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Round Compression for Parallel Matching Algorithms Krzysztof Onak - - PowerPoint PPT Presentation

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Round Compression for Parallel Matching Algorithms Krzysztof Onak IBM T.J. Watson Research Center Joint work with Artur Czumaj (U of Warwick), Jakub acki (Google), Aleksander M adry (MIT), Slobodan Mitrovi c (EPFL), and Piotr

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SLIDE 1

Round Compression for Parallel Matching Algorithms

Krzysztof Onak

IBM T.J. Watson Research Center

Joint work with Artur Czumaj (U of Warwick), Jakub Ł ˛ acki (Google), Aleksander M ˛ adry (MIT), Slobodan Mitrovi´ c (EPFL), and Piotr Sankowski (U of Warsaw)

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 1 / 25

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SLIDE 2

Mandatory “Big Data” Slides

Massive Data Systems

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 2 / 25

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SLIDE 3

Outline

1

Model of Computation

2

Graph Matchings in MPC

3

Review of Distributed Algorithms

4

Our Algorithm

5

Further Research

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 3 / 25

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SLIDE 4

Outline

1

Model of Computation

2

Graph Matchings in MPC

3

Review of Distributed Algorithms

4

Our Algorithm

5

Further Research

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 4 / 25

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SLIDE 5

Model: Massive Parallel Computation (MPC)

Introduced by Karloff, Suri, and Vassilvitskii (2010) M machines S space per machine Input: N items

Machine Machine Machine Machine Machine Machine Machine Machine Machine

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 5 / 25

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SLIDE 6

Model: Massive Parallel Computation (MPC)

Introduced by Karloff, Suri, and Vassilvitskii (2010) M machines S space per machine Input: N items

Machine Machine Machine Machine Machine Machine Machine Machine Machine

  • Initially: each machine receives N/M items

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 5 / 25

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SLIDE 7

Model: Massive Parallel Computation (MPC)

Introduced by Karloff, Suri, and Vassilvitskii (2010) M machines S space per machine Input: N items

Machine Machine Machine Machine Machine Machine Machine Machine Machine

  • Initially: each machine receives N/M items
  • Single round:
  • 1. Each machine performs computation
  • 2. Each machine sends and receives at most O(S) data

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 5 / 25

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SLIDE 8

Model: Massive Parallel Computation (MPC)

  • Inspired by MapReduce [Dean, Ghemawat 2004]
  • Sleak abstraction that hides details of MapReduce

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 6 / 25

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SLIDE 9

Model: Massive Parallel Computation (MPC)

  • Inspired by MapReduce [Dean, Ghemawat 2004]
  • Sleak abstraction that hides details of MapReduce
  • Total space considerations:

[Beame 2009: Problem 27 at sublinear.info] [Beame, Koutris, Suciu 2013] [Andoni, Nikolov, Onak, Yaroslavtsev 2014]

  • Karloff et al. allow for N1−ǫ machines with N1−ǫ space

⇒ near quadratic total space N2−2ǫ

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 6 / 25

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SLIDE 10

Model: Massive Parallel Computation (MPC)

  • Inspired by MapReduce [Dean, Ghemawat 2004]
  • Sleak abstraction that hides details of MapReduce
  • Total space considerations:

[Beame 2009: Problem 27 at sublinear.info] [Beame, Koutris, Suciu 2013] [Andoni, Nikolov, Onak, Yaroslavtsev 2014]

  • Karloff et al. allow for N1−ǫ machines with N1−ǫ space

⇒ near quadratic total space N2−2ǫ

  • A refined version asks for near-linear total space:

M × S = ˜ O(N).

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 6 / 25

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SLIDE 11

Model: Massive Parallel Computation (MPC)

  • Inspired by MapReduce [Dean, Ghemawat 2004]
  • Sleak abstraction that hides details of MapReduce
  • Total space considerations:

[Beame 2009: Problem 27 at sublinear.info] [Beame, Koutris, Suciu 2013] [Andoni, Nikolov, Onak, Yaroslavtsev 2014]

  • Karloff et al. allow for N1−ǫ machines with N1−ǫ space

⇒ near quadratic total space N2−2ǫ

  • A refined version asks for near-linear total space:

M × S = ˜ O(N).

  • Goals:
  • Small number of rounds
  • Small space per machine
  • Fast local computation

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 6 / 25

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SLIDE 12

This Talk: MPC for Graphs

Input: edges of an m-edge graph on n vertices S space per machine M = O(m/S) machines

Machine Machine Machine Machine Machine Machine Machine Machine Machine

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 7 / 25

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SLIDE 13

This Talk: Matching Algorithms

Why study graph matchings?

