You say its only constant? Then something must be hyperfinite! And - - PowerPoint PPT Presentation

You say it’s only constant? Then something must be hyperfinite!

And I say your title is too long.

Hendrik Fichtenberger July 20, 2019

Property Testing in a Nutshell

planar

1

Property Testing in a Nutshell

planar non-planar

1

Property Testing in a Nutshell

planar non-planar non-planar

1

Property Testing in a Nutshell

planar non-planar non-planar non-planar

1

Property Testing in a Nutshell

planar non-planar non-planar non-planar time complexity: Ω(|𝑊|)

1

Property Testing in a Nutshell

planar non-planar non-planar non-planar time complexity: Ω(|𝑊|)

1

Property Testing in a Nutshell

planar non-planar non-planar non-planar time complexity: Ω(|𝑊|) very quite slightly

1

Property Testing in a Nutshell

planar non-planar non-planar non-planar time complexity: Ω(|𝑊|) very quite slightly

1

Property Testing in a Nutshell

planar non-planar non-planar non-planar 𝜗 1

# edge edits |𝑊|+|𝐹|

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Property Testing in a Nutshell

planar non-planar non-planar non-planar reject w.p. > 2

3

accept w.p. > 2

3

𝜗 1

# edge edits |𝑊|+|𝐹|

1

Property Testing in a Nutshell

planar non-planar non-planar non-planar reject w.p. > 2

3

accept w.p. > 2

3

𝜗 1

# edge edits |𝑊|+|𝐹|

𝜗-close 𝜗-far 𝜗-far

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Property Testing in a Nutshell

planar non-planar non-planar non-planar reject w.p. > 2

3

accept w.p. > 2

3

𝜗 1

# edge edits |𝑊|+|𝐹|

𝜗-close 𝜗-far 𝜗-far complexity: # queries to adjacency list entries

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Property Testing of Bounded Degree Graphs

bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar

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Property Testing of Bounded Degree Graphs

bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar, degree-regular, cycle-free, subgraph-free, connected, minor-free, hyperfinite, ...

2

Property Testing of Bounded Degree Graphs

bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar, degree-regular, cycle-free, subgraph-free, connected, minor-free, hyperfinite, ... bipartite, expander Θ(√𝑜)

2

Property Testing of Bounded Degree Graphs

bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar, degree-regular, cycle-free, subgraph-free, connected, minor-free, hyperfinite, ... bipartite, expander Θ(√𝑜) big picture?

2

Property Testing of Bounded Degree Graphs

bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar, degree-regular, cycle-free, subgraph-free, connected, minor-free, hyperfinite, ... bipartite, expander Θ(√𝑜) big picture? bounded-degree graphs: litule known about constant-time testability

2

𝑙-Disks and Frequency Vectors

disk𝑙(𝑤): subgraph induced by BFS(𝑤) of depth 𝑙

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𝑙-Disks and Frequency Vectors

disk1( ) disk𝑙(𝑤): subgraph induced by BFS(𝑤) of depth 𝑙

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𝑙-Disks and Frequency Vectors

disk2( ) disk1( ) disk𝑙(𝑤): subgraph induced by BFS(𝑤) of depth 𝑙

3

𝑙-Disks and Frequency Vectors

disk2( ) disk1( ) disk𝑙(𝑤): subgraph induced by BFS(𝑤) of depth 𝑙 freq𝑙(𝐻): for each 𝑙-disk isomorphism type calculate its share of vertices freq2( ) = ( 0.4 0.6 ⋮ ) ∑ 1

3

𝑙-Disks and Frequency Vectors

disk2( ) disk1( ) disk𝑙(𝑤): subgraph induced by BFS(𝑤) of depth 𝑙 freq𝑙(𝐻): for each 𝑙-disk isomorphism type calculate its share of vertices freq2( ) = ( 0.4 0.6 ⋮ ) ∑ 1 frequency vector is a locality feature

3

Constant-Qvery Testers

accept reject 1 1 Π Π Π Π Π Theorem [GR’09, …] Every property tester with constant query complexity 𝑟 ∶= 𝑟(𝜗) can be transformed into an algorithm that

• 1. computes an approximation ̃

freqΘ(𝑟)(𝐻) of freqΘ(𝑟)(𝐻)

