A Log-Star Distributed Maximal Independent Set Algorithm for Growth-Bounded Graphs Johannes Schneider Roger Wattenhofer

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A Log-Star Distributed Maximal Independent Set
Algorithmfor Growth-Bounded GraphsJohannes
SchneiderRoger Wattenhofer
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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) lt 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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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 gt 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
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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 lt 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
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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 gt 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.

w
v
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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.

w
v
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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
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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 13log 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.

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12
10
11
W
12
10
13
13
11
13
12
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Distance lt 13
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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 logn) 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