  • Non-trivial appealing packing problem
  • Great testbed for many new algorithmic ideas
  • Helpful to understand the power of the model
  • They have practical applications

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 8 / 25

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SLIDE 14

Outline

1

Model of Computation

2

Graph Matchings in MPC

3

Review of Distributed Algorithms

4

Our Algorithm

5

Further Research

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 9 / 25

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SLIDE 15

Graph Problems

Maximum Matching: find maximum set of vertex disjoint edges

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 10 / 25

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SLIDE 16

Large Matchings: Rounds vs. Space

  • Space n1+Ω(1): (1 + ǫ)-approximation in O(1) rounds

[Lattanzi, Moseley, Suri, Vassilvitskii 2011] [Ahn, Guha 2015]

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 11 / 25

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SLIDE 17

Large Matchings: Rounds vs. Space

  • Space n1+Ω(1): (1 + ǫ)-approximation in O(1) rounds

[Lattanzi, Moseley, Suri, Vassilvitskii 2011] [Ahn, Guha 2015]

  • Space nΩ(1): 2-approximation in O(log n) rounds
  • Simulate classic MIS or Maximal Matching

PRAM/distributed algorithms:

  • Luby (1986)
  • Alon, Babai, Itai (1986)
  • Israeli, Itai (1986)

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 11 / 25

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SLIDE 18

Large Matchings: Rounds vs. Space

  • Space n1+Ω(1): (1 + ǫ)-approximation in O(1) rounds

[Lattanzi, Moseley, Suri, Vassilvitskii 2011] [Ahn, Guha 2015]

  • Space nΩ(1): 2-approximation in O(log n) rounds
  • Simulate classic MIS or Maximal Matching

PRAM/distributed algorithms:

  • Luby (1986)
  • Alon, Babai, Itai (1986)
  • Israeli, Itai (1986)
  • Tools for simulation:
  • Karloff, Suri, and Vassilvitskii (2010)
  • Goodrich, Sitchinava, and Zhang (2011)

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 11 / 25

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SLIDE 19

Large Matchings: Rounds vs. Space

  • Space n1+Ω(1): (1 + ǫ)-approximation in O(1) rounds

[Lattanzi, Moseley, Suri, Vassilvitskii 2011] [Ahn, Guha 2015]

  • Space nΩ(1): 2-approximation in O(log n) rounds
  • Simulate classic MIS or Maximal Matching

PRAM/distributed algorithms:

  • Luby (1986)
  • Alon, Babai, Itai (1986)
  • Israeli, Itai (1986)
  • Tools for simulation:
  • Karloff, Suri, and Vassilvitskii (2010)
  • Goodrich, Sitchinava, and Zhang (2011)
  • For O(n) space, round complexity becomes Θ(log n)

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 11 / 25

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SLIDE 20

Our Results

For O(n) space, improve O(1)-approximation to poly(log log n) rounds

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 12 / 25

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SLIDE 21

Our Results

For O(n) space, improve O(1)-approximation to poly(log log n) rounds

More detailed version:

For n/α space, O(1)-approximation in O

  • (log log n)2 + log α
  • rounds

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 12 / 25

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SLIDE 22

Our Results

For O(n) space, improve O(1)-approximation to poly(log log n) rounds

More detailed version:

For n/α space, O(1)-approximation in O

  • (log log n)2 + log α
  • rounds

(Works well even if space slightly sublinear in n)

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 12 / 25

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SLIDE 23

Our Results

For O(n) space, improve O(1)-approximation to poly(log log n) rounds

More detailed version:

For n/α space, O(1)-approximation in O

  • (log log n)2 + log α
  • rounds

(Works well even if space slightly sublinear in n) Interesting space regime:

  • often just enough to fit a solution on a single machine
  • gold standard for space in semi-streaming algorithms
  • reasonable middle ground

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 12 / 25

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SLIDE 24

Highlights of Our Approach

Starting point: O(1)-approximation distributed algorithm (in the LOCAL model)

  • It uses Θ(log n) rounds
  • So would direct simulation

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 13 / 25

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SLIDE 25

Highlights of Our Approach

Starting point: O(1)-approximation distributed algorithm (in the LOCAL model)

  • It uses Θ(log n) rounds
  • So would direct simulation

Round compression: Repeatedly compress a superconstant number of rounds

  • f the original algorithm into a constant number of MPC

rounds

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 13 / 25

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SLIDE 26

Highlights of Our Approach

Starting point: O(1)-approximation distributed algorithm (in the LOCAL model)

