Compact and Low Delay Routing Labeling Scheme for Unit Disk Graphs

1. Compact and Low Delay Routing Labeling Scheme for Unit Disk Graphs Chenyu Yan, Yang Xiang, and Feodor F. Dragan (WADS 2009) Kent State University, Kent, OH, USA

2. Unit Disk Graphs • Unit Disk Graphs are the intersection graphs of equal sized circles in the plane. Model wireless networks Algorithmic Lab, Kent State University

3. Previous works • Routing schemes in UDG • Heuristic, no guarantee of delivery or optimality of routing path • Shortest path routing, routing table is large and hard to build and maintain. • Sparse spanners for UDG • Bounded degree • Planar • Small stretch factor Algorithmic Lab, Kent State University

4. Stretch Factor • Length stretch factorThe length of the routing path (or shortest path in a spanner) over the minimum length of the path in the original graph • Hop stretch factor The hop of the routing path (or shortest path in a spanner) over the minimum hop of the path in the original graph Algorithmic Lab, Kent State University

5. Spanner Length Stretch Factor Hop Stretch Factor Gabriel Graph - Yao Graph - Unit Del - LDel Very Large Constant Known Sparse spanners for UDGs Algorithmic Lab, Kent State University

6. Unit Delaunay Triangulation and Greedy Routing • [KG’92] showed that Unit Delaunay triangulation is a length t-spanner for t≈2.42. • (Localized) Unit Delaunay triangulation with Greedy Routing(no guarantee of delivery). • Face greedy routing by [BMSU’99] guarantees delivery (4m moves) Algorithmic Lab, Kent State University

7. Our Objectives • Design a compact labeling scheme for Unit Disk Graphs, such that routing decision can be done in very short time and routing path is guaranteed to have constant hop (and also constant length) stretch factor. • To achieve the above goal, we would like to see if there exist collective tree spanners for a Unit Disk Graph. Algorithmic Lab, Kent State University

8. New results on collective tree spanners of Unit Disk Graphs Definition: A graph G admits a system of  collective tree (t, r)-spanners if there is a system T(G)of at most  spanning trees of G such that for any two vertices x, y of G a spanning tree T T(G) exists such that dT(x,y) ≤ t dG(x,y)+r. Theorem: Any Unit Disk Graph admits a system of at most2log3/2n+2collective tree (3,12)-spanners. Construction is in O((C+m) log n) time where C is the number of crossings in G. Algorithmic Lab, Kent State University

9. x P2 ≤ 2n/3 r ≤ 2n/3 ≤ 2n/3 S ≤ 2n/3 P1 y Planar Graphs √n balanced separator Two shortest paths balanced separator O(√n) trees giving +0 O(log n) trees giving x3 Lipton&Tarjan ? Alber&Fiala Unit Disk Graphs Algorithmic Lab, Kent State University

10. Finding a Balanced Separator in a Unit Disk Graph • Build a layering spanning treeT for G. • Convert the Unit Disk Graph Ginto a planar graphGp and T into a spanning tree Tpfor Gp. • Apply Lipton&Tarjan’s separator theorem to the planar graph Gp and spanning tree Tp to find a balanced separator Sp for Gp. • (The most important Step) From Sp, reconstruct a balanced separator S for G. Algorithmic Lab, Kent State University

11. Step 1: Build a layering spanning tree T for G r Algorithmic Lab, Kent State University

12. Step 2:Convert the Unit Disk Graph G into a planar graph Gp and T into a spanning tree Tpfor Gp Intersection between two non tree edges r r Intersection between a tree edge an a non-tree edge Intersection between two tree edges Algorithmic Lab, Kent State University

13. Step 3:Apply Lipton&Tarjan’s separator theorem to the planar graph Gp and spanning tree Tp to find a balanced separator Sp for Gp r Algorithmic Lab, Kent State University

14. Step 4:From Sp, reconstruct a balanced separator S for G r r a a c c d d Our algorithm will decide either to put acd or abd into P1 to make S=N3[P1∪ P2 ] a balanced separator. b b Algorithmic Lab, Kent State University

15. Challenging problem: an edge has multiple crossings in G • Our algorithm can deal with this case. For Example: Li-1 Li-1 Li Li Algorithmic Lab, Kent State University

16. Separator theorem • S=N3G [P1UP2] is a balanced separator for G with 2/3-split, i.e., removal of S from G leaves no connected component with more than 2/3n vertices x P2 ≤ 2n/3 r ≤ 2n/3 P1 y Algorithmic Lab, Kent State University

17. Constructing two spanning trees for a balanced separator r r T2=BFS( P2 ) T1=BFS( P1 ) Algorithmic Lab, Kent State University

18. Lemma for the two spanning trees • Let x, y be two arbitrary vertices of G and P(x,y) be a (hop-) shortest path between x and y in G. If P(x,y)∩S ≠ ø, then • dT1(x,y) ≤ 3dG(x,y)+12or • dT2(x,y) ≤ 3dG(x,y)+12 Algorithmic Lab, Kent State University

19. Constructing two spanning trees per level of decomposition • For each layer of the decomposition tree, construct local spanning trees (shortest path trees in the subgraph) r r Algorithmic Lab, Kent State University

20. Theorem for collective tree spanners • Any unit disk graphG with n vertices and m edges admits a system T(G)of at most 2log3/2n+2collective tree(3,12)-spanners, i.e., for any two vertices x and y in G, there exists a spanning tree T T(G) with dT(x,y) ≤ 3dG(x,y)+12 Algorithmic Lab, Kent State University

21. Applications: Distance Labeling Scheme and Routing Labeling Scheme • Distance Labeling Scheme: The family of n-vertex unit disk graphs admits an O(log2n) bit hop (3,12)-approximate distance labeling scheme with O(log n) time distance decoder. • Routing Labeling Scheme: The family of n-vertex unit disk graphs admits an O(log2n) bit routing labeling scheme. The Scheme has hop (3,12)-route-stretch. Once computed by the sender in O(log n) time, headers never change, and the routing decision is made in constant time per vertex. Algorithmic Lab, Kent State University

22. Extension to routing labeling scheme with bounded length route-stretch • The family of n-vertex unit disk graphs admits an O(log2n) bit routing or distance labeling scheme. The scheme has (5,13) lengthstretch factor andO(log n) time distance decoder or routing initializing time. Routing decisions other than initializing are made in constant time. Algorithmic Lab, Kent State University

23. Open questions • Does there exist a balanced separatorofformS=NG [P1UP2]? • Does there exist a distance or a routinglabeling scheme that can be locally constructed for Unit Disk Graphs? Algorithmic Lab, Kent State University

24. Thank You! Algorithmic Lab, Kent State University