Lower Bounds for Local Algorithms Jukka Suomela Aalto University, - - PowerPoint PPT Presentation

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Lower Bounds for Local Algorithms Jukka Suomela Aalto University, Finland ADGA Austin, Texas 12 October 2014 LOCAL model Input: simple undirected graph G communication network nodes labelled with 54 unique O

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SLIDE 1

Lower Bounds for
 Local Algorithms

Jukka Suomela Aalto University, Finland

  • ADGA · Austin, Texas · 12 October 2014
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SLIDE 2

LOCAL model

  • Input: simple undirected graph G
  • communication network
  • nodes labelled with


unique O(log n)-bit
 identifiers 3 54 23 12

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SLIDE 3

LOCAL model

  • Input: simple undirected graph G
  • Output: each node v produces a local output
  • graph colouring: colour of node v
  • vertex cover: 1 if v is in the cover
  • matching: with whom v is matched
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SLIDE 4

LOCAL model

  • Nodes exchange messages with each other,


update local states

  • Synchronous communication rounds
  • Arbitrarily large messages
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SLIDE 5

LOCAL model

  • Time = number of communication rounds
  • until all nodes stop and


produce their local outputs

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SLIDE 6

LOCAL model

  • Time = number of communication rounds
  • Time = distance:
  • in t communication rounds,


all nodes can learn everything
 in their radius-t neighbourhoods

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SLIDE 7

LOCAL model

time t = 2

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SLIDE 8

LOCAL model

A: 1

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SLIDE 9

LOCAL model

  • Everything trivial in time diam(G)
  • all nodes see whole G,


can compute any function of G

  • What can be solved much faster?
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SLIDE 10

Distributed
 time complexity

  • Smallest t such that the problem


can be solved in time t

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SLIDE 11

Distributed
 time complexity

  • n = number of nodes
  • Δ = maximum degree
  • Δ < n
  • Time complexity t = t(n, Δ)
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SLIDE 12

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1)

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SLIDE 13

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1) t = O(Δ + log* n)

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SLIDE 14

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1) t = O(log n)

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SLIDE 15

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1) t = O(log* Δ)

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SLIDE 16

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1)

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SLIDE 17

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1) All problems

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SLIDE 18

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1) Maximal matching

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SLIDE 19

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1) Bipartite
 maximal matching

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SLIDE 20

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1) Linear
 programming
 approximation

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SLIDE 21

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1) Weak colouring
 (odd-degree graphs)

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SLIDE 22

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1) Dominating sets
 (planar graphs)

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SLIDE 23
  • ur focus today

n >> Δ

Landscape

Δ log* Δ log n log Δ n log* n O(1) O(1)

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SLIDE 24

Typical state of the art

Δ log* Δ log Δ log* n O(1) O(1) no yes tight bounds
 as a function of n positive: O(log* n) negative: o(log* n)

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SLIDE 25

positive: O(Δ)

Typical state of the art

Δ log* Δ log Δ log* n O(1) O(1) yes no ? ? ? negative: o(log Δ) exponential gap
 as a function of Δ

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SLIDE 26

positive: O(Δ)

Typical state of the art

Δ log* Δ log Δ log* n O(1) O(1) yes ? ? ? negative: nothing exponential gap
 as a function of Δ — or much worse

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SLIDE 27

fairly well
 understood Δ log* Δ log Δ log* n O(1) O(1) poorly
 understood

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SLIDE 28

Example:
 LP approximation

  • O(log Δ): possible
  • Kuhn et al. (2004, 2006)
  • o(log Δ): not possible
  • Kuhn et al. (2004, 2006)
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SLIDE 29

Example:
 Maximal matching

  • O(Δ + log* n): possible
  • Panconesi & Rizzi (2001)
  • O(Δ) + o(log* n): not possible
  • Linial (1992)
  • o(Δ) + O(log* n): unknown
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SLIDE 30

Example:
 (Δ+1)-colouring

  • O(Δ + log* n): possible
  • Barenboim & Elkin (2008), Kuhn (2008)
  • O(Δ) + o(log* n): not possible
  • Linial (1992)
  • o(Δ) + O(log* n): unknown
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SLIDE 31

Example: Bipartite maximal matching

  • O(Δ): trivial
  • Hańćkowiak et al. (1998)
  • o(Δ): unknown
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SLIDE 32

Example:
 Semi-matching

  • O(Δ5): possible
  • Czygrinow et al. (2012)
  • o(Δ5): unknown
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SLIDE 33

Example:
 Semi-matching

  • O(Δ5): possible
  • Czygrinow et al. (2012)
  • o(Δ5): unknown
  • o(Δ): unknown
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SLIDE 34

Example:
 Weak colouring

  • O(log* Δ): possible (in odd-degree graphs)
  • Naor & Stockmeyer (1995)
  • o(log* Δ): unknown
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SLIDE 35

fairly well
 understood Δ log* Δ log Δ log* n O(1) O(1) poorly
 understood

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SLIDE 36

Orthogonal challenges?

