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Experimental analysis of simple, distributed
vertex coloring algorithms
Irene Finocchi Alessandro Panconesi Riccardo
Silvestri DSI, University of Rome La Sapienza
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The vertex coloring problem

• Goal finding good colorings, i.e., using few
  colors

• Finding or even approximating the minimum number
  of colors is hard

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Distributed vertex coloring

• Good colorings in a distributed setting
• O(D)-colorings, where D maximum degree
• to be computed in O(polylog n) time

• Each graph is D1-colorable
• Sequential algorithm trivial
• Deterministic distributed algorithm working in
  O(polylog n) time open problem!

• Brooks-Vizing colorings colorings using many
  fewer than D colors
• G square or triangle free ? ?(D/log
  D)-colorable

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Distributed coloring algorithms

• Model of computation synchronous,
  message-passing
• Vertices processors operate in parallel
• Routing messages costs order of magnitude more
  than performing local computations

• Characteristics of the algorithms
• Each vertex manages a list of colors (palette)
• The computation proceeds in rounds

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The basic round r for vertex u
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Important parameters
Algorithms with very different characteristics
can be obtained changing these parameters
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D1 colorings theory
Key lemma At any round an uncolored vertex
colors with probability 1/4 Johansson99,Luby93

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D1 colorings Luby vs. Trivial
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D1 colorings Luby vs. Trivial
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Brooks-Vizing colorings theory
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A new conflict resolution rule
For each color c Gc (conflict graph) graph
induced by vertices with tentative color c
The hungarian approach for independent set
computation given a random permutation p of the
vertices of Gc, a vertex enters the independent
set iff it comes before its neighbors in p
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Expected number of colored vertices
n number of vertices of Gc d average degree
of Gc
Hungarian folklore
s palette shrinking factor
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Parameter settings
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Wake up probability rounds
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Shrinking factor colors
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Shrinking factor colors
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Shrinking factor rounds
TH saves approx. 70-80 rounds over GP and HGP
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Conclusions open problems
Deterministic algorithms?
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The vertex coloring problem

• Minimize the number of colors

• Each graph is D1-colorable (greedy approach)
• D maximum degree

• Brooks-Vizing colorings colorings using many
  fewer than D colors
• G square or triangle free ? ?(D/log
  D)-colorable

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Luby vs. Trivial intuitive explanation
Only for the very first rounds...
w wake up probability l palette size d of
uncolored neighbors
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Luby vs. Trivial intuitive explanation
Only for the very first rounds...
w wake up probability p
palette size d of uncolored neighbors
Prv colors
w e-d w/p
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D1 colorings Luby vs. Trivial
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Distributed coloring algorithms

• Model of computation synchronous,
  message-passing
• Vertices processors operate in parallel
• Routing messages costs order of magnitude more
  than performing local computations

• Characteristics of the algorithms
• Randomized
• Each vertex manages a list of colors (palette)
• The computation proceeds in rounds
• All the algorithms can be cast into a general
  framework

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Wake up probability rounds

• How does the wake up probability affect the
  running time?
• Can it be constant?
• Which is the best choice?