  • It uses Θ(log n) rounds
  • So would direct simulation

Round compression: Repeatedly compress a superconstant number of rounds

  • f the original algorithm into a constant number of MPC

rounds Vertex sampling: Previous algorithms used edge sampling

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 13 / 25

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SLIDE 27

Outline

1

Model of Computation

2

Graph Matchings in MPC

3

Review of Distributed Algorithms

4

Our Algorithm

5

Further Research

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 14 / 25

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SLIDE 28

Vertex Cover [Parnas, Ron 2007]

  • Distributed O(log n)-approximation algorithm

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 15 / 25

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SLIDE 29

Vertex Cover [Parnas, Ron 2007]

V1

  • Distributed O(log n)-approximation algorithm
  • Algorithm:
  • Remove vertices of degree at least n/2 ⇒ V1
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SLIDE 30

Vertex Cover [Parnas, Ron 2007]

V1 V2

  • Distributed O(log n)-approximation algorithm
  • Algorithm:
  • Remove vertices of degree at least n/2 ⇒ V1
  • Remove vertices of degree at least n/4 ⇒ V2
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SLIDE 31

Vertex Cover [Parnas, Ron 2007]

V1 V2 V3

  • Distributed O(log n)-approximation algorithm
  • Algorithm:
  • Remove vertices of degree at least n/2 ⇒ V1
  • Remove vertices of degree at least n/4 ⇒ V2
  • Remove vertices of degree at least n/8 ⇒ V3
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SLIDE 32

Vertex Cover [Parnas, Ron 2007]

. . .

V1 V2 V3

  • Distributed O(log n)-approximation algorithm
  • Algorithm:
  • Remove vertices of degree at least n/2 ⇒ V1
  • Remove vertices of degree at least n/4 ⇒ V2
  • Remove vertices of degree at least n/8 ⇒ V3
  • . . .
  • Remove vertices of degree at least n/2i ⇒ Vi
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SLIDE 33

Vertex Cover [Parnas, Ron 2007]

. . .

V1 V2 V3

  • Distributed O(log n)-approximation algorithm
  • Algorithm:
  • Remove vertices of degree at least n/2 ⇒ V1
  • Remove vertices of degree at least n/4 ⇒ V2
  • Remove vertices of degree at least n/8 ⇒ V3
  • . . .
  • Remove vertices of degree at least n/2i ⇒ Vi
  • . . .
  • Remove vertices of degree at least 1 ⇒ Vlog n

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 15 / 25

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SLIDE 34

Vertex Cover [Parnas, Ron 2007]

. . .

V1 V2 V3

  • Distributed O(log n)-approximation algorithm
  • Algorithm:
  • Remove vertices of degree at least n/2 ⇒ V1
  • Remove vertices of degree at least n/4 ⇒ V2
  • Remove vertices of degree at least n/8 ⇒ V3
  • . . .
  • Remove vertices of degree at least n/2i ⇒ Vi
  • . . .
  • Remove vertices of degree at least 1 ⇒ Vlog n
  • C := Vi is a vertex cover
  • f size O(log n) · OPT

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 15 / 25

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SLIDE 35

O(1) Approximation [Onak, Rubinfeld 2010]

  • Originally developed for dynamic graph algorithms

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 16 / 25

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SLIDE 36

O(1) Approximation [Onak, Rubinfeld 2010]

Vi

  • Originally developed for dynamic graph algorithms
  • Modified Parnas-Ron partition, in phase i:
  • Select Vi as before

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 16 / 25

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SLIDE 37

O(1) Approximation [Onak, Rubinfeld 2010]

Vi Mi

  • Originally developed for dynamic graph algorithms
  • Modified Parnas-Ron partition, in phase i:
  • Select Vi as before
  • Find a matching Mi of vertices Vi of size Ω(|Vi|)

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 16 / 25

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SLIDE 38

O(1) Approximation [Onak, Rubinfeld 2010]

Vi Mi V ′

i

  • Originally developed for dynamic graph algorithms
  • Modified Parnas-Ron partition, in phase i:
  • Select Vi as before
  • Find a matching Mi of vertices Vi of size Ω(|Vi|)
  • Let V ′

i be the additionally matched vertices

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 16 / 25

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SLIDE 39

O(1) Approximation [Onak, Rubinfeld 2010]