• 2. accepts ifg ‖̃

freqΘ(𝑟)(𝐻) − freqΘ(𝑟)(𝐻′)‖1 ≤

1 Θ(𝑟) for any 𝐻′ ∈ Π

Goldreich, Ron, STOC’09 4

Hyperfinite Graphs

In every planar graph, there exists a set of √𝑜 separators Definition (𝝑, 𝐭)-hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇-hyperfinite: (𝜗, 𝜍(𝜗))-hyperfinite for all 𝜗 ∈ (0, 1]

5

Hyperfinite Graphs

In every planar graph, there exists a set of √𝑜 separators Definition (𝝑, 𝐭)-hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇-hyperfinite: (𝜗, 𝜍(𝜗))-hyperfinite for all 𝜗 ∈ (0, 1]

5

Hyperfinite Graphs

In every planar graph, there exists a set of √𝑜 separators > 1

3|𝑊|

> 1

3|𝑊|

Definition (𝝑, 𝐭)-hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇-hyperfinite: (𝜗, 𝜍(𝜗))-hyperfinite for all 𝜗 ∈ (0, 1]

5

Hyperfinite Graphs

In every planar graph, there exists a set of √𝑜 separators remove 𝜗𝑜 edges components of size 𝜗−2 Definition (𝝑, 𝐭)-hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇-hyperfinite: (𝜗, 𝜍(𝜗))-hyperfinite for all 𝜗 ∈ (0, 1]

5

Hyperfinite Graphs

𝜍(0.5) 𝜍(0.5) 𝜍(0.5) 𝜗 = 0.5 Definition (𝝑, 𝐭)-hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇-hyperfinite: (𝜗, 𝜍(𝜗))-hyperfinite for all 𝜗 ∈ (0, 1]

5

Hyperfinite Graphs

𝜗 = 0.5 𝜍(0.5) 𝜍(0.5) 𝜍(0.5) Definition (𝝑, 𝐭)-hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇-hyperfinite: (𝜗, 𝜍(𝜗))-hyperfinite for all 𝜗 ∈ (0, 1]

5

Hyperfinite Graphs

𝜗 = 0.5 𝜗 = 0.6 𝜍(0.6) 𝜍(0.6) 𝜍(0.6) Definition (𝝑, 𝐭)-hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇-hyperfinite: (𝜗, 𝜍(𝜗))-hyperfinite for all 𝜗 ∈ (0, 1]

5

The Story so Far

Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍-hyperfinite

Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far

Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍-hyperfinite [GR’09]

Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far

Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍-hyperfinite [GR’09] [NS’11]

Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far

Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍-hyperfinite [GR’09] [NS’11]

Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far

Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍-hyperfinite [GR’09] [NS’11] e.g. Π = connectivity contains expanders

Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far

Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍-hyperfinite [GR’09] [NS’11] e.g. Π = connectivity contains expanders ∃ Π′ ⊆ Π: Π′ is 𝜍′-hyperfinite [FPS’19]

Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

Small Frequency-Preserver Graphs

freq𝑙(𝐻) freq𝑙(𝐼) 𝜀 𝐻 𝐼 𝑜 𝑁(𝜀, 𝑙) Theorem [Alon’11] For every 𝜀, 𝑙 > 0, there exists 𝑁(𝜀, 𝑙) such that for every 𝐻 there exists 𝐼 of size at most 𝑁(𝜀, 𝑙) and ‖freq𝑙(𝐻) − freq𝑙(𝐼)‖1 < 𝜀.

Alon, ’11, see: Lovász, Large Networks and Graph Limits, Proposition 19.10 7

Small Frequency-Preserver Graphs

freq𝑙(𝐻) freq𝑙(𝐼) 𝜀 𝐻 𝐼 𝑜 𝑁(𝜀, 𝑙) For ρ-hyperfinite graphs M(δ,k) is at most Theorem [Alon’11] For every 𝜀, 𝑙 > 0, there exists 𝑁(𝜀, 𝑙) such that for every 𝐻 there exists 𝐼 of size at most 𝑁(𝜀, 𝑙) and ‖freq𝑙(𝐻) − freq𝑙(𝐼)‖1 < 𝜀.

Alon, ’11, see: Lovász, Large Networks and Graph Limits, Proposition 19.10 7

Putuing Everything Into The Property Testing Blender

in Π 𝑜

• rel. 𝚸

size hypf. ?

