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New Classes of Distributed Time Complexity Alkida Balliu Joint work - - PowerPoint PPT Presentation

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New Classes of Distributed Time Complexity Alkida Balliu Joint work with: Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiinen, Dennis Oliveti, Jukka Suomela HALG 2018 New Classes of Distributed Time Complexity 1 / 9 LOCAL Model Distributed

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SLIDE 1

New Classes of Distributed Time Complexity

Alkida Balliu Joint work with: Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Dennis Oliveti, Jukka Suomela

HALG 2018 New Classes of Distributed Time Complexity 1 / 9

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SLIDE 2

LOCAL Model

Distributed Unlimited bandwidth Unlimited computational power Nodes have IDs

HALG 2018 New Classes of Distributed Time Complexity 2 / 9

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SLIDE 3

Locally Checkable Labellings (LCLs)

Introduced by Naor and Stockmeyer in 1995 Constant-size input labels Constant-size output labels The maximum degree is constant Validity of the output is locally checkable

HALG 2018 New Classes of Distributed Time Complexity 3 / 9

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SLIDE 4

LCLs on Cycles

1 log∗n n

HALG 2018 New Classes of Distributed Time Complexity 4 / 9

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SLIDE 5

LCLs on General Graphs

1 log∗n n loglog∗n log n

? ? ? ?

n1

/ 2

n1

/ 3

no

( 1 )

. . .

n1

/ 4

? ? ?

HALG 2018 New Classes of Distributed Time Complexity 5 / 9

slide-6
SLIDE 6

LCLs on General Graphs?

1 log∗n n loglog∗n log n n1

/ 2

n1

/ 3

no

( 1 )

. . .

n1

/ 4

HALG 2018 New Classes of Distributed Time Complexity 6 / 9

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SLIDE 7

Motivation

∆–colouring in general graphs can be done in O(polylog n) rounds [Panconesi, Srinivasan 1995] 4–colouring in 2–dimensional balanced grids can be done in O(polylog n) rounds

1 polylog n √n

HALG 2018 New Classes of Distributed Time Complexity 7 / 9

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SLIDE 8

Motivation

∆–colouring in general graphs can be done in O(polylog n) rounds [Panconesi, Srinivasan 1995] 4–colouring in 2–dimensional balanced grids can be done in O(polylog n) rounds

1 log∗n polylog n √n

[Brandt et al. 2017]

HALG 2018 New Classes of Distributed Time Complexity 7 / 9

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SLIDE 9

Motivation

∆–colouring in general graphs can be done in O(polylog n) rounds [Panconesi, Srinivasan 1995] 4–colouring in 2–dimensional balanced grids can be done in O(polylog n) rounds

1 log∗n polylog n √n

HALG 2018 New Classes of Distributed Time Complexity 7 / 9

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SLIDE 10

LCLs on General Graphs (Our Results)

1 log∗n n loglog∗n log n n1

/ 2

n1

/ 3

no

( 1 )

. . .

n1

/ 4

New (unpublished) results

HALG 2018 New Classes of Distributed Time Complexity 8 / 9

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SLIDE 11

For More Details...

New Classes of Distributed Time Complexity

Alkida Balliu, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Dennis Olivetti, Jukka Suomela

Aalto University, Finland

Context and Goals

  • Study locally checkable labelling (LCL) problems in the

LOCAL model

  • Understanding the complexity landscape of LCL problems on

general graphs

The LOCAL Model

  • Synchronous model
  • Nodes have IDs
  • No limits on bandwidth or computational power

Locally Checkable Labellings

  • Introduced by Naor and Stockmeyer in 1995 [2]
  • ∆–bounded degree graphs (where ∆ is a constant)
  • Constant-size input and output labels
  • Validity of the output is locally checkable

Example: Vertex Colouring

LCLs on Cycles and Paths

  • Θ(1): trivial problems
  • Θ(log∗ n): local problems (symmetry breaking)
  • Θ(n): global problems

Landscape of Complexities on Cycles and Paths 1 log∗n n

LCLs on Trees

  • Any no(1) rounds algorithm can be converted to an O(log n)

rounds algorithm [3]

  • There are problems of complexity Θ(n1/k) [3]

Landscape of Complexities on Trees 1 log∗n n loglog∗n log n ? ? ? ? n1

/ 2

n1

/ 3

no

( 1 ) . . .

n1

/ 4

? ? Conjecture on Trees

1 log∗n n loglog∗n log n n1

/ 2

n1

/ 3

no

( 1 ) . . .

n1

/ 4

? Towards Proving the Conjecture on Trees [4]

1 log∗n n loglog∗n log n n1

/ 2

n1

/ 3

no

( 1 ) . . .

n1

/ 4

? ? ? ? ?

