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

Johannes Schneider – 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

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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