• freq. v.

change

• riginal

start w/ 𝐻 ∈ Π

8

Putuing Everything Into The Property Testing Blender

in Π 𝑜

• rel. 𝚸

size hypf. ?

• freq. v.

change

• riginal

start w/ 𝐻 ∈ Π 𝑃(1)

• freq. pres.

? ? < 𝜀/2

8

Putuing Everything Into The Property Testing Blender

in Π 𝑜

• rel. 𝚸

size hypf. ?

• freq. v.

change

• riginal

start w/ 𝐻 ∈ Π 𝑃(1)

• freq. pres.

? ? < 𝜀/2 𝑜 (0, 𝑁(𝜀, 𝑙))- hyperfinite blow-up 𝜗-close to Π = 0

8

Putuing Everything Into The Property Testing Blender

in Π 𝑜

• rel. 𝚸

size hypf. ?

• freq. v.

change

• riginal

start w/ 𝐻 ∈ Π 𝑃(1)

• freq. pres.

? ? < 𝜀/2 𝑜 (0, 𝑁(𝜀, 𝑙))- hyperfinite blow-up 𝜗-close to Π = 0 in Π 𝑜 (𝜗, 𝑁(𝜀, 𝑙))- hyperfinite modify < 𝜀/2

8

Open (Bl)ending

Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍-hyperfinite [GR’09] [NS’11] e.g. Π = connectivity contains expanders ∃ Π′ ⊆ Π: Π′ is 𝜍′-hyperfinite [FPS’19]

9

Connection Between Hyperfinite Graphs and 𝑙-Disks

freq𝑙(𝐻) freq𝑙(𝐼)

10𝜗 𝑒

𝐻 𝐼 Theorem [BSS’08] If 𝐻 is 𝜍(𝜗)-hyperfinite, then all 𝐼 with ‖freq𝑙(𝐻) − freq𝑙(𝐼)‖1 < 10𝜗

𝑒

are 𝜍(𝑔 (𝜗))-hyperfinite for some function 𝑔 .

Benjamini, Schramm, Shapira, STOC’08 10

…still blending…

𝐻 pick 𝐻 ∈ Π

11

…still blending…

𝐻 pick 𝐻 ∈ Π (𝜗1, 𝑁(𝜀1, 𝑙1))-hyperfinite

11

…still blending…

(𝜗′

1, 𝑁(𝜀1, 𝑙2))-hyperfinite

𝐻 pick 𝐻 ∈ Π (𝜗1, 𝑁(𝜀1, 𝑙1))-hyperfinite (𝜗2, 𝑁(𝜀2, 𝑙2))-hyperfinite

11

…still blending…

(𝜗′

1, 𝑁(𝜀1, 𝑙2))-hyperfinite

(𝜗′

2, 𝑁(𝜀2, 𝑙2))-hyperfinite

(𝜗′

1, 𝑁(𝜀1, 𝑙2))-hyperfinite

𝐻 pick 𝐻 ∈ Π (𝜗1, 𝑁(𝜀1, 𝑙1))-hyperfinite (𝜗2, 𝑁(𝜀2, 𝑙2))-hyperfinite (𝜗3, 𝑁(𝜀3, 𝑙3))-hyperfinite

11

…still blending…

(𝜗′

1, 𝑁(𝜀1, 𝑙2))-hyperfinite

(𝜗′

2, 𝑁(𝜀2, 𝑙2))-hyperfinite

(𝜗′

1, 𝑁(𝜀1, 𝑙2))-hyperfinite

𝐻 pick 𝐻 ∈ Π (𝜗1, 𝑁(𝜀1, 𝑙1))-hyperfinite (𝜗2, 𝑁(𝜀2, 𝑙2))-hyperfinite (𝜗3, 𝑁(𝜀3, 𝑙3))-hyperfinite define 𝜍

11

…still blending…

(𝜗′

1, 𝑁(𝜀1, 𝑙2))-hyperfinite

(𝜗′

2, 𝑁(𝜀2, 𝑙2))-hyperfinite

(𝜗′

1, 𝑁(𝜀1, 𝑙2))-hyperfinite

𝐻 pick 𝐻 ∈ Π (𝜗1, 𝑁(𝜀1, 𝑙1))-hyperfinite (𝜗2, 𝑁(𝜀2, 𝑙2))-hyperfinite (𝜗3, 𝑁(𝜀3, 𝑙3))-hyperfinite define 𝜍 for every 𝑜

11