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

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 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) 2 / 50

Model of Computation: LOCAL ◮ Communication graph 3 / 50

Model of Computation: LOCAL T = 0 ◮ Synchronous communication 4 / 50

Model of Computation: LOCAL T = 1 ◮ Synchronous communication 5 / 50

Model of Computation: LOCAL T = 2 ◮ In T rounds gather radius- T neighbourhood 6 / 50

Model of Computation: LOCAL ◮ Constant-time algorithms 7 / 50

Model of Computation: LOCAL ◮ Each node maps neighbourhood to output 8 / 50

Fractional Graph Colouring 9 / 50

Fractional Graph Colouring output input 0 0 . 5 1 1 . 5 2 2 . 5 3 10 / 50

Fractional Graph Colouring output input independent set 0 0 . 5 1 1 . 5 2 2 . 5 3 11 / 50

Fractional Graph Colouring output input independent set 0 0 . 5 1 1 . 5 2 2 . 5 3 12 / 50

Fractional Graph Colouring output input independent set 0 0 . 5 1 1 . 5 2 2 . 5 3 13 / 50

Fractional Graph Colouring output input independent set ≥ 1 ≥ 1 ≥ 1 ≥ 1 ≥ 1 0 0 . 5 1 1 . 5 2 2 . 5 3 14 / 50

Fractional Graph Colouring output input independent set ≥ 1 ≥ 1 ≥ 1 ≥ 1 ≥ 1 0 0 . 5 1 1 . 5 2 2 . 5 3 length of schedule 15 / 50

Fractional Graph Colouring output input ≥ 1 ≥ 1 ≥ 1 ≥ 1 ≥ 1 0 0 . 5 1 1 . 5 2 2 . 5 3 16 / 50

Fractional Graph Colouring output input independent set ≥ 1 ≥ 1 ≥ 1 ≥ 1 ≥ 1 0 0 . 5 1 1 . 5 2 2 . 5 3 length of schedule 17 / 50

Our Result Again 0 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

Lower Bound ∆ = 2 ≥ 1 ≥ 1 ≥ 1 0 0 . 5 1 1 . 5 2 2 . 5 3 ∆ + 1 19 / 50

Finding a Fractional Graph Colouring 20 / 50

Finding a Fractional Graph Colouring ◮ Impossible to break symmetry in an anonymous cycle 21 / 50

Finding a Fractional Graph Colouring ◮ Nodes must produce an empty schedule 22 / 50

Finding a Fractional Graph Colouring 44 31 9 17 91 6 61 8 7 88 16 75 3 65 5 39 66 87 95 76 43 34 12 ◮ Standard assumption: numeric identifiers 23 / 50

Finding a Fractional Graph Colouring 44 31 9 17 91 6 61 8 7 88 16 75 3 65 5 39 66 87 95 76 43 34 12 ◮ Standard assumption: numeric identifiers 24 / 50

Finding a Fractional Graph Colouring 44 31 9 17 91 6 61 8 7 88 16 75 3 65 5 39 66 87 95 76 43 34 12 ◮ Standard assumption: numeric identifiers ◮ FGC is the first example where numeric identifiers give a constant-time algorithm 25 / 50

Why Numeric Identifiers Do Not Help? 26 / 50

Numeric Identifiers Not Needed 44 31 9 17 91 6 61 8 7 88 16 75 3 65 5 39 66 87 95 76 43 34 12 ◮ Naor & Stockmeyer (1995) studied when numeric identifiers are necessary ◮ LCL-problems 27 / 50

LCL-problems ◮ Maximal Independent Set 28 / 50

LCL-problems ◮ Vertex Cover 29 / 50

LCL-problems ◮ Maximal Matching 30 / 50

LCL-problems ◮ Fractional Graph Colouring 31 / 50

Numeric Identifiers Not Needed 44 31 9 17 91 6 61 8 7 88 16 75 3 65 5 39 66 87 95 76 43 34 12 ◮ Naor & Stockmeyer (1995): In LCL-problems numeric identifiers not necessary ◮ Technicality: applies if output bounded 32 / 50

No FGC with Comparisons ◮ Identifiers arranged in an ascending order 33 / 50

No FGC with Comparisons ◮ Some nodes must produce an empty schedule 34 / 50

Why Numeric Identifiers Help with FGC? 35 / 50

Non-constant output 0 1 2 3 ◮ In FGC natural encoding of solution not bounded in size 36 / 50

Non-constant output 0 1 2 3 0 1 2 3 37 / 50

Non-constant output 0 1 2 3 0 1 2 3 38 / 50

The Algorithm 39 / 50

Algorithm Design Idea random bits indentifiers derandomisation randomised deterministic algorithm algorithm independent set 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 40 / 50

A Randomised Algorithm ◮ A randomised algorithm for the independent set 1011 problem 0100 ◮ Each node gets a random 0011 bit string 1010 0101 0101 1001 0010 0101 0110 1110 0011 41 / 50

A Randomised Algorithm ◮ Local maxima join the independent set 1011 ◮ Guarantee: each node v 0100 joins with probability at 0011 least 1010 0101 1 − ε deg( v ) + 1 0101 1001 0010 0101 0110 1110 0011 42 / 50

Deterministic Algorithm (Oversimplified) identifiers 1 2 3 4 5 6 7 time 0 1 ◮ 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 43 / 50

Deterministic Algorithm (Oversimplified) time t identifiers 1 2 3 4 5 6 7 time 0 1 3 61 18 71 6 randomised algorithm identifiers random bits independent set 44 / 50

Tradeoffs 45 / 50

Granularity Tradeoff ◮ Any two can be kept constant in bounded degree graphs ◮ Constant running time and length of schedule ◮ This work granularity unbounded ◮ granularity of schedule grows with size of the O ( 1 ) length network running time O ( 1 ) 46 / 50

Running Time Tradeoff ◮ Any two can be kept constant in bounded degree graphs ◮ Constant length of schedule and granularity ◮ find a O ( 1 ) granularity (∆ + 1)-colouring in O (log ∗ n ) rounds O ( 1 ) length running time log ∗ n 47 / 50

Length of Schedule Tradeoff ◮ Any two can be kept constant in bounded degree graphs ◮ Constant running time and granularity ◮ node of colour c ( v ) is O ( 1 ) granularity active during time interval poly ( n ) length � � c ( v ) − 1 , c ( v ) running time O ( 1 ) ◮ length of schedule poly( n ) 48 / 50

Time-Length-Granularity Tradeoff—Summary ◮ Impossible to have constant running time, length and granularity at the same time ◮ Naor & Stockmeyer O ( 1 ) granularity O ( 1 ) length running time O ( 1 ) 49 / 50

Our Result 0 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