[PPT] - A Near-Optimal Algorithm for Testing Isomorphism of Two Unknown PowerPoint Presentation

SLIDE 1

A Near-Optimal Algorithm for Testing Isomorphism

f Two Unknown Graphs

Krzysztof Onak IBM T.J. Watson Research Center Joint work with Xiaorui Sun (Columbia)

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 1/22

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Graph Isomorphism

Problem: Are two graphs identical?

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 2/22

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Graph Isomorphism

Problem: Are two graphs identical?

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 2/22

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Graph Isomorphism

Problem: Are two graphs identical?

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 2/22

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Graph Isomorphism

Problem: Are two graphs identical? Status: Not known to be in P or NP-hard Best known algorithm: 2 ˜

O(√n) time (early 1980’s)

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 2/22

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Property Testing Version

This Talk: Dense graph model

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 3/22

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Property Testing Version

This Talk: Dense graph model

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Requirements: accept with probability 9/10 if graphs isomorphic reject with probability 9/10 if for any matching of vertices at least ǫn2 edges disagree

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 3/22

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Property Testing Version

This Talk: Dense graph model

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Requirements: accept with probability 9/10 if graphs isomorphic reject with probability 9/10 if for any matching of vertices at least ǫn2 edges disagree This talk: focus on the complexity as a function of n, assume that ǫ is a small constant (say, ǫ = 10−9)

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 3/22

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Query Complexity

Fischer, Matsliah (2006): (one sided error = never reject if isomorphic)

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 4/22

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Query Complexity

Fischer, Matsliah (2006): (one sided error = never reject if isomorphic) Remaining open case: neither graph known, two-sided testing

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 4/22

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Query Complexity

Fischer, Matsliah (2006): (one sided error = never reject if isomorphic) Remaining open case: neither graph known, two-sided testing Our result: Algorithm that makes n · 2O(√log n) queries

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 4/22

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Review of Fischer-Matsliah Techniques

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 5/22

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Core sets

Core set = list of polylog n vertices (v1, v2, . . . , vk)

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 6/22

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Core sets

11010 10011

Core set = list of polylog n vertices (v1, v2, . . . , vk) Every vertex u has a label l ∈ {0, 1}k with respect to core set:

li =

1

if vu and u connected

therwise

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 6/22

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Core sets

11010 10011

Core set = list of polylog n vertices (v1, v2, . . . , vk) Every vertex u has a label l ∈ {0, 1}k with respect to core set:

li =

1

if vu and u connected

therwise

Intuition: a large random core set partitions vertices into sets of similar vertices

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 6/22

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Important Theorem

H G

If G and H have core sets (v1, . . . , vk) and (w1, . . . , wk), respectively, such that:

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 7/22

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Important Theorem

11010 11010

H G

If G and H have core sets (v1, . . . , vk) and (w1, . . . , wk), respectively, such that: the distributions on labels are ǫ1-close

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 7/22

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Important Theorem

10011 10011 11010 11010

H G

If G and H have core sets (v1, . . . , vk) and (w1, . . . , wk), respectively, such that: the distributions on labels are ǫ1-close if you take two random vertices in G and two random vertices in H with the same labels, then w.p. 1 − ǫ2, the connectivity is the same

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 7/22

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Important Theorem

10011 10011 11010 11010

H G

If G and H have core sets (v1, . . . , vk) and (w1, . . . , wk), respectively, such that: the distributions on labels are ǫ1-close if you take two random vertices in G and two random vertices in H with the same labels, then w.p. 1 − ǫ2, the connectivity is the same then G and H are ǫ3-close to being isomorphic

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 7/22

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Important Theorem

10011 10011 11010 11010

H G

If G and H have core sets (v1, . . . , vk) and (w1, . . . , wk), respectively, such that: the distributions on labels are ǫ1-close if you take two random vertices in G and two random vertices in H with the same labels, then w.p. 1 − ǫ2, the connectivity is the same then G and H are ǫ3-close to being isomorphic Generic tester: Search for such a pair of core sets and accept if found

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 7/22

SLIDE 21

The Fischer-Matsliah Algorithm

Select ˜

O(n3/4) random vertices V ⋆

G from G

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 8/22

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The Fischer-Matsliah Algorithm

Select ˜

O(n3/4) random vertices V ⋆

G from G

and query their connections to ˜

O(n1/2) random vertices

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 8/22

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The Fischer-Matsliah Algorithm

