edge color pcos

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Contribution Edge Coloring Experiment and Results Conclusion

Two Edge Coloring Algorithms Using a Simple

Matching Discovery Automata

J. Paul Daigle & Sushil K. Prasad

Department of Computer ScienceGeorgia State University

05-16-2012
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Outline

1 Contribution

2 Edge Coloring

3 Experiment and Results

4 Conclusion
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Outline

1 Contribution

2 Edge Coloring


4 Conclusion
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2 Novel Algorithms

Constant approximation for edge coloring of a graph

Probabilistic, distributed algorithm

Previous best time complexity: 2+ 1 for acyclic graphs

Our algorithm time: O() for all graphs
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Definitions

Structure and Assumptions

Our algorithm assumes a Message Passing Modelofdistributed computing. Each node in the distributed computer is

analogous to a vertex in the input graph, with communication

links between nodes being analogous to edges in the input

graph.

C ib i Ed C l i E i d R l C l i
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Definitions

Structure and AssumptionsOur algorithm assumes a Message Passing Modelof

distributed computing. Each node in the distributed computer is

analogous to a vertex in the input graph, with communication

links between nodes being analogous to edges in the input

graph.

Message Passing Model

The message passing model of distributed computing assumes

that:1 The distributed computer is represented by a network

2 Compute nodes can only communicate with their direct neighbors

3 Compute nodes can communicate with every neighbor in each round

4

Communication rounds are synchronized

C t ib ti Ed C l i E i t d R lt C l i
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Definitions

Structure and Assumptions

Our algorithm assumes a Message Passing Modelof

distributed computing. Each node in the distributed computer is

analogous to a vertex in the input graph, with communicationlinks between nodes being analogous to edges in the input

graph.

Message Passing ModelThe time complexity of our algorithm is based on the number of

synchronized commucation rounds required to find a solution.

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Distributed Matching

Computation by edges

We adapt sequential algorithms that meet the following criteria:

One computation is required for each edge of the graph

The order of the computations is arbitrary

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Computation by edges

We adapt sequential algorithms that meet the following criteria:

One computation is required for each edge of the graphThe order of the computations is arbitrary

Matching

A matching on a graph G(V,E) is a setE | e(u, v) E, e(v,w) / E e(u,w) / E.

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Computation by edgesWe adapt sequential algorithms that meet the following criteria:

One computation is required for each edge of the graph

The order of the computations is arbitrary

Matching

A matching on a graph G(V,E) is a setE | e(u, v) E, e(v,w) / E e(u,w) / E.

Equivalence to Sequential Evaluation

A simultaneous computation between the endpoints of the

edges in a matching has identical results to any arbitrary

sequency of computations between those same endpoints.

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Matching Automata

each node in the network proceeds syn-chronously through the stages of the au-tomata

C: Choose roleby coin cossstart

I: Invite a neigh-bor

L: Listen for localinvites

W: Wait for re-sponse

R: Respond to 1invite

U: Update sub-problem solutions

D: Done

heads

tails

no invitation received

finished all subprob-lems

unresolved subprob-

lems

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Matching Automata

initially, every node is in the choose state







D: Done

heads

tails



unresolved subprob-

lems

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g g p

Matching Automata

Each node chooses randomly to be in theInvite or Listen State







D: Done

heads

tails



unresolved subprob-

lems

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g g p

Matching Automata

Each node in the Invite state chooses aneighbor at random to send an invitationto, nodes in the Listen state listen for invi-tations from their neighbors







D: Done

heads

tails



unresolved subprob-

lems

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Matching Automata

After sending invitations, Invite nodestransition to the Waiting state to wait forreplies







D: Done

heads

tails



unresolved subprob-

lems

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Matching Automata

Nodes that received no invitations go tothe Update state







D: Done

heads

tails



unresolved subprob-

lems

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Matching Automata

Nodes that received invitations reply toexactly one of those invitations







D: Done

heads

tails



unresolved subprob-

lems

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Matching Automata

Those replies are heard by nodes in thewait state: the pairings between invitorsand invitees compose a matching on theproblem graph




W: Wait for re-sponseW: Wait for re-sponse



D: Done

heads

tails



unresolved subprob-

lems

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Matching Automata

All nodes are in the update state. Nodeswhich have formed Invitor/Invitee pairsupdate the affected edge and share thenew information with their neighbors







D: Done

heads

tails



unresolved subprob-

lems

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Matching Automata

Nodes that have no more edges to pro-cess are done







D: Done

heads

tails



unresolved subprob-

lems

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Matching Automata

Nodes with more edges to process returnto the choose state

C: Choose roleby coin cossstartC: Choose roleby coin cossstart






D: Done

heads

tails



unresolved subprob-

lems

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Outline

1 Contribution

2 Edge Coloring


4 Conclusion

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Definitions I

Edge Coloring

An edge coloring of a graph is an assignment of colors to the

edges of a graph in such a way that no two adjacent edges are

assigned the same color.