  • n: “symmetry breaking”
  • fairly well understood
  • Cole & Vishkin (1986), Linial (1992),


Ramsey theory …

  • Δ: “local coordination”
  • poorly understood
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SLIDE 37

“symmetry breaking” Δ log* Δ log Δ log* n O(1) O(1) “local coordination”

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SLIDE 38

Orthogonal challenges

  • Example: maximal matching, O(Δ + log* n)
  • Restricted versions:
  • pure symmetry breaking, O(log* n)
  • pure local coordination, O(Δ)
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SLIDE 39

Orthogonal challenges

  • Example: maximal matching, O(Δ + log* n)
  • Pure symmetry breaking:
  • input = cycle
  • no need for local coordination
  • O(log* n) is possible and tight
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SLIDE 40

Orthogonal challenges

  • Example: maximal matching, O(Δ + log* n)
  • Pure local coordination:
  • input = 2-coloured graph
  • no need for symmetry breaking
  • O(Δ) is possible — is it tight?
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SLIDE 41

Maximal matching
 in 2-coloured graphs

  • Trivial algorithm:
  • black nodes send proposals


to their neighbours, one by one

  • white nodes accept the first


proposal that they get

  • “Coordination” ≈ one by one traversal
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SLIDE 42

Maximal matching
 in 2-coloured graphs

  • Trivial algorithm:
  • black nodes send proposals


to their neighbours, one by one

  • white nodes accept the first


proposal that they get

  • Clearly O(Δ), but is this tight?
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SLIDE 43

Maximal matching
 in 2-coloured graphs

  • General case:
  • upper bound: O(Δ)
  • lower bound: Ω(log Δ) — Kuhn et al.
  • Regular graphs:
  • upper bound: O(Δ)
  • lower bound: nothing!
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SLIDE 44

Linear-in-Δ bounds

  • Many combinatorial problems seem to


require “local coordination” , takes O(Δ) time?

  • Lacking: linear-in-Δ lower bounds
  • known for restricted algorithm classes


(Kuhn & Wattenhofer 2006)

  • not previously known for the LOCAL model
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SLIDE 45

Recent progress

  • Maximal fractional matching
  • O(Δ): possible
  • SPAA 2010
  • o(Δ): not possible
  • PODC 2014
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SLIDE 46
  • Edges labelled with integers {0, 1}
  • Sum of incident edges at most 1
  • Maximal matching:


cannot increase the value of any label

Matching

1

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SLIDE 47
  • Edges labelled with real numbers [0, 1]
  • Sum of incident edges at most 1
  • Maximal fractional matching:


cannot increase the value of any label

Fractional
 matching

0.4 0.6 0.3 0.3

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SLIDE 48
  • Possible in time O(Δ)
  • does not require symmetry breaking
  • d-regular graph: label all edges with 1/d
  • Nontrivial part: graphs that are not regular…

Maximal fractional matching

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SLIDE 49
  • Not possible in time o(Δ), independently of n
  • note: we do not say anything about e.g.


possibility of solving in time o(Δ) + O(log* n)

  • Key ingredient of the proof:


analyse many different models of
 distributed computing

Maximal fractional matching

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SLIDE 50

ID: unique identifiers

Nodes have unique identifiers,


  • utput may depend on them

2 9 7 4 3 54 23 12

≠

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SLIDE 51

OI: order invariant

Output does not change if we change
 identifiers but keep their relative order 2 9 7 4 3 54 23 12

=

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SLIDE 52

PO: ports & orientation

No identifiers Node v labels
 incident edges
 with 1, …, deg(v) Edges oriented 2 1 1 2 1 3 2 1