Select ˜

O(n3/4) random vertices V ⋆

G from G

and query their connections to ˜

O(n1/2) random vertices

Select ˜

O(n1/4) random vertices V ⋆

H from H

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 8/22

SLIDE 24

The Fischer-Matsliah Algorithm

Select ˜

O(n3/4) random vertices V ⋆

G from G

and query their connections to ˜

O(n1/2) random vertices

Select ˜

O(n1/4) random vertices V ⋆

H from H

and query their connections to all other vertices

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 8/22

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The Fischer-Matsliah Algorithm

Select ˜

O(n3/4) random vertices V ⋆

G from G

and query their connections to ˜

O(n1/2) random vertices

Select ˜

O(n1/4) random vertices V ⋆

H from H

and query their connections to all other vertices Enumerate over all possible core sets from V ⋆

G and V ⋆ H

(quasipoly(n) options)

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 8/22

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The Fischer-Matsliah Algorithm

Select ˜

O(n3/4) random vertices V ⋆

G from G

and query their connections to ˜

O(n1/2) random vertices

Select ˜

O(n1/4) random vertices V ⋆

H from H

and query their connections to all other vertices Enumerate over all possible core sets from V ⋆

G and V ⋆ H

(quasipoly(n) options) For each pair of candidates: Step 1: See if distributions of labels similar (Use the ˜

O(√n)-sample tester of Batu et al. (2001))

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 8/22

SLIDE 27

The Fischer-Matsliah Algorithm

Select ˜

O(n3/4) random vertices V ⋆

G from G

and query their connections to ˜

O(n1/2) random vertices

Select ˜

O(n1/4) random vertices V ⋆

H from H

and query their connections to all other vertices Enumerate over all possible core sets from V ⋆

G and V ⋆ H

(quasipoly(n) options) For each pair of candidates: Step 1: See if distributions of labels similar (Use the ˜

O(√n)-sample tester of Batu et al. (2001))

Step 2: Select polylog n pairs of random vertices in G and see if the connectivity of their labels almost the same in H

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 8/22

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First Attempts

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 9/22

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Better Label Distribution Testing?

Idea: Use a different distribution tester Select ˜

O(√n) vertices from G and H

Query connections to random subset of size ˜

O(n2/3)

Use the distribution tester of Batu et al. (2000) Query Complexity: ˜

O(n7/6)

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 10/22

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Better Label Distribution Testing?

Idea: Use a different distribution tester Select ˜

O(√n) vertices from G and H

Query connections to random subset of size ˜

O(n2/3)

Use the distribution tester of Batu et al. (2000) Query Complexity: ˜

O(n7/6)

Problem: Unclear how to do Step 2 Only a subset of labels known Sampling subsets and comparing discovered identical labels introduces a bias

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 10/22

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Better Label Distribution Testing?

Idea: Use a different distribution tester Select ˜

O(√n) vertices from G and H

Query connections to random subset of size ˜

O(n2/3)

Use the distribution tester of Batu et al. (2000) Query Complexity: ˜

O(n7/6)

Problem: Unclear how to do Step 2 Only a subset of labels known Sampling subsets and comparing discovered identical labels introduces a bias Ignore this issue for now

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 10/22

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Exponent Less than 7/6?

Could the Ω(n2/3) distribution testing lower bound [Valiant 2008] imply a Ω(n7/6) lower bound? vs.

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 11/22

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Exponent Less than 7/6?

Could the Ω(n2/3) distribution testing lower bound [Valiant 2008] imply a Ω(n7/6) lower bound? vs. We have additional information

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 11/22

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Exponent Less than 7/6?

Could the Ω(n2/3) distribution testing lower bound [Valiant 2008] imply a Ω(n7/6) lower bound? vs. We have additional information: Estimate in advance distances between adjacency vectors in both graphs We care about random core sets, where labels preserve distances between adjacency vectors We should be able to distinguish heavy and light elements

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 11/22

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Exponent Less than 7/6?

Could the Ω(n2/3) distribution testing lower bound [Valiant 2008] imply a Ω(n7/6) lower bound? vs. We have additional information: Estimate in advance distances between adjacency vectors in both graphs We care about random core sets, where labels preserve distances between adjacency vectors We should be able to distinguish heavy and light elements Nice consistent clustering of slightly different adjacency vectors still difficult

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 11/22

SLIDE 36

Our Algorithm

Krzysztof Onak – A Near-Optimal Algorithm for Testing Graph Isomorphism – p. 12/22

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