Vi V ′

i

  • Originally developed for dynamic graph algorithms
  • Modified Parnas-Ron partition, in phase i:
  • Select Vi as before
  • Find a matching Mi of vertices Vi of size Ω(|Vi|)
  • Let V ′

i be the additionally matched vertices

  • Remove from the graph both Vi and V ′

i

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 16 / 25

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SLIDE 40

O(1) Approximation [Onak, Rubinfeld 2010]

  • Originally developed for dynamic graph algorithms
  • Modified Parnas-Ron partition, in phase i:
  • Select Vi as before
  • Find a matching Mi of vertices Vi of size Ω(|Vi|)
  • Let V ′

i be the additionally matched vertices

  • Remove from the graph both Vi and V ′

i

  • Output: vertex cover C =

i(Vi ∪ V ′ i )

matching M =

i Mi

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 16 / 25

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SLIDE 41

O(1) Approximation [Onak, Rubinfeld 2010]

  • Originally developed for dynamic graph algorithms
  • Modified Parnas-Ron partition, in phase i:
  • Select Vi as before
  • Find a matching Mi of vertices Vi of size Ω(|Vi|)
  • Let V ′

i be the additionally matched vertices

  • Remove from the graph both Vi and V ′

i

  • Output: vertex cover C =

i(Vi ∪ V ′ i )

matching M =

i Mi

  • Analysis:
  • |C| and |M| are within a constant factor
  • minimum vertex cover size ≥ maximum matching size
  • C and M are constant-factor approximations

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 16 / 25

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SLIDE 42

O(1) Approximation [Onak, Rubinfeld 2010]

  • Originally developed for dynamic graph algorithms
  • Modified Parnas-Ron partition, in phase i:
  • Select Vi as before
  • Find a matching Mi of vertices Vi of size Ω(|Vi|)
  • Let V ′

i be the additionally matched vertices

  • Remove from the graph both Vi and V ′

i

  • Output: vertex cover C =

i(Vi ∪ V ′ i )

matching M =

i Mi

  • Analysis:
  • |C| and |M| are within a constant factor
  • minimum vertex cover size ≥ maximum matching size
  • C and M are constant-factor approximations
  • Goal: efficiently emulate this algorithm in MPC

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 16 / 25

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SLIDE 43

Outline

1

Model of Computation

2

Graph Matchings in MPC

3

Review of Distributed Algorithms

4

Our Algorithm

5

Further Research

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 17 / 25

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SLIDE 44

Emulating the Peeling Algorithm

  • Needed to emulate a phase of the peeling algorithm:

1 (Approximate) vertex degrees 2 A random neighbor for each high degree vertex

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 18 / 25

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SLIDE 45

Emulating the Peeling Algorithm

  • Needed to emulate a phase of the peeling algorithm:

1 (Approximate) vertex degrees 2 A random neighbor for each high degree vertex

  • Then we can:

1 Find the set of high degree vertices 2 Find a matching for constant fraction of them

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 18 / 25

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SLIDE 46

Emulating the Peeling Algorithm

  • Needed to emulate a phase of the peeling algorithm:

1 (Approximate) vertex degrees 2 A random neighbor for each high degree vertex

  • Then we can:

1 Find the set of high degree vertices 2 Find a matching for constant fraction of them

  • Our plan:
  • Partition vertices at random into a number of groups
  • Ensure that graphs induced by each group fit
  • nto a single machine
  • Ensure that enough neighbors on the machine to

satisfy the properties above

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 18 / 25

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SLIDE 47

Random Vertex Partitioning

  • Phase 1:
  • Partition vertices at random into √n groups
  • Each group should have O((√n)2) = O(n) edges
  • In each group, degrees scale down by factor of √n
  • Can still find high degree vertices and their random

neighbors

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 19 / 25

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SLIDE 48

Random Vertex Partitioning

  • Phase 1:
  • Partition vertices at random into √n groups
  • Each group should have O((√n)2) = O(n) edges
  • In each group, degrees scale down by factor of √n
  • Can still find high degree vertices and their random

neighbors

  • Phase 2:
  • Now maximum degree roughly n/2
  • Repeat the same by partitioning vertices into √n

groups

. . .

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 19 / 25

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SLIDE 49

Random Vertex Partitioning

  • Phase 1:
  • Partition vertices at random into √n groups
  • Each group should have O((√n)2) = O(n) edges
  • In each group, degrees scale down by factor of √n
  • Can still find high degree vertices and their random

neighbors

  • Phase 2:
  • Now maximum degree roughly n/2
  • Repeat the same by partitioning vertices into √n

groups

. . .

