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Deterministic Local Algorithms, Unique Identifiers, and Fractional - - PowerPoint PPT Presentation

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Deterministic Local Algorithms, Unique Identifiers, and Fractional Graph Colouring Henning Hasemann, Juho Hirvonen, Joel Rybicki, and Jukka Suomela TU Braunschweig University of Helsinki SIROCCO 2012 30 June 2012 1 / 50 Our Result 0 0 . 5

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SLIDE 1

Deterministic Local Algorithms, Unique Identifiers, and Fractional Graph Colouring

Henning Hasemann, Juho Hirvonen, Joel Rybicki, and Jukka Suomela TU Braunschweig University of Helsinki SIROCCO 2012 30 June 2012

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SLIDE 2

Our Result

0.5 1 1.5 2 2.5 3 3.5 α(∆ + 1)

There is a deterministic distributed algorithm that runs in 1 communication round that, for any α > 1, finds a fractional graph colouring of length at most α(∆ + 1)

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SLIDE 3

Model of Computation: LOCAL

◮ Communication graph

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SLIDE 4

Model of Computation: LOCAL

T = 0

◮ Synchronous communication

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SLIDE 5

Model of Computation: LOCAL

T = 1

◮ Synchronous communication

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SLIDE 6

Model of Computation: LOCAL

T = 2

◮ In T rounds gather radius-T neighbourhood

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SLIDE 7

Model of Computation: LOCAL

◮ Constant-time algorithms

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SLIDE 8

Model of Computation: LOCAL

◮ Each node maps neighbourhood to output

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SLIDE 9

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Fractional Graph Colouring

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SLIDE 10

Fractional Graph Colouring

0.5 1 1.5 2 2.5 3

input

  • utput

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SLIDE 11

Fractional Graph Colouring

0.5 1 1.5 2 2.5 3

independent set input

  • utput

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SLIDE 12

Fractional Graph Colouring

0.5 1 1.5 2 2.5 3

independent set input

  • utput

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SLIDE 13

Fractional Graph Colouring

0.5 1 1.5 2 2.5 3

independent set input

  • utput

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SLIDE 14

Fractional Graph Colouring

0.5 1 1.5 2 2.5 3

independent set

≥ 1 ≥ 1 ≥ 1 ≥ 1 ≥ 1

input

  • utput

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SLIDE 15

Fractional Graph Colouring

0.5 1 1.5 2 2.5 3

independent set length of schedule

≥ 1 ≥ 1 ≥ 1 ≥ 1 ≥ 1

input

  • utput

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SLIDE 16

Fractional Graph Colouring

0.5 1 1.5 2 2.5 3 ≥ 1 ≥ 1 ≥ 1 ≥ 1 ≥ 1

input

  • utput

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SLIDE 17

Fractional Graph Colouring

0.5 1 1.5 2 2.5 3

independent set length of schedule

≥ 1 ≥ 1 ≥ 1 ≥ 1 ≥ 1

input

  • utput

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SLIDE 18

Our Result Again

0.5 1 1.5 2 2.5 3 3.5 α(∆ + 1)

There is a deterministic distributed algorithm that runs in 1 communication round that, for any α > 1, finds a fractional graph colouring of length at most α(∆ + 1)

18 / 50

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SLIDE 19

Lower Bound

0.5 1 1.5 2 2.5 3 ≥ 1 ≥ 1 ≥ 1

∆ = 2 ∆ + 1

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SLIDE 20

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Finding a Fractional Graph Colouring

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SLIDE 21

Finding a Fractional Graph Colouring

◮ Impossible to break symmetry in an anonymous cycle

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SLIDE 22

Finding a Fractional Graph Colouring

◮ Nodes must produce an empty schedule

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SLIDE 23

Finding a Fractional Graph Colouring

44 31 9 17 91 6 61 8 7 88 16 75 3 5 12 34 43 76 65 39 66 87 95

◮ Standard assumption: numeric identifiers

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SLIDE 24

Finding a Fractional Graph Colouring

44 31 9 17 91 6 61 8 7 88 16 75 3 5 12 34 43 76 65 39 66 87 95

◮ Standard assumption: numeric identifiers

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SLIDE 25

Finding a Fractional Graph Colouring

44 31 9 17 91 6 61 8 7 88 16 75 3 5 12 34 43 76 65 39 66 87 95

◮ Standard assumption: numeric identifiers ◮ FGC is the first example where numeric identifiers give a

constant-time algorithm

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SLIDE 26

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Why Numeric Identifiers Do Not Help?

