LCL Problems Janne H. Korhonen on Grids Aalto University LCL - - PowerPoint PPT Presentation

▶
lcl problems
SMART_READER_LITE
LIVE PREVIEW

LCL Problems Janne H. Korhonen on Grids Aalto University LCL - - PowerPoint PPT Presentation

Aug 13, 2022 286 likes •599 views

LCL Problems Janne H. Korhonen on Grids Aalto University LCL Problems Janne H. Korhonen on Grids Aalto University joint work with: Tuomo Lempiinen, Patric R.J. stergrd, Christopher Purcell, Jukka Suomela (Aalto) Sebastian

slide-1
SLIDE 1

LCL Problems

  • n Grids

Janne H. Korhonen

Aalto University

slide-2
SLIDE 2

joint work with:

  • Tuomo Lempiäinen, Patric R.J. Östergård,


Christopher Purcell, Jukka Suomela (Aalto)

  • Sebastian Brandt, Przemysław Uznański


(ETH Zürich)

  • Juho Hirvonen (Paris Diderot)
  • Joel Rybicki (U. Helsinki)

(arXiv:1702.05456)

LCL Problems

  • n Grids

Janne H. Korhonen

Aalto University

slide-3
SLIDE 3

Setting: LOCAL model, 2D grids

slide-4
SLIDE 4

Introduction

1.

slide-5
SLIDE 5
  • graph = computing network
  • graph = input instance
  • input: local information at each node
  • output: local structure of solution

Setting:

LOCAL model

slide-6
SLIDE 6
  • LOCAL model
  • unique identifiers
  • synchronous communication rounds
  • time measure: number of rounds
  • unlimited local computation, unlimited message size
  • This work: deterministic algorithms

Setting:

LOCAL model

slide-7
SLIDE 7
  • LCL = locally checkable labelling
  • Naor and Stockmeyer (1995)
  • constant-size labels
  • validity of a solution checkable in O(1)-radius

neighbourhood of each node

  • maximal independent set, maximal matching, vertex

colouring, edge colouring ...

Setting:

LCL problems

slide-8
SLIDE 8
  • Directed cycles
  • Cole and Vishkin (1986), 


Linial (1992)…

  • Well-understood classification
  • Θ(1) time: “trivial”
  • Θ(log* n) time: “local” (3-colouring)
  • Θ(n) time: “global” (2-colouring)
  • classification decidable

Background:

LCLs on cycles

slide-9
SLIDE 9
  • General (bounded-degree) graphs
  • lots of ongoing work
  • challenge: expander graphs
  • A more complicated landscape
  • gaps: nothing is ω(log* n) and o(log n)
  • intermediate problems: Θ(log n) complexity
  • Brandt et al. (2016), Chang et al. (2016),


Ghaffari and Su (2017), …

Background:

LCLs on general graphs

slide-10
SLIDE 10
  • Oriented grids (2D)
  • toroidal grid, n × n nodes, unique identifiers
  • consistent orientations north/east/south/west
  • Generalisation of directed cycles (1D)
  • Closer to real-world systems than expander-like

worst-case constructions? This work:

2D grids

slide-11
SLIDE 11

This work:

2D grids

  • Vertex colouring (deterministic)
  • 2-colouring: global, Θ(n) rounds
  • 3-colouring: ???
  • 4-colouring: ???
  • 5-colouring: local, Θ(log* n) rounds
slide-12
SLIDE 12
  • Vertex colouring (deterministic)
  • 2-colouring: global, Θ(n) rounds
  • 3-colouring: global, Θ(n) rounds
  • 4-colouring: local, Θ(log* n) rounds
  • 5-colouring: local, Θ(log* n) rounds

This work:

2D grids

slide-13
SLIDE 13

Classification of LCL problems on grids

2.

slide-14
SLIDE 14

Main theorem:

Classification on grids

  • LCL problems on 2D grids have exactly three

possible deterministic complexities:

  • Θ(1) time: “trivial”
  • Θ(log* n) time: “local”
  • Θ(n) time: “global”
  • Why?
  • o(log* n) time implies O(1) time (Naor–Stockmeyer)
  • o(n) time implies O(log* n) time (this work)
slide-15
SLIDE 15

