Distributed Coloring

Laurent Viennot

INRIA – Paris Diderot University

Outline

• Distributed coloring• Basic algorithm : First Free• LOCAL model• Deterministic O (∆ log n) coloring• Maximal Independent Set (MIS)• Randomized O (log n) coloring• Coloring the ring• Best known algorithms

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Distributed Coloring

Design a distributed algorithm to color the graph of theunderlying network.

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Setting

• Nodes of a network form a graph G .• n is the number of vertices in V (G ).• ∆ is the maximal degree of a node in G .• Each node has an ID in 1, . . . , n .

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c-coloring problem

• We want c colors.• Each node u has to communicate with its neighbors and

finally outputs a color cu in 1, . . . , c at some time tu .• We want cu 6= cv for u and v neighbors.• We want maxu tu to be small.

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c-coloring problem

• We want c colors.• Each node u has to communicate with its neighbors and

finally outputs a color cu in 1, . . . , c at some time tu .• We want cu 6= cv for u and v neighbors.• We want maxu tu to be small.

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c-coloring problem

• We want c colors.• Each node u has to communicate with its neighbors and

finally outputs a color cu in 1, . . . , c at some time tu .• We want cu 6= cv for u and v neighbors.• We want maxu tu to be small.

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What for ?

• Basic building block for other algorithms.• A direct application : scheduling communications in a

wireless network.• An intriguing fundamental problem.

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What for ?

• Basic building block for other algorithms.• A direct application : scheduling communications in a

wireless network.• An intriguing fundamental problem.

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Why is it easy ?

χ(G )-coloring is hard but ∆ + 1-coloring is easy :

Procedure FirstFree (u)Node u collects current colors cv for v ∈ N (u).Set cu to the first free color in 1, . . . ,∆.

Apply FirstFree (u) to each node in turn.

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Why is it easy ?

χ(G )-coloring is hard but ∆ + 1-coloring is easy :

Procedure FirstFree (u)Node u collects current colors cv for v ∈ N (u).Set cu to the first free color in 1, . . . ,∆.

Apply FirstFree (u) to each node in turn.

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Why is it easy ?

χ(G )-coloring is hard but ∆ + 1-coloring is easy :

Procedure FirstFree (u)Node u collects current colors cv for v ∈ N (u).Set cu to the first free color in 1, . . . ,∆.

Apply FirstFree (u) to each node in turn.

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Why is it hard ?

FirstFree (1) ; . . . ; FirstFree (n) takes time n .

Two neighbors cannot run safely FirstFree at the sametime.

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What can you do with FirstFree ?

From a c-coloring to a ∆ + 1-coloring in time c :

For i := 1 to c doIf cu = i then FirstFree (u)

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What can you do with FirstFree ?

From a c-coloring to a ∆ + 1-coloring in time c :

For i := 1 to c doIf cu = i then FirstFree (u)

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Fast distributed coloring

The IDs give an initial n-coloring.

How can you quickly reduce the number of colors ?

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LOCAL Model

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A simple model

Infinite computational power at each node but sending amessage costs 1.

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A simple model

• Each node u of a graph G is given an input X (u).• Each node u has to compute an output Y (u).• Nodes work synchronously in rounds of 1 unit of time.• At each round, u computes and then sends a message to

each neighbor.• Running time is the number of rounds to obtain Y (u) for

all u .

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A simple model

• Each node u of a graph G is given an input X (u).• Each node u has to compute an output Y (u).• Nodes work synchronously in rounds of 1 unit of time.• At each round, u computes and then sends a message to

each neighbor.• Running time is the number of rounds to obtain Y (u) for

all u .

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A simple model

• Each node u of a graph G is given an input X (u).• Each node u has to compute an output Y (u).• Nodes work synchronously in rounds of 1 unit of time.• At each round, u computes and then sends a message to

each neighbor.• Running time is the number of rounds to obtain Y (u) for

all u .

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A locality measure

How far do you need to communicate ?

The running time t is the distance up to which a node’scomputation depends on.

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Running an algorithm A after communicating...

Send X (u) to neighbors and receive their inputs.Send X (u) and IDs of neighbors up to distance t .Receive in t rounds all information in the ball B (u , t ).Emulate A for all nodes in B (u , t ).Output the value obtained for u .

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Randomized is deterministic

Consider that the random bits u is going to use are part ofX (u).

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∆ + 1-coloring in O (∆ log n) time

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Using FirstFree

Let b1 · · · b` be the binary representation of ID (u).Let cu := 1.For i := ` down to 1 do

If bi = 1 then cu := cu + ∆ + 1P (u) := b1 · · · bi−1Run FirstFree ignoring neighbors v with P (v) 6= P (u).

Proof : after iteration i we get a valid coloring for eachsubset V (b1 · · · bi−1) of nodes with prefix b1 · · · bi−1.