Formally, given a graph, G(V,E), and set of colors C, an edgecoloring of G is a mapping:

f(e) = E C| e(u, v), e

(v,w) E, c Cf(e) = c = f(e) = c

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Definitions II

0

1

2

3

4

5

6

7

8

Figure: Edge Colorings: assigning a color to the solid edge makesthat color unavailable to the dashed edges, but not to the dottededges

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Definitions III

Strong Edge Coloring

A strong edge coloring of a graph is an assignment of colors to

the edges of a graph such that no two edges that can be

connected by a common edge are assigned the same color.Formally, given a graph G(V,E) and a set of colors C, a strongedge coloring is a mapping:

f(e) = C E| e(u, v), e(v,w), e(w,x) E, c C

f(e) = c = f(e) = c f(e) = c

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Definitions IV

Strong Directed Edge ColoringA strong edge coloring is an assignment of colors to the edges

of the graph such that no two edges that can be connected by a

common edge are assigned the same color.

In the directed case, a strong edge coloring is a mapping:

f(e) = C E| e(u, v)

e(v, u)

e(w, v)

e(w,x) E

c C

f(e) = c = f(e) = c,f(e) = c,f(e) = c

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Definitions V

0

1

2

3

4

5

6

7

8

Figure: Edge Colorings: assigning a color to the solid edge makesthat color unavailable to the dashed edges, but not to the dottededges

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Solving Edge Coloring with distributed matching

Strategy

Find a sequential strategy that depends on edge evaluation

Use the matching automata to execute the strategy in

parrallel

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Edge Coloring Algorithm I

Sequential Strategy

1 Number the colors

2 Choose an edge

3 For each endpoint, find the used colors

4 Remove those colors from the available colors

5

Assign the lowest available color to the edge

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Edge Coloring Algorithm II

Distributed Modification1 Assume each node knows its neighbors free colors

2 Choose to invite or receive

3 An invitator node u sends an invitation for a specific node v

to use a specific color c to color edge u, v.4 c is the lowest indexed free color common to u and v

5 a response from a node v indicates the id of a single invitor

v and the color c to be used.

6 uand

vupdate their neighbors that

cis no longer a freecolor

A key point about step 6 is that a free color, in this context,

means free for u or for v. The neighbors of u and v are still free

to use that color.

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Edge Coloring Algorithm III

Modification for strong directed coloring

1 all steps up to step 5 are the same.

2 a responding node v will not respond to an invitation to use

a color c if there is more than one invitation to use c in vs

neighborhood.

3 v can respond to a single invitor v with the color c to be

used.

4 u and v update their neighbors that c is no longer a free

color

In a strong directed coloring, there is a conflict if an origin

vertex uses a color that is used by a terminal neighbor, but not

if the color is being used by an origin neighbor. The meaning of

a free color is therefore one that is not being used by any

terminal neighbor.

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Outline

1 Contribution

2 Edge Coloring


4 Conclusion

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Design

Simulator

Simulation program written in Ruby

Discrete time simulation

Code is available for download

Graph Types

Experiments with the edge coloring algorithm were performed

on Erdos-Renyi, Small World, and Scale Free graphs, with

large variations in Average and number of nodes.Experiments were conducted on Erdos-Renyi graphs alone for

the directed edge coloring algorithm.

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Results

20 30 40 50

40

60

80

100

120

Average

Roun

ds

200 Nodes

400 Nodes

Figure: Edge Coloring of Erdos-Renyi Graph

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Results

50 100 150 200 250

100

200

300

400

500

Average

Round

s

100 Nodes

400 Nodes

Figure: Edge Coloring of Scale-Free Graphs

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Results

10 20 30 40

50

100

150

Average

Rounds

16 Nodes

64 Nodes

256 Nodes

Figure: Edge Coloring of Small World Graphs


R l
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Results

16 18 20 22 24 26 28 30

80

100

120

140

Average

Rounds

200 Nodes

400 Nodes

Figure: Strong Edge Coloring of Directed Erdos-Renyi Graph
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O tli


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Outline

1 Contribution

2 Edge Coloring


4 Conclusion


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Conclusion

Probabilistic O() constant-approximate algorithm forEdge Coloring

Probabilistic O() solution for Strong Directed EdgeColoring

Framework successfully adapted to solve two new

problems

Additional research/analysis required to understand

behavior under different graph structures
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edge color pcos

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