Log-Star Distributed Maximal Independent Set Algorithm for Growth-Bounded Graphs

1. A Log-Star Distributed Maximal Independent Set Algorithmfor Growth-Bounded GraphsJohannes SchneiderRoger Wattenhofer TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

2. Motivation Maximal Independent Set (MIS) algorithms allow to get Connected Dominating Sets (CDS) and Minimum Dominating Sets (MDS) for wireless multi-hop networks MDS and CDS are useful for Routing Media access control Coverage … Compute CDS/MDS with little communication to save valuable time and energy

3. Model and Definitions Maximal Independent Set (MIS) Node v in MIS or ≥1 neighbor in MIS Nodes u,v in MIS cannot be adjacent Unit Disk Graph (UDG) Geometrical graph Edge between nodes u,v if dist(u,v) < 1 Growth bounded Maximum size of an independent set in the neighborhood of a node is at most 5 Every node has an ID in [1,n] A node communicates with neighbors in synchronized rounds without interference Definition log* How often one has to take the logarithm to get 1 Example: log*16 = 3 since log 16 = 4; loglog 16 = 2; logloglog 16 = 1

4. Algorithm • Every node performs competitions (with breaks) until it (or a neighbor) is in the MIS • Competition • First one based on ID to obtain result r • Node v picks neighbor u with smallest ID • If ID_v ≤ ID_u • result r_v is 0 • If ID_v > ID_u • result r_v is the maximum position where ID_v has a 1 and ID_u has a 0. • Example: Position 4 3 2 1 • ID_v 1 1 0 1 • ID_u 1 0 1 0 •  r_v = 11 (binary) ID_a 10 r_a 0 ID_u 1010 r_u 100 ID_v 1101 r_v 11 ID_d 1100 r_d 11

5. What to do with the result of a competition? • Node v changes its state depending on its result and those of neighbors. • Dominator • If result r_v < r_u for all neighbors u • Joins the MIS • Neighbors are dominated and stay quiet • Ruler • if result r_v ≤ r_u for all neighbors u • and at least one has same result • All neighbors become ruled (if not dominated or rulers themselves) • Ruled nodes stay quiet until all neighbors become ruled or dominated. • Rulers immediately become competitors again and compete again based on IDs • Competitor • None of above conditions applies • Compete again based on the result of the last competition 100 0 110 101 110 111 10 110 10 111

6. How many competitions? How often must a competitor compete before changing its state? at most log* n times The result of log* n consecutive competitions must be 1. Proof The result of the 1st competition is in [0,log n] The result gives an index of a bit of the ID An ID in [1,n] => needs log n bits … 2nd … in [0,loglog n] Since the previous result has up to loglog n bits a.s.o. Once a node has result 1, it must change its state. Either its own result is a minimum or a neighbor has smallest result possible, i.e. 0.

7. How often can a node be before changing to ? Let S be the set of connected competitors with v in S A node not in S cannot join before v is ruled or dominated v

8. How often can a node be before changing to ? S shrinks with every transition When v becomes a ruler, one 2-hop neighbor w in S is not reachable by a path of rulers! Node w (and all its neighbors) cannot be in S any more. w v

9. How many of such 2-hop neighbors W exist? For the UDG there exist only 13 such 2 hop neighbors W for a node v. w v

10. After a competitor has become a ruler 13 times (without becoming ruled), no 2 hop neighbor can be reached by a path of rulers. Thus all neighbors of ruler v, that are still rulers form a clique. In the next competition based on the ID, the ruler of the clique with the smallest ID becomes a dominator! How often can a node be before changing to ? 101 10 101 10 1 100 1 100

11. After log* n competitions a competitor changes its state. If dominated or dominator it is done A competitor can become a ruler at most 13 times in a row. After 13·log* n competitions every node gets a dominator within distance 13. Within distance 13 there are at most 132 nodes in an independent set, thus the maximum comptetions the algorithm needs are 133·log* n. How many competitions for an arbitrary node? 10 12 13 11 |W| 10 12 13 13 11 12 13 11 … … … … … Distance <= 13

12. Related work How many rounds of communication to get a MIS? Lower bounds on ring (log* n) [Lineal92] on general graphs (log n/loglog n) [Kuhn05] Upper bounds On general graphs O(log n) [Luby86] … a CDS? Lower bounds on UDG (log* n) [Lenzen08] Upper bounds on UDG O(loglog n log*n) [VicariGfeller07] on UDG with distance information O(log* n) [Kuhn05] Here: MIS, CDS, MDS and Coloring on UDG in O(log* n)

13. Thanks for your attention