Main theorem:

Normalisation/speed-up

  • Theorem: Any deterministic o(n)-time algorithm

can be translated to a “normal form”:

  • 1. fixed Θ(log* n)-time symmetry breaking component
  • 2. problem-specific O(1)-time component

92 33 77 57 49 26 71 79 8 62 48 24 31 21 15 30 60 67 5 17 95 23 47 87 80 25 38 20 64 45 61 91 51 69 1 74 55 3 98 88 99 58 53 63 40 16 2 39 1 1 1 1 1 1 1 1 1 1 1 1 1 1

O(log* n) O(1) MIS f

slide-16
SLIDE 16

Main theorem:

Normalisation, proof ideas

  • For any problem P of complexity o(n), there are

constants k and r and function f such that P can be solved as follows:

  • input: 2D grid G with unique identifiers
  • find a maximal independent set in Gk
  • discard unique identifiers
  • apply function f to each r × r neighbourhood
slide-17
SLIDE 17

Main theorem:

Normalisation, proof ideas

  • Why does this work?
  • o(n) algorithm A cannot see the whole graph
  • symmetry breaking gives locally unique identifiers
  • pretend that instance has constant size


→ A still has to produce valid output

  • Compare with speed-up for general graphs
  • o(log n) → O(log* n)
  • Chang et al. (2016)
slide-18
SLIDE 18

Vertex colouring
 upper and lower bounds

3.

slide-19
SLIDE 19
  • 4-colouring is local
  • why?

Local problems:

4-colouring

  • First proof: prior work and normalisation
  • Δ-colouring is polylog(n) for constant Δ
  • Panconesi and Srinivasan (1995)
  • normalisation → O(log* n)
slide-20
SLIDE 20
  • 4-colouring is local
  • why?

Local problems:

4-colouring

  • Second proof: synthesis
  • guess it is local, use computers to find normal form
  • turns out it is enough to find an MIS in G3, then

consider 7 × 5 tiles

  • algorithm ≈ mapping {0, 1}7 × 5 → {1, 2, 3, 4}
  • only 2079 possible tiles, easy to find a solution
slide-21
SLIDE 21
slide-22
SLIDE 22
slide-23
SLIDE 23
slide-24
SLIDE 24
  • Can be applied to any LCL problem
  • However, classification on grids is undecidable
  • synthesis works if the problem is local
  • cannot give a negative answers for global problems
  • constants quite small in practice
  • more examples in the paper

Local problems:

More on synthesis

slide-25
SLIDE 25
  • 2-colouring is global
  • 3-colouring is global
  • not local, but lower bound non-trivial
  • 3-colouring algorithm on grids solves

“sum coordination” on n-cycles

  • “sum coordination” is global
  • reduction has topological flavour

Global problems:

2-colouring and 3-colouring

slide-26
SLIDE 26
  • Human-designed local algorithm

for 4-colouring on d-dimensional grids

  • Connection to finitary colourings
  • infinite 2D grids
  • same proof techniques
  • upper and lower bounds
  • Holroyd et al. (2016)

Vertex colouring:

Some remarks

slide-27
SLIDE 27

Conclusions

4.

slide-28
SLIDE 28
  • Generalisations
  • d-dimensional grids: everything generalises
  • bounded neighbourhood growth: similar speed-up
  • randomised algorithms?
  • LCL landscape on general graphs still open
  • Θ(n1/2) problems exist (this work)
  • Θ(n1/k) problems exist (Chang and Pettie 2017)
  • more gap theorems
slide-29
SLIDE 29
  • Generalisations
  • d-dimensional grids: everything generalises
  • bounded neighbourhood growth: similar speed-up
  • randomised algorithms?
  • LCL landscape on general graphs still open
  • Θ(n1/2) problems exist (this work)
  • Θ(n1/k) problems exist (Chang and Pettie 2017)
  • more gap theorems

Thanks! Questions?

slide-30
SLIDE 30