LCLs on General Graphs

  • There are problems with complexity Θ(log n)
  • Any o(log log∗ n) rounds algorithm can be converted to an

O(1) rounds algorithm (same techniques of [2])

  • Any o(log n) rounds algorithm can be converted to an

O(log∗ n) rounds algorithm [5]

  • Many problems require Ω(log n) and O(poly log n) rounds

Landscape of Complexities on General Graphs 1 log∗n n loglog∗n log n ? ? ? ? n1

/ 2

n1

/ 3

no

( 1 ) . . .

n1

/ 4

? ? ? Conjectures

1 log∗n n loglog∗n log n n1

/ 2

n1

/ 3

no

( 1 ) . . .

n1

/ 4

A Motivating Example

  • ∆–colouring in general graphs can be done in O(polylog n)

rounds

  • 4–colouring a 2–dimensional balanced grid can be done in

O(polylog n) rounds

  • In 2–dimensional grids, there is a gap between ω(log∗ n)

and o(√n) [6]

  • Implication: 4–colouring a 2–dimensional balanced grid can

be done in O(log∗ n) rounds

Our Results

Complexities on General Graphs [1] 1 log∗n n loglog∗n log n ? n1

/ 2

n1

/ 3

no

( 1 ) . . .

n1

/ 4

Latest (Unpublished) News [4] 1 log∗n n loglog∗n log n n1

/ 2

n1

/ 3

no

( 1 ) . . .

n1

/ 4

Low vs High Complexities 1 loglog∗n log∗n 2logq

/ plog∗n

logp

/ qlog∗n

(log∗n)q/

p

v 2logq

/ pn

logp

/ qn

log∗n log n n n1

/ 2

n1

/ 3

no

( 1 ) . . .

n1

/ 4

nq/

p

Proof Ideas

  • Start from an LCL problem Π on cycles
  • Build a speed-up construction
  • Example: exponential speed-up function (2ℓ, where ℓ is the

level of the grid-like structure)

A Valid LCL

An LCL problem must be defined on any graph, not just on some “relevant” instances Local Checkability of the Input Graph On Correct Instances

  • T(n) = Θ(log∗ n) for 3–vertex colouring on cycles
  • T(n) = Θ(n) for 2–vertex colouring on cycles
  • Problem Π can be solved in o(T(n)) rounds using the

shortcuts On Incorrect Instances Hardness Balance

  • On incorrect instances, it should be easy to prove that there

is an error

  • On correct instances, it should be impossible, or hard, to

prove that there is an error

Open Problems

  • What happens between Ω(log log∗ n) and O(log∗ n) on

trees?

  • What are meaningful subclasses of LCL problems worth

studying?

References

[1] A. Balliu, J. Hirvonen, J. H. Korhonen, T. Lempiäinen,

  • D. Olivetti, and J. Suomela, “New classes of distributed time

complexity,” in STOC 2018 (to appear). [2] M. Naor and L. Stockmeyer, “What can be computed locally?,” SIAM Journal on Computing, 1995. [3] Y. Chang and S. Pettie, “A time hierarchy theorem for the LOCAL model,” in FOCS 2017. [4] A. Balliu, S. Brandt, D. Olivetti, and J. Suomela, “Almost global problems in the LOCAL model,” 2018 (unpublished). https://arxiv.org/abs/1805.04776. [5] Y. Chang, T. Kopelowitz, and S. Pettie, “An exponential separation between randomized and deterministic complexity in the LOCAL model,” in FOCS 2016. [6] S. Brandt, J. Hirvonen, J. H. Korhonen, T. Lempiäinen, P. R. Östergård, C. Purcell, J. Rybicki, J. Suomela, and P. Uznański, “LCL problems on grids,” in PODC 2017.

HALG 2018 New Classes of Distributed Time Complexity 9 / 9