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SLIDE 53

EC: edge colouring

No identifiers No orientations Edges coloured
 with O(Δ) colours 2 1 1 3

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SLIDE 54

2 1 1 2 1 3 2 1 2 1 1 3 a b c d a < b < c < d 3 12 23 54

ID PO OI EC

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SLIDE 55

2 1 1 2 1 3 2 1 2 1 1 3 a b c d a < b < c < d 3 12 23 54

ID PO OI EC

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SLIDE 56

Simulation argument

  • Trivial: ID → OI → PO
  • for any problem
  • We show: EC → PO → OI → ID
  • for maximal fractional matching


in “loopy graphs”

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SLIDE 57

Proof overview

  • EC model is very limited
  • maximal fractional matching requires


Ω(Δ) time in EC, even for “loopy graphs”

  • Simulation argument: EC → PO → OI → ID
  • maximal fractional matching requires


Ω(Δ) time in ID, at least for “loopy graphs”

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SLIDE 58

EC

  • Recursively construct a degree-i graph


where this algorithm takes time i

  • Focus on “loopy graphs”
  • highly symmetric
  • forces algorithm to produce “tight” outputs


(all nodes saturated, “perfect matching”) 2 1 1 3

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SLIDE 59

EC → PO

“Unhelpful” port numbering & orientation 2 2 1 3

PO EC

1 6 5 3 4

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SLIDE 60

PO → OI

“Unhelpful”
 total order can be easily
 constructed given
 a port numbering
 and orientation

8 7 18

12

6 5 24

11

20 33 32

26

19

4 3 16

10

2 1 23

9

15 41 40

29

17

14 13 39

25

31 53 52

45

38 51 50

44

37

22 21 34

28

30 49 48

43

36 47 46

42

35

27

1 2 3 4 4 2 3 1

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SLIDE 61

OI → ID

“Unhelpful” unique identifiers Ramsey-like argument:
 for any algorithm we can find unique identifiers
 that do not help in comparison with total order

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EC → PO → OI → ID

  • In general: stronger models help
  • In our case: we can always come up


with situations in which ID model
 is not any better than EC model

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SLIDE 63

What about


  • ther problems?
  • Now we have a linear-in-Δ lower bound


for maximal fractional matching

  • Can we use the same techniques to prove


lower bounds for other problems?

  • e.g., maximal matching?
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SLIDE 64

General recipe

  • 1. Find a suitable “simple model”
  • 2. Prove a lower bound for the simple model
  • keep input “symmetric”
  • keep output “tight” and “fragile”
  • local changes have non-local consequences
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SLIDE 65

General recipe

  • 1. Find a suitable “simple model”
  • 2. Prove a lower bound for the simple model
  • 3. Amplify the lower bound
  • simple model → OI (some thinking required)
  • OI → ID (standard techniques)
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SLIDE 66

What about
 maximal matchings?

  • Could we use the same techniques to show


that o(Δ) + O(log* n) is not sufficient
 for maximal matching?

  • Two obstacles…
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SLIDE 67

What about
 maximal matchings?

  • Obstacle 1 — final step:
  • final step OI → ID based on


a Ramsey argument

  • works great for t independent of n
  • fails if t ≈ log* n
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SLIDE 68

What about
 maximal matchings?

  • Obstacle 2 — starting point:
  • O(log* n) time enough to find


e.g. graph colouring

  • cannot assume “symmetric” input
  • difficult to force “tight” and “fragile” output
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SLIDE 69

What about
 maximal matchings?

  • Two hard, interlinked obstacles
  • How to proceed:
  • get rid of obstacle 1 — log* n
  • focus on obstacle 2 — asymmetry
  • Start with bipartite maximal matchings
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SLIDE 70

Maximal matching
 in 2-coloured graphs

  • Can be solved in time O(Δ) independently of n
  • Can focus on just one obstacle: asymmetry
  • Most of the other machinery already exists!
  • we just need tight bounds for simple models
  • should be easy to generalise to LOCAL model
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SLIDE 71

Maximal matching
 in 2-coloured graphs

  • Until we have lower bounds:


reductions, conditional lower bounds

  • many other problems are at least


as hard as bipartite maximal matching

  • locally optimal semi-matching in time T


→ bipartite maximal matching in time T

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SLIDE 72

Summary

  • Distributed time complexity, LOCAL model
  • O(log* n): “symmetry breaking”

, OK

  • O(Δ): “local coordination”

, poorly understood

  • Maximal fractional matching solved,


next step: bipartite maximal matching