A Log-Star Distributed Maximal Independent Set Algorithm for Growth-Bounded Graphs Johannes Schneider Roger Wattenhofer TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

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

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

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

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 100 110 101 110 111 10 10 110 111

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.

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

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

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

How often can a node be before changing to ? 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! 101 10 101 10 1 100 1 100

How many competitions for an arbitrary node? 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. |W| 12 13 10 11 13 12 11 13 12 10 13 11 … … … … … Distance <= 13

Related work How many rounds of communication to get a MIS? … a CDS? 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? on UDG (log* n) [Lenzen08] 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)

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