Local Algorithms on Grids Jukka Suomela Aalto University - - PowerPoint PPT Presentation

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Local Algorithms on Grids Jukka Suomela Aalto University - - PowerPoint PPT Presentation

Nov 17, 2022 462 likes •1.01k views

Local Algorithms on Grids Jukka Suomela Aalto University arXiv:1702.05456 LCL Problems on Grids , joint work with: Janne H Korhonen, Tuomo Lempiinen, Christopher Purcell, Patric RJ stergrd (Aalto) Sebastian Brandt,

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Local Algorithms

  • n Grids

Jukka Suomela · Aalto University

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arXiv:1702.05456

“LCL Problems on Grids”, joint work with:

  • Janne H Korhonen, Tuomo Lempiäinen,

Christopher Purcell, Patric RJ Östergård (Aalto)

  • Sebastian Brandt, Przemysław Uznański (ETH)
  • Juho Hirvonen (Paris Diderot)
  • Joel Rybicki (Helsinki)
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2-colouring 3-colouring 4-colouring global global local

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Introduction

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Setting

  • Distributed graph algorithms
  • Input graph = computer network
  • node = computer, edge = communication link
  • unknown topology
  • Each node outputs its own part of solution
  • e.g. graph colouring: node outputs its own colour
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Setting

  • Deterministic distributed algorithms,

LOCAL model of computing

  • unique identifiers
  • synchronous communication rounds
  • time = number of rounds until all nodes stop
  • unlimited message size,

unlimited local computation

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Setting

  • Deterministic distributed algorithms,

LOCAL model of computing

  • Time = distance
  • Algorithm with running time T:

mapping from radius-T neighbourhoods to local outputs

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LCL problems

  • LCL = locally checkable labelling
  • Naor–Stockmeyer (1995)
  • Valid solution can be detected by checking

O(1)-radius neighbourhood of each node

  • maximal independent set, maximal matching,

vertex colouring, edge colouring …

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LCL problems

  • All LCL problems can be solved with

O(1)-round nondeterministic algorithms

  • guess a solution, verify it in O(1) rounds
  • Key question: how fast can we solve them

with deterministic algorithms?

  • cf. P vs. NP
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Traditional settings

  • Directed cycles
  • Cole–Vishkin (1986), Linial (1992)…
  • well understood
  • General (bounded-degree) graphs
  • lots of ongoing work…
  • typical challenge:

expander-like constructions

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Our setting today

  • 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?

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1D grids

  • Vertex colouring
  • 2-colouring: global, Θ(n) rounds
  • 3-colouring: local, Θ(log* n) rounds
  • Cole–Vishkin (1986), Linial (1992)
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Why is 3-colouring Θ(log* n)?

  • Upper bound: one-round colour reduction
  • input: colouring with 2k colours
  • output: colouring with 2k colours
  • Lower bound: speed-up lemma
  • given: algorithm for k-colouring in time T
  • construct: algorithm for 2k-colouring in time T − 1
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1D grids

  • Vertex colouring
  • 2-colouring: global, Θ(n) rounds
  • 3-colouring: local, Θ(log* n) rounds
  • Cole–Vishkin (1986), Linial (1992)
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2D grids

  • Vertex colouring
  • 2-colouring: global, Θ(n) rounds
  • 3-colouring: ???
  • 4-colouring: ???
  • 5-colouring: local, Θ(log* n) rounds
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2D grids

  • Vertex colouring
  • 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
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Classification of LCL problems

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LCL problems on grids

  • O(1) time: “trivial”
  • o(log* n) time implies O(1) time (Naor–Stockmeyer)
  • Θ(log* n) time: “local”
  • Θ(n) time: “global”
  • Why nothing between local and global?
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Normalisation

  • Setting: LCL problems, 2D grids
  • Theorem: Any o(n)-time algorithm can be

translated to a “normal form”:

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

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Normalisation in more detail…

  • 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
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Some proof ideas…

  • Given: A solves P in time o(n) in n × n grids
  • Solving P in time O(log* N) in N × N grids:
  • pick suitable n = O(1), k = O(1)
  • find a maximal independent set (MIS) in Gk
  • use MIS to find locally unique identifiers for

n × n neighbourhoods

  • simulate A in n × n local neighbourhoods
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LCL problems on grids

  • O(1) time: “trivial”
  • o(log* n) time implies O(1) time (Naor–Stockmeyer)
  • Θ(log* n) time: “local”
  • o(n) time implies O(log* n) time (normalisation)
  • Θ(n) time: “global”
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Vertex colouring

  • Every LCL problem is trivial, local, or global
  • Why is 4-colouring in 2D grids “local”?
  • Why is 3-colouring in 2D grids “global”?
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4-colouring

  • n grids
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4-colouring

  • Lucky guess: maybe it is local?
  • Try to 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
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3-colouring

  • n grids
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3-colouring

  • Inherently different from 4-colouring:
  • cannot be solved locally
  • But also different from 2-colouring:
  • nontrivial to argue that the problem is global
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2-colouring 3-colouring 4-colouring global global local

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Proof idea

  • Assume: a local algorithm

for 3-colouring in n × n grids

  • Implication: a local algorithm

for “sum coordination” in n-cycles

  • But we can prove that this problem is global
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even × even

  • dd × odd

Consider any feasible 3-colouring…

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even × even

  • dd × odd

We can convert it into a greedy solution in constant time

(eliminate colour 2 whenever possible, then colour 3)

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even × even

  • dd × odd

Greedy solution: boundaries + 2-coloured regions

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even × even

  • dd × odd

Parity changes at each boundary

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even × even

  • dd × odd

Parity changes at each boundary

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even × even

  • dd × odd

Wrap around: same parity Wrap around:

  • pposite parity
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even × even

  • dd × odd

Boundaries can be oriented with local rules

(keep orange on right, white on left)

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even × even

  • dd × odd

Pick any row, label boundary crossings with +1 / −1

up = +1, down = −1 −1

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even × even

  • dd × odd

Sum of crossings: even Sum of crossings:

  • dd

−1

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even × even

  • dd × odd

Sum of crossings: even Sum of crossings:

  • dd

−1

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even × even

  • dd × odd

Boundaries are closed curves: constant sum

up = +1, down = −1 −1 −1 −1 −1

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even × even

  • dd × odd

Locality: sum only depends on grid dimensions, not on IDs

(otherwise we could construct one instance with non-constant sum) −1 −1 −1 −1

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Sum coordination

  • What any 3-colouring algorithms has to

solve for every row of the grid:

  • label nodes with {+1, 0, −1}
  • there is some function q so that the sum of labels is

q(n) in any n-cycle, regardless of unique identifiers

  • q(n) odd iff n is odd: cannot label everything with 0
  • |q(n)| not too large: cannot label everything with +1
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Sum coordination

  • What any 3-colouring algorithms has to

solve for every row of the grid

  • Requires global coordination
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Conclusions

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2-colouring 3-colouring 4-colouring global global local

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Conclusions: LCLs on grids

  • Only three complexity classes in 2D grids:

trivial O(1), local Θ(log* n), global Θ(n)

  • 4-colouring is local: algorithm synthesis
  • 3-colouring is global: sum coordination
  • Can be generalised to d-dimensional grids!
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