Experimental analysis of simple, distributed vertex coloring algorithms

Irene FinocchiAlessandro Panconesi Riccardo Silvestri

DSI, University of Rome “La Sapienza”

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

The vertex coloring problem

(u,v) E c(u) c(v)

Given G = (V,E) find c :VN such that

• 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 San Francisco, January 2002

Distributed vertex coloring

• Each graph is +1-colorable

– Sequential algorithm: trivial

– Deterministic distributed algorithm working in O(polylog n) time: open problem!

• Brooks-Vizing colorings = colorings using “many fewer” than colors

G square or triangle free (log)-colorable

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

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Distributed coloring algorithms

• Characteristics of the algorithms

– Each vertex manages a list of colors (palette)

– The computation proceeds in rounds

• Model of computation: synchronous, message-passing

– Vertices = processors operate in parallel

– Routing messages costs order of magnitude more than performing local computations

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

The basic round r for vertex u1. Wake up! Wake up with probability wr

2. Try! Pick a tentative color c from palette Lu

(uniformly at random)

c-color iff no awaken neighbor has selected c astentative color

3. Conflict resolution

4. Deliverance? If 3 succeded exit, otherwise remove from Lu the final colors of the neighbors

If Lu= introduce fresh new colors(up to the greatest color of a neighbor +1)

5. Feed the hungry!

Go back to sleep6. Back to square one

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Important parameters

Palette size: can it be << ?

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?

Algorithms with very different characteristics can be obtained changing these parameters

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

+1 colorings: theory

Key lemma: At any round an uncolored vertex colors with

probability ≥1/4 [Johansson’99,Luby’93]

wr=1

Palette = [1, +1]Trivial algorithm

wr=1/2

Palette = [1, +1]Luby’s algorithm

A +1-coloring is computed in O(log n) rounds with high probability

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

+1 colorings: Luby vs. Trivial

How fast are the +1-coloring algorithms

for different wake up probabilities?

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

+1 colorings: Luby vs. Trivial

0

10

20

30

40

50

60

70

80

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1

WakeUp Probability

Rounds

D+1-coloring

Instance: G(1000,0.1)

Luby

Trivial

+1-coloring

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Brooks-Vizing colorings: theory

Grable & Panconesi 2000 (GP)

Reduced palette: /s(s = shrinking factor)

palette sizewr = uncolored neighbors

G triangle-free

-regular

>> log n

With high probability GP

colors G with O(/log )

colors within O(log n) rounds

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

A new conflict resolution rule

For each color c:

Gc (conflict graph) = graph induced

by vertices with tentative color c

Hungarian step (HS)

conflict resolution by computing an independent set on Gc

The hungarian approach for independent set computation:

given a random permutation of the vertices of Gc, a vertex

enters the independent set iff it comes before its neighbors in

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Expected number of colored vertices

n = number of vertices of Gc

= average degree of Gc

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

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

folklore]

s = palette shrinking factor

] = s (if all vertices wake up)2. [Easy to prove]

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Parameter settings

GPGrable

PanconesiGive up!

palette size

uncolored neighborsReduced:

/s

HGPHungarian GP

Hungarianpalette size

uncolored neighborsReduced:

/s

HungarianConstantReduced:

/sCH

Constantlyhungarian

Hungarian1Reduced:

/sTH

Triviallyhungarian

Palette Wake up prob. Conflict res.Algorithms

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Wake up probability & rounds

0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Wake up probability

Rounds

GP

HGP

CH

Instance: G(500,0.25) s=3

TH

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Shrinking factor & colors

How good are colorings? (As a function of the initial palette size)

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Shrinking factor & colors

30

50

70

90

110

130

150

170

0 2 4 6 8 10 12 14 16

Shrinking Factor

Colors

GP

HGP

TH

+1

: (500,0.25)Instance G

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Shrinking factor & rounds

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12 14 16

Shrinking Factor

Rounds

GP

HGP

TH

+1

: (500,0.25)Instance G

TH saves approx. 70-80%rounds over GP and HGP

13th ACM-SIAM Symposium on Discrete Algorithms San Francisco, January 2002

Conclusions & open problems

+1-coloringsTrivial is the algorithm of choice

Tight analysis?

Brooks-Vizing colorings

TH is the algorithm of choice

Rigorous theoretical analysis?

Deterministic algorithms?