Distributed Coloring

1
Distributed Coloring

• Discrete Mathematics and Algorithms Seminar
• Melih Onus
• November 10 2005

2
Outline

• Vertex Coloring
• Model
• Lubys Algorithm
• Coloring Constant Degree Oriented Graphs
• Coloring Oriented Graphs
• Conclusion Open Problems

3
Vertex Coloring

• Given a graph G, find a coloring of the vertices
  so that no two neighbors in G have the same color

Proper Coloring
Improper Coloring
4
Model

• G(V, E), V represents the set of processors and E
  represents communication links
• Communication links are bidirectional
• Processors are synchronized
• Each node knows n, ? and its neighbors
• The edges in E have an orientation (edge u, v
  is oriented either as u ? v or v ? u)

5
How can we use orientation?

• If nodes u and v choose the same color during any
  round of algorithm, in the existing algorithms
  both nodes remain uncolored
• With orientation, u can be colored provided that
  there is no edge w ? v and node w also chooses
  the same color

u
v
u
v
With existing algorithms both remain uncolored
Using orientation, u gets colored red
6
Lubys algorithm

• In each round
• Each uncolored node chooses a color uniformly
  random
• If there is no conflict, node is colored with
  that color
• Distributed ?1-coloring algorithm
• Works in O(log n) rounds w.h.p.

7
Lubys algorithm (Example)
u
v
u
v
a
a
b
c
b
c
Round 1
Round 2
v
u
v
u
a
a
b
c
b
c
Round 3
Round 4
8
Coloring Constant Degree Oriented Graphs

• Special case Constant degree graphs
• Algorithm Color-RandomIn each round
• Each uncolored node v chooses an available color
  cv uniformly at random
• If no neighbor node u with higher priority ( u ?
  v) chooses the same color cv, node v is colored
  with cv

u ? v u has higher priority
9
Algorithm Color-Random (Example)
u
v
u
v
a
a
b
c
b
c
Round 1
Round 2
u
v
a
b
c
Round 3
10
Algorithm Color-Random

• For constant degree graph with n nodes provided
  with -acyclic orientation, our algorithm
  obtains a ?1 coloring in O( ) rounds,
  w.h.p..

An orientation of the edges of a graph is said to
be m-acyclic if and only if the orientation does
not have cycles of length at most m.
11
Analysis(Part I)

• Lemma After O((logn)1/2) rounds, every path of
  length (logn)1/2 has at least one colored node,
  w.h.p..
• Proof

p p p p p p p
p p p p p
p
p(logn)1/2
(logn)1/2
Each node has constant number of neighbors, so
the probability that a node is not colored at a
round is at most p (constant).
The probability that none of the nodes at the
path is colored at a round is at most p(logn)1/2.
12
Analysis(Part I)
p p p p p p p
p p p p p
p(logn)1/2
(logn)1/2 nodes
p(logn)1/2
c(logn)1/2 rounds
p(logn)1/2
p(logn)1/2
pclogn 1/n-clogp
13
Analysis(Part II)

• After O((logn)1/2) rounds, every connected
  component of uncolored nodes have diameter at
  most (logn)1/2, w.h.p..
• Orientation is (logn)1/2-acyclic, so there can be
  no cycles on connected component of uncolored
  nodes.
• This provides a topological ordering.

14
Analysis(Part II)
1
0
0
1
2
2
Maximum label is (logn)1/2.
3
4
All nodes will be colored after (logn)1/2 rounds.
0 if no entering
edge v?u 1maxvv?u label v otherwise
label(u)
15
Lowerbound

• For every Las Vegas algorithm A, there is
  infinite family of oriented graphs G such that A
  has complexity of at least ?((logn)1/2), on
  expectation, to compute a proper vertex coloring.

A Las Vegas algorithm is a randomized algorithm
that always produces a correct result, with the
only variation being its runtime.
16
Coloring Oriented Graphs

• General Case Arbitrary degree graphs
• Algorithm Color-Wait
• For each round
• If u is uncolored and does not have any uncolored
  neighbor w such that w ? u then node u is colored
  with the lowest available color

17
Algorithm Color-Wait (Example)
no node with entering edge for node u
no uncolored node with entering edge for node v
u
u
v
v
a
a
c
b
c
b
no node with entering edge for node b
Round 1
Round 2
no uncolored node with entering edge for node c
v
u
a
c
b
Round 3
18
Coloring Oriented Graphs

• While there are uncolored nodes
• Use Algorithm Color-Random for loglog n rounds
• If ? ?((logn)1/2loglogn)
• Use Algorithm Color-Random for (8/?4)
  (logn)1/2/loglogn rounds
• Else
• Use Algorithm Color-Random for 4(logn)1/2 rounds
• Use Algorithm Color-Wait (logn)1/2 rounds

Phase I
Phase II
Phase III
? constant, ?gt0, ? gt log?1/2n loglog n
19
Phase I
Use Algorithm Color-Random for loglog n rounds

• Lemma After phase I, the number of uncolored
  neighbors of any node reduces to log n w.h.p..

20
Phase II
If ? ?((logn)1/2loglogn) Use Algorithm
Color-Random for (8/?4) (logn)1/2/loglogn
rounds Else Use Algorithm Color-Random for
4(logn)1/2 rounds

• Lemma After phase II, every path of length
  (logn)1/2 has at least one colored node, w.h.p..

21
Phase III
Use Algorithm Color-Wait (logn)1/2 rounds

• After phase III, all nodes will be colored.

22
Coloring Oriented Graphs

• Given an -acyclic oriented graph G(V,E)
  of maximum degree ?, for any constant ?gt0 a
  (1?)?-vertex coloring of G can be obtained in
  O(log ?) O( log log n) rounds, with high
  probability.

23
Results
for any constant ?gt0
24
Conclusion Open Problems

• Distributed coloring algorithm
• Acyclic orientations, better bounds
• Deterministic distributed algorithms for
  ?1-coloring that run in polylogarithmic number
  of rounds

25
References

• K. Kothapalli, C. Scheideler, M. Onus, C.
  Schindelhauer. Distributed coloring with O(logn)
  bits. submitted to IPDPS 06.
• M. Luby. A simple parallel algorithm for the
  maximal independent set problem. STOC 1985.