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

3. Algorithmic Lab, Kent State University 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

4. Algorithmic Lab, Kent State University Stretch Factor Length stretch factor The 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

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

6. Algorithmic Lab, Kent State University 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)

7. Algorithmic Lab, Kent State University 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.

8. Algorithmic Lab, Kent State University 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 d T (x,y) ≤ t d G (x,y)+r. Theorem: Any Unit Disk Graph admits a system of at most 2log 3/2 n+2 collective tree (3,12)- spanners. Construction is in O((C+m) log n) time where C is the number of crossings in G.

9. Algorithmic Lab, Kent State University Planar Graphs Two shortest paths balanced separator Unit Disk Graphs r x y P1P1 P2P2 ≤ 2n/3 S √n balanced separator O(log n) trees giving x 3 O(√n) trees giving +0 ? Lipton&Tarjan Alber&Fiala

10. Algorithmic Lab, Kent State University Finding a Balanced Separator in a Unit Disk Graph 1.Build a layering spanning tree T for G. 2.Convert the Unit Disk Graph G into a planar graph G p and T into a spanning tree T p for G p. 3.Apply Lipton&Tarjan’s separator theorem to the planar graph G p and spanning tree T p to find a balanced separator S p for G p. 4.(The most important Step) From S p, reconstruct a balanced separator S for G.

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

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

13. Algorithmic Lab, Kent State University Step 3: Apply Lipton&Tarjan’s separator theorem to the planar graph G p and spanning tree T p to find a balanced separator S p for G p r

14. Algorithmic Lab, Kent State University Step 4: From S p, reconstruct a balanced separator S for G r a d c b r a d c b Our algorithm will decide either to put acd or abd into P 1 to make S=N 3 [P 1 ∪ P 2 ] a balanced separator.

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

16. Algorithmic Lab, Kent State University Separator theorem S=N 3 G [P 1 UP 2 ] 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 r x y P1P1 P2P2 ≤ 2n/3

17. Algorithmic Lab, Kent State University Constructing two spanning trees for a balanced separator r r T 1 =BFS( P 1 ) T 2 =BFS( P 2 )

18. Algorithmic Lab, Kent State University 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 –d T1 (x,y) ≤ 3d G (x,y)+12 or –d T2 (x,y) ≤ 3d G (x,y)+12

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

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

21. Algorithmic Lab, Kent State University Applications: Distance Labeling Scheme and Routing Labeling Scheme Distance Labeling Scheme: The family of n- vertex unit disk graphs admits an O(log 2 n) 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(log 2 n) 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.

22. Algorithmic Lab, Kent State University Extension to routing labeling scheme with bounded length route- stretch The family of n-vertex unit disk graphs admits an O(log 2 n) bit routing or distance labeling scheme. The scheme has (5,13) length stretch factor and O(log n) time distance decoder or routing initializing time. Routing decisions other than initializing are made in constant time.

23. Algorithmic Lab, Kent State University Open questions Does there exist a balanced separator of form S=N G [P 1 UP 2 ]? Does there exist a distance or a routing labeling scheme that can be locally constructed for Unit Disk Graphs?

24. Algorithmic Lab, Kent State University Thank You!