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SLIDE 27

Numeric Identifiers Not Needed

44 31 9 17 91 6 61 8 7 88 16 75 3 5 12 34 43 76 65 39 66 87 95

◮ Naor & Stockmeyer (1995) studied when numeric identifiers

are necessary

◮ LCL-problems

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SLIDE 28

LCL-problems

◮ Maximal Independent Set

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SLIDE 29

LCL-problems

◮ Vertex Cover

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SLIDE 30

LCL-problems

◮ Maximal Matching

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SLIDE 31

LCL-problems

◮ Fractional Graph Colouring

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SLIDE 32

Numeric Identifiers Not Needed

44 31 9 17 91 6 61 8 7 88 16 75 3 5 12 34 43 76 65 39 66 87 95

◮ Naor & Stockmeyer (1995): In LCL-problems numeric

identifiers not necessary

◮ Technicality: applies if output bounded 32 / 50

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SLIDE 33

No FGC with Comparisons

◮ Identifiers arranged in an ascending order

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SLIDE 34

No FGC with Comparisons

◮ Some nodes must produce an empty schedule

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SLIDE 35

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Why Numeric Identifiers Help with FGC?

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SLIDE 36

Non-constant output

1 2 3

◮ In FGC natural encoding of solution not bounded in size

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SLIDE 37

Non-constant output

1 2 3 1 2 3

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SLIDE 38

Non-constant output

1 2 3 1 2 3

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SLIDE 39

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The Algorithm

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SLIDE 40

Algorithm Design Idea

derandomisation randomised deterministic algorithm algorithm random bits independent set indentifiers FGC

◮ Use a randomised independent set algorithm as a black box ◮ Iterate over possible random bit strings for the black box to

get a deterministic algorithm

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SLIDE 41

A Randomised Algorithm

1011 1010 0011 0101 0101 0100 1001 0010 1110 0011 0101 0110

◮ A randomised algorithm

for the independent set problem

◮ Each node gets a random

bit string

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SLIDE 42

A Randomised Algorithm

1011 1010 0011 0101 0101 0100 1001 0010 1110 0011 0110 0101

◮ Local maxima join the

independent set

◮ Guarantee: each node v

joins with probability at least 1 − ε deg(v) + 1

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SLIDE 43

Deterministic Algorithm (Oversimplified)

1

time identifiers

1 2 3 4 5 6 7 ◮ Simulate the random algorithm by iterating over all

combinations of inputs

◮ Encoding of the output grows with size of the network ◮ By Naor & Stockmeyer, dependence on n is necessary

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SLIDE 44

Deterministic Algorithm (Oversimplified)

1 2 3 4 5 6 7

time

time t

independent set

1 61

identifiers random bits

3 18 6 71

identifiers randomised algorithm

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SLIDE 45

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Tradeoffs

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SLIDE 46

Granularity Tradeoff

running time length O(1) O(1) granularity unbounded

◮ Any two can be kept

constant in bounded degree graphs

◮ Constant running time

and length of schedule

◮ This work ◮ granularity of schedule

grows with size of the network

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SLIDE 47

Running Time Tradeoff

running time length O(1) O(1) granularity log∗n

◮ Any two can be kept

constant in bounded degree graphs

◮ Constant length of

schedule and granularity

◮ find a

(∆ + 1)-colouring in O(log∗ n) rounds

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SLIDE 48

Length of Schedule Tradeoff

running time length O(1) O(1) granularity poly(n)

◮ Any two can be kept

constant in bounded degree graphs

◮ Constant running time

and granularity

◮ node of colour c(v) is

active during time interval

  • c(v) − 1, c(v)
  • ◮ length of schedule

poly(n)

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SLIDE 49

Time-Length-Granularity Tradeoff—Summary

granularity running time length O(1) O(1) O(1)

◮ Impossible to have

constant running time, length and granularity at the same time

◮ Naor & Stockmeyer 49 / 50

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SLIDE 50

Our Result

0.5 1 1.5 2 2.5 3 3.5 α(∆ + 1)

There is a deterministic distributed algorithm that runs in 1 communication round that, for any α > 1, finds a fractional graph colouring of length at most α(∆ + 1)

50 / 50