1. Handling Spatial Data In P2P Systems Verena Kantere, Timos Sellis, Yannis Kouvaras

2. Problem Definition Assumptions: Structure P2P Overlays Spatial Data in Peers Problem: Necessity for indexing and routing techniques for such an environment

3. Technique Requirements We need a technique that: Guarantees the retrieval of any existing spatial information in the system Achieves satisfying index size for any node Achieves a satisfying length for any search path Provides easier access to popular information Preserves proximity of areas

4. Related Work P2PR-tree (Workshop of EDBT ‘04) Quad-tree approach (Poster in ICDE’05) kd-tree approach (WebDB’04)

5. Partitioning Space Spatial Coding

6. Partitioning Space cont’d The code of an area A is: A  a z a x a y a z : is the granularity level, i.e. the size in terms of cells. a x : is the average of the digit d 1 of all the area cells. a y : is the average of the digit d 2 of all the area cells. Distance Metrics: d 1 (A 1, A 2 ) = |a 1x -a 2x |+|a 1y -a 2y | d 2 (A 1, A 2 ) = ||a 1x -a 2x |-|a 1y -a 2y || D(A 1, A 2 )  d 1 d 2 D zoom (A 1, A 2 )  D(A 1, A 2 )/(0.5*|A 1.a z +A 2.a z |)

7. Distributed Indexing for Spatial Areas Distributed Indexing Algorithm Step 1: Indexing the grids of each level Phase A: Indexing areas of the same grid Phase B: Indexing areas of other grids of the same level Step 2: Indexing the grids of other levels

8. Grid Example

9. Direct Neighbors & Data Hashing Peers have 2 neighbors on each dimension: Horizontal Vertical Perpendicular Neighbors are the closest peers on each direction. Areas are stored to the closest neighbor following a priorities of dimensions

10. Direct Neighbors: Examples

11. Routing of Spatial Queries Basic Routing Algorithm Step 1: Find a peer corresponding to the same size as the sought area. Step 2: Find a peer hosting the sought area or an overlapping area of the latter Step 3: Find the peer hosting the sought area

12. Routing Example

13. Details…

14. Complexity Characteristics Complexity for both pure Chord and our method is O(3log 2 n). But with our method: Peers with big areas are favored with small indexes Shorter search paths for bigger areas The search path grows with the relative distance

15. Assignment of Peer IDs The peer ID plays a significant role in our approach  joining peers can ask for specific ids: it is adequate to assign an id that belongs to the same or similar level as the requested id The above constraint loosens with the size of the area (higher levels are more connected)

16. Preliminary Experiments

17. Preliminary Experiments (2)