Johannes Schneider – 1

A Log-Star Distributed Maximal Independent Set Algorithm

for Growth-Bounded Graphs

Johannes SchneiderRoger Wattenhofer

Johannes Schneider – 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

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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

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• 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)

Algorithm

ID_a 10 r_a 0

ID_u 1010r_u 100

ID_v 1101r_v 11 ID_d 1100

r_d 11

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What to do with the result of a competition?

0

101

10 10

100

111

110

110

110111

• 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

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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.

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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

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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.

vw

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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.

vw

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• 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

1 100

101 10

1 100

How often can a node be before changing to ?

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• 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?

… … …

…

…

Distance <= 13

|W| 13 13 13 13 12 12 12 11 11 11 10 10

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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)

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Thanks for your attention