Linear-in- lower bounds in the LOCAL model Mika Gs, University of - - PowerPoint PPT Presentation

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Linear-in- lower bounds in the LOCAL model Mika Gs, University of - - PowerPoint PPT Presentation

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Linear-in- lower bounds in the LOCAL model Mika Gs, University of Toronto Juho Hirvonen , Aalto University & HIIT Jukka Suomela, Aalto Univesity & HIIT PODC 16.7.2014 This work The first linear-in- lower bound for a

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Linear-in-Δ lower bounds in the LOCAL model

Mika Göös, University of Toronto Juho Hirvonen, Aalto University & HIIT Jukka Suomela, Aalto Univesity & HIIT

  • PODC 16.7.2014
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This work

The first linear-in-Δ lower bound for a natural graph problem in the LOCAL model Fractional maximal matching:

  • There is no o(Δ)-algorithm, independent of n
  • There is an O(Δ)-algorithm, independent of n

(Δ = maximum degree, n = number of vertices)

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Matching

Matching assigns weight 1 to matched edges and weight 0 to the rest

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1 1

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Fractional matching

FM is a linear relaxation of matching: weights of the incident edges sum up to at most 1

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0.4 0.1 0.3 0.3 0.1 0.4

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Maximal fractional matching

A node is saturated, if the sum of the weights of the incident edges is equal to one

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0.4 0.1 0.3 0.3 0.1 0.4

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Maximal fractional matching

The fractional matching is maximal, if no two unsaturated nodes are adjacent

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0.4 0.1 0.3 0.3 0.1 0.4 0.4 0.1 0.3 0.2 0.1 0.4

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Standard LOCAL model

  • Synchronous communication
  • No bandwidth restrictions
  • Running time = number of communication rounds
  • Both deterministic and randomized algorithms

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This work

The first linear-in-Δ lower bound for a natural graph problem in the LOCAL model Fractional maximal matching:

  • There is no o(Δ)-algorithm, independent of n
  • There is an O(Δ)-algorithm, independent of n

(Δ = maximum degree, n = number of vertices)

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Prior work

Coordination problems:

  • Maximal matching
  • Maximal independent set
  • (Δ+1)-coloring

Algorithms O(Δ+ log*n) also O(polylog(n) Lower bounds Ω(log* n) and Ω(log Δ)

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[Linial ’92] [Kuhn et al. ’05]

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SLIDE 10

Prior work

Coordination problems:

  • Maximal matching
  • Maximal independent set
  • (Δ+1)-coloring

Algorithms O(Δ+ log*n) also O(polylog(n) Lower bounds Ω(log* n) and Ω(log Δ)

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[Linial ’92] [Kuhn et al. ’05]

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SLIDE 11

The Proof

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The Proof

A short guide

  • Step 0: Introduce models EC, PO, OI and ID
  • Step 1: Ω(Δ)-lower bound in the EC-model
  • Step 2: Simulation result EC↝PO↝OI↝ID
  • Step 3: ID ↝ Randomized algorithms

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The Proof

A short guide

  • Step 0: Introduce models EC, PO, OI and ID
  • Step 1: Ω(Δ)-lower bound in the EC-model
  • Step 2: Simulation result EC↝PO↝OI↝ID
  • Step 3: ID ↝ Randomized algorithms

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Edge coloring (EC)

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EC ↝ PO ↝ OI ↝ ID ↝ R

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Port-numbering and orientation (PO)

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EC ↝ PO ↝ OI ↝ ID ↝ R

1 2 2 1 2 1 3 3 1 2 1 4

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Port-numbering and orientation (PO)

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EC ↝ PO ↝ OI ↝ ID ↝ R

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Unique Identifiers (ID)

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EC ↝ PO ↝ OI ↝ ID ↝ R

1 18 71 19 8

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Order Invariant (OI)

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EC ↝ PO ↝ OI ↝ ID ↝ R

1 18 71 19 8 2 15 41 31 5

≈

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The Proof

A short guide

  • Step 0: Introduce models EC, PO, OI and ID
  • Step 1: Ω(Δ)-lower bound in the EC-model
  • Step 2: Simulation result EC↝PO↝OI↝ID
  • Step 3: ID ↝ Randomized algorithms

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Loopy graphs

A graph is k-loopy, if it has at least k self-loops at each node

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EC ↝ PO ↝ OI ↝ ID ↝ R k=2 k=3

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Loopy graphs

Loopy graphs are a compact representation of simple graphs with lots of symmetry

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EC ↝ PO ↝ OI ↝ ID ↝ R

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Loopy graphs

A loopy graph can be unfolded to get a simple graph

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EC ↝ PO ↝ OI ↝ ID ↝ R

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Loopy graphs

A loopy graph can be unfolded to get a simple graph

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EC ↝ PO ↝ OI ↝ ID ↝ R

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Loopy graphs

loopy graphs ≈ infinite trees

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EC ↝ PO ↝ OI ↝ ID ↝ R

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Loopy graphs

Key observation: a maximal fractional matching must saturate all nodes of a loopy graph!