  • Can do this for roughly log(√n) = 1

2 log n phases:

  • Stuck when max degree gets roughly √n
  • Why? Current high degree vertices see no neighbors

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 19 / 25

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SLIDE 50

Ideal Algorithm

  • Maybe do not repartition each time?
  • No clear reason why this would not work
  • However, vertices are no longer randomly partitioned
  • Not clear how to analyze this

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 20 / 25

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SLIDE 51

Ideal Algorithm

  • Maybe do not repartition each time?
  • No clear reason why this would not work
  • However, vertices are no longer randomly partitioned
  • Not clear how to analyze this
  • Where would this take us?
  • We would compress 1

2 log n phases into O(1) MPC

rounds

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 20 / 25

slide-52
SLIDE 52

Ideal Algorithm

  • Maybe do not repartition each time?
  • No clear reason why this would not work
  • However, vertices are no longer randomly partitioned
  • Not clear how to analyze this
  • Where would this take us?
  • We would compress 1

2 log n phases into O(1) MPC

rounds

  • In the end, max degree at most √n
  • Can now partition into only n1/4 groups and graphs

induced by each group will fit onto a single machine

  • Would be able to simulate the next 1

4 log n phases

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 20 / 25

slide-53
SLIDE 53

Ideal Algorithm

  • Maybe do not repartition each time?
  • No clear reason why this would not work
  • However, vertices are no longer randomly partitioned
  • Not clear how to analyze this
  • Where would this take us?
  • We would compress 1

2 log n phases into O(1) MPC

rounds

  • In the end, max degree at most √n
  • Can now partition into only n1/4 groups and graphs

induced by each group will fit onto a single machine

  • Would be able to simulate the next 1

4 log n phases

  • Then 1

8 log n phases, 1 16 log n phases, . . .

  • After O(log log n) MPC rounds, we would be done

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 20 / 25

slide-54
SLIDE 54

Actual Solution

  • We do not know how to analyze this approach directly
  • We tweak the peeling algorithm
  • Show independence and near-uniformity of surviving

vertices

  • We get O
  • (log log n)2

rounds

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 21 / 25

slide-55
SLIDE 55

Outline

1

Model of Computation

2

Graph Matchings in MPC

3

Review of Distributed Algorithms

4

Our Algorithm

5

Further Research

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 22 / 25

slide-56
SLIDE 56

Recent Follow-Up Work

Assadi (arXiv 2017)

  • Round compression for the Parnas-Ron algorithm
  • O(log n)-approximation to vertex cover in O(log log n)

MPC round

  • Bounding technique of Assadi and Khanna (2017)

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 23 / 25

slide-57
SLIDE 57

Recent Follow-Up Work

Assadi (arXiv 2017) Assadi, Bateni, Bernstein, Mirrokni, and Stein (arXiv 2017)

  • Improve round complexity to O(log log n)
  • Approximation improved to 1 + ǫ [McGregor 2005]
  • (2 + ǫ)-approximation for vertex cover
  • No round compression, but still vertex sampling
  • Apply techniques developed for dynamic matching

[Bernstein, Stein 2015]

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 23 / 25

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SLIDE 58

Recent Follow-Up Work

Assadi (arXiv 2017) Assadi, Bateni, Bernstein, Mirrokni, and Stein (arXiv 2017)

Ghaffari, Gouleakis, Konrad, Mitrovi´ c, Rubinfeld (PODC 2018)

  • Improve round complexity to O(log log n)
  • Simulate a parallel fractional algorithm
  • Explore connections to congested clique model
  • Also O(log log n)-round algorithm for Maximal

Independent Set

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 23 / 25

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SLIDE 59

Follow-Up Questions

  • Round compression for other problems?

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 24 / 25

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SLIDE 60

Follow-Up Questions

  • Round compression for other problems?
  • Any good reason why log log n seems to be a barrier?
  • MPC hard to prove unconditional lower bounds
  • Show reductions to/from other problems?
  • Limitations of natural sampling techniques?

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 24 / 25

slide-61
SLIDE 61

Follow-Up Questions

  • Round compression for other problems?
  • Any good reason why log log n seems to be a barrier?
  • MPC hard to prove unconditional lower bounds
  • Show reductions to/from other problems?
  • Limitations of natural sampling techniques?
  • Show that a simple very greedy algorithm just works?

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 24 / 25

slide-62
SLIDE 62

Questions?

Full version: https://arxiv.org/abs/1707.03478

Krzysztof Onak (IBM Research) Round Compression for Parallel Matching Algorithms 25 / 25