Time : log n use of FirstFree from 2(∆ + 1) to ∆ + 1 colors.

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Using FirstFree

Let b1 · · · b` be the binary representation of ID (u).Let cu := 1.For i := ` down to 1 do

If bi = 1 then cu := cu + ∆ + 1P (u) := b1 · · · bi−1Run FirstFree ignoring neighbors v with P (v) 6= P (u).

Proof : after iteration i we get a valid coloring for eachsubset V (b1 · · · bi−1) of nodes with prefix b1 · · · bi−1.

Time : log n use of FirstFree from 2(∆ + 1) to ∆ + 1 colors.

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Using FirstFree

Let b1 · · · b` be the binary representation of ID (u).Let cu := 1.For i := ` down to 1 do

If bi = 1 then cu := cu + ∆ + 1P (u) := b1 · · · bi−1Run FirstFree ignoring neighbors v with P (v) 6= P (u).

Proof : after iteration i we get a valid coloring for eachsubset V (b1 · · · bi−1) of nodes with prefix b1 · · · bi−1.

Time : log n use of FirstFree from 2(∆ + 1) to ∆ + 1 colors.

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Maximal Independent Set (MIS)

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Problem

Each node u outputs Y (u) ∈ 0, 1 such that nodes withoutput 1 form a Maximal Independent Set in G .

• Independent : two nodes in the MIS cannot beneighbors.

• Maximal : no node can be added in the MIS, i.e. eachnode not in the MIS has a neighbor in the MIS.

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Problem

Each node u outputs Y (u) ∈ 0, 1 such that nodes withoutput 1 form a Maximal Independent Set in G .

• Independent : two nodes in the MIS cannot beneighbors.

• Maximal : no node can be added in the MIS, i.e. eachnode not in the MIS has a neighbor in the MIS.

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Problem

Each node u outputs Y (u) ∈ 0, 1 such that nodes withoutput 1 form a Maximal Independent Set in G .

• Independent : two nodes in the MIS cannot beneighbors.

• Maximal : no node can be added in the MIS, i.e. eachnode not in the MIS has a neighbor in the MIS.

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From c-coloring to MIS

A c-coloring is a partition of V (G ) into independent sets.

Run FirstFree, and color class 1 becomes a MIS.

A c-coloring algorithm in time T gives a MIS algorithm intime T + c .

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From c-coloring to MIS

A c-coloring is a partition of V (G ) into independent sets.

Run FirstFree, and color class 1 becomes a MIS.

A c-coloring algorithm in time T gives a MIS algorithm intime T + c .

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From c-coloring to MIS

A c-coloring is a partition of V (G ) into independent sets.

Run FirstFree, and color class 1 becomes a MIS.

A c-coloring algorithm in time T gives a MIS algorithm intime T + c .

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From MIS to ∆ + 1-coloring

A MIS of the cartesian product G × K∆+1 gives a∆ + 1-coloring.

A T (n ,∆) MIS algorithm gives a ∆ + 1-coloring algorithm intime T (n(∆ + 1), 2∆).

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From MIS to ∆ + 1-coloring

A MIS of the cartesian product G × K∆+1 gives a∆ + 1-coloring.

A T (n ,∆) MIS algorithm gives a ∆ + 1-coloring algorithm intime T (n(∆ + 1), 2∆).

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Randomized O (log n) MIS [Luby’86, Alon&B.&I.’86]

Y := ∅ /* candidate MIS solution */A := V (G ) /* set of active nodes */While A 6= ∅ do

I := IndependentSet (G|A )Y := Y ∪ IA := A \ (I ∪ N (I ))

Procedure IndependentSet (u)I := ∅ /* candidate independent set */Add u to I with probability 1

2 deg(u)If ∃v ∈ N (u) ∩ I s.t. (deg(v), ID (v)) < (deg(u), ID (u))then

remove u from I .Return I

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Coloring the Ring

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Best Known Algorithms

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Unbounded degree

• Deterministic : 2O (√

log n) MIS [Panconesi&S.’96]• Randomized : O (log n) MIS [Luby’86, Alon&B.&I.’86]

Bounded degree (deterministic)

• O (∆ + log∗ n) time ∆ + 1-coloring [Kuhn’09, Bar.&E.’09]

Bounded arboricity (deterministic)

• O (a√

log n + a log a) MIS [Bar.&E.’08]• O (log a log n) time O (a1+ε)-coloring [Bar.&E.’10]• O (aε log n) time O (a)-coloring [Bar.&E.’10]

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Best lower bounds

O (∆)-coloring

• (Ring) Ω(log∗ n) [Linial’92]• Reduce colors algorithms Ω(∆/ log2 ∆ + log∗ n)

[Kuhn&W.’06]

MIS

• Ω(√

log nlog log n

)[Kuhn&M.&W.’04]

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Bibliography

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