Christian Sohler 1

University of Dortmund

Testing Expansion in