Experimental analysis of simple, distributed vertex coloring algorithms
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Experimental analysis of simple, distributed vertex coloring algorithms. Irene Finocchi Alessandro Panconesi Riccardo Silvestri DSI, University of Rome “La Sapienza”. Given G = (V,E) find c :V N such that.  (u,v)  E c(u)  c(v). The vertex coloring problem.

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Experimental analysis of simple, distributed vertex coloring algorithms

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Experimental analysis of simple distributed vertex coloring algorithms

Experimental analysis of simple, distributed vertex coloring algorithms

Irene Finocchi

Alessandro Panconesi

Riccardo Silvestri

DSI, University of Rome “La Sapienza”


Experimental analysis of simple distributed vertex coloring algorithms

Given G = (V,E) findc :VN such that

 (u,v)  E c(u)  c(v)

The vertex coloring problem

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

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

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

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 D+1-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

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

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

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

1.Wake up!

Wake up with probability wr

2.Try!

Pick a tentative color c from palette Lu

(uniformly at random)

3.Conflict

resolution

c-color iff no awaken neighbor has selected c as

tentative color

4.Deliverance?

If 3 succeded exit, otherwise remove from Lu the final

colors of the neighbors

5.Feed the

hungry!

If Lu= introduce fresh new colors

(up to the greatest color of a neighbor +1)

6.Back to

square one

Go back to sleep

The basic round r for vertex u

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

Palette size: can it be << D?

Wake up probability: how do we choose it? Can it be constant?

Conflict resolution: the give up rule is not very smart... can we improve it?

Important parameters

Algorithms with very different characteristics can be obtained changing these parameters

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

wr=1

Palette = [1, D+1]

wr=1/2

Palette = [1, D+1]

Trivial algorithm

Luby’s algorithm

A D+1-coloring is computed in O(log n) rounds

with high probability

D+1 colorings: theory

Key lemma: At any round an uncolored vertex colors with probability ≥1/4 [Johansson’99,Luby’93]

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

How fast are the D+1-coloring algorithms

for different wake up probabilities?

D+1 colorings: Luby vs. Trivial

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

D+1-coloring

D+1 colorings: Luby vs. Trivial

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

Reduced palette: D/s

(s = shrinking factor)

palette size

wr =

uncolored neighbors

Grable & Panconesi 2000 (GP)

G triangle-free

D-regular

D >> log n

With high probability GP

colors G with O(D/log D)

colors within O(log n) rounds

Brooks-Vizing colorings: theory

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

Hungarian step (HS)

conflict resolution by computing an independent set on Gc

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

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

E[|independent set|]=n/(d+1)

1.

E[d] = s (if all vertices wake up)

[Easy to prove]

2.

E[|colored vertices|] = n/(s+1)

Expected number of colored vertices

n = number of vertices of Gc

d= average degree of Gc

[Hungarian folklore]

s = palette shrinking factor

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

Palette

Wake up prob.

Conflict res.

Algorithms

GP

Grable

Panconesi

Reduced: D/s

palette size

Give up!

uncolored neighbors

Reduced: D/s

palette size

HGP

Hungarian GP

Hungarian

uncolored neighbors

CH

Constantly

hungarian

Reduced: D/s

Constant

Hungarian

TH

Trivially

hungarian

Reduced: D/s

1

Hungarian

Parameter settings

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

Wake up probability & rounds

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

How good are colorings?

(As a function of the initial palette size)

Shrinking factor & colors

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

Shrinking factor & colors

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

Shrinking factor & rounds

TH saves approx. 70-80%

rounds over GP and HGP

13th ACM-SIAM Symposium on Discrete Algorithms


Experimental analysis of simple distributed vertex coloring algorithms

Trivial is the algorithm of choice

D+1-colorings

Tight analysis?

TH is the algorithm of choice

Brooks-Vizing colorings

Rigorous theoretical analysis?

Conclusions & open problems

Deterministic algorithms?

13th ACM-SIAM Symposium on Discrete Algorithms


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