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EC ↝ PO ↝ OI ↝ ID ↝ R

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EC lower bound

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G H

EC ↝ PO ↝ OI ↝ ID ↝ R

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EC lower bound

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EC ↝ PO ↝ OI ↝ ID ↝ R

GG GH HH

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EC lower bound

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EC ↝ PO ↝ OI ↝ ID ↝ R

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The Proof

A short guide to the proof

  • Step 0: Introduce models EC, PO, OI and ID
  • Step 1: Ω(Δ)-lower bound in the EC-model
  • Step 2: Simulation result EC↝PO↝OI↝ID
  • Step 3: ID ↝ Randomized algorithms

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EC ↝ PO

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EC ↝ PO

Assume we have an o(Δ)-time algorithm A for maximal edge packing in the PO model

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EC ↝ PO ↝ OI ↝ ID ↝ R

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EC ↝ PO

Transform EC graph into PO graph by replacing each edge with two oriented edges

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EC ↝ PO ↝ OI ↝ ID ↝ R

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EC ↝ PO

Simulate the PO-algorithm A and combine the weights of the corresponding edges

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EC ↝ PO ↝ OI ↝ ID ↝ R

0.15 0.1 0.3 0.2 0.25 0.1 0.45 0.45

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EC ↝ PO

We get an o(Δ)-algorithm in the EC-model, which is a contradiction

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EC ↝ PO ↝ OI ↝ ID ↝ R

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PO ↝ OI

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PO ↝ OI

  • Similar technology as Göös et al. (2012)
  • Now we do not need any approximation

guarantees

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EC ↝ PO ↝ OI ↝ ID ↝ R

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PO ↝ OI

Assume we have a PO-algorithm A We use port numbers and orientation to get a local

  • rdering

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EC ↝ PO ↝ OI ↝ ID ↝ R

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PO ↝ OI

Take the universal cover of G

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EC ↝ PO ↝ OI ↝ ID ↝ R

v v

G U(G)

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EC ↝ PO ↝ OI ↝ ID ↝ R Canonically

  • rdered tree

17 19

2 1 23 9 14 13 39 11 6 5 24 42 36 46 47 38 50 51 20 32 33 26

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53 45 52 31 49 48 30 41 40 15 29 43 44 25

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28 22 34 21

27

8 18 7 12 4 16 3 10

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EC ↝ PO ↝ OI ↝ ID ↝ R

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2 1 23 9 14 13 39 11 6 5 24 42 36 46 47 38 50 51 20 32 33 26

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53 45 52 31 49 48 30 41 40 15 29 43 44 25

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28 22 34 21

27

8 18 7 12 4 16 3 10

17

v

Embed U(G)

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PO ↝ OI

It is possible to make a PO-graph an OI-graph locally Use this to simulate A

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EC ↝ PO ↝ OI ↝ ID ↝ R

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OI ↝ ID

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OI ↝ ID

Use the OI ↝ ID lemma of Naor and Stockmeyer (1995) (essentially Ramsey’s Theorem) The idea is to force any ID-algorithm A to behave like an OI-algorithm on some inputs

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EC ↝ PO ↝ OI ↝ ID ↝ R

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OI ↝ ID

Trick: consider an algorithm A* that simulates A and

  • utputs 1 at saturated nodes and 0 elsewhere to

apply the Lemma This forces all nodes to be saturated in A in loopy neighborhoods Any change must propagate outside A’s run time

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EC ↝ PO ↝ OI ↝ ID ↝ R

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The Proof

A short guide

  • Step 0: Introduce models EC, PO, OI and ID
  • Step 1: Ω(Δ)-lower bound in the EC-model
  • Step 2: Simulation result EC↝PO↝OI↝ID
  • Step 3: ID ↝ Randomized algorithms

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Randomized algorithms

Idea: Reduce random algorithms back to deterministic ones Again use a lemma of Naor and Stockmeyer (1995)

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EC ↝ PO ↝ OI ↝ ID ↝ R

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Summary

This work Fractional maximal matching has complexity Θ(Δ) Open questions What is the complexity of maximal matching? What is the complexity of 2-colored maximal matching?

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