R ( C v ) Data Structures for Moving Objects Center for Geometric Computing 13

Re-Clustering ✫ Reorganize the children of a node if the rectangles of their children overlap a lot. u 2 u 3 u 3 w 2 u 1 w 1 v v ✫ Collect all the grandchildren of the node ✫ Reconstruct a 2-level R-tree on them Data Structures for Moving Objects Center for Geometric Computing 14 Experimental Results Synthetic Data ✫ 100,000–500,000 points inside 1000 × 1000 km 2 area with different distributions ✫ Points are inserted/deleted dynamically, at any time at least 80% points present ✫ Three range of speed: 45 km/h, 75km/h, 180 km/h 50 S1 (approx.) S1 (exact) 45 S2 (approx.) 200 S2 (exact) 40 S3 (approx.) 180 S3 (exact) 35 160 Search I/O 30 140 25 120 Search I/O 20 100 80 15 60 10 40 5 20 0 100 150 200 250 300 0 Time 100 150 200 250 300 350 400 450 500 Number of points (x 1000) Data Structures for Moving Objects Center for Geometric Computing 15

Experimental Results Realistic Data ✫ Extracted the roads map around Durham, NC, within 120 miles cen- tered at Durham ( ≈ 250 , 000 polygonal chains) ✫ Computed a planar map of the road network ✫ Chose source and destinations randomly with some distribution ✫ Computed a good path using Dijkstra’s algorithm — minimize the length + number of turns ✫ Used Douglas Peucker algorithm to simplify the paths Data Structures for Moving Objects Center for Geometric Computing 16 Experimental Results 20 18 16 14 12 Avg. Query I/O 10 8 6 4 2 0 50 100 150 200 250 300 Time Data Structures for Moving Objects Center for Geometric Computing 17

Tradeoffs in Performance ✫ Accuracy vs efficiency • Maintain approximate structures ✫ Query vs events • Combine KDS and time-oblivious approaches ✫ Time Responsive Approach: • Near-future queries are more critical than far-future queries • Fast query time for near-future queries • Measure future by the number of events occurred � • # events: ∆ , query: ∆ /n + k Data Structures for Moving Objects Center for Geometric Computing 18 Concluding Remarks ✫ Incorporating more realistic motions • Use dynamic systems, e.g., Kalman, particle filters, to model tra- jectories • How does one perform geometric computation in this model? • Geometric computation under uncertainty ✫ Hierarchical representation of motion ✫ Kinetic data structure for clustering, similarity searching Data Structures for Moving Objects Center for Geometric Computing 19