1. A Log-Star Distributed Maximal Independent Set Algorithm for Growth-Bounded Graphs Johannes Schneider Roger Wattenhofer
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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
ID_v
ID_u
 r_v = 11 (binary)
ID_a 10
r_a 0
ID_u 1010
r_u
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
110
101
110
111 10 10
110
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. 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

11. 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

12. 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)

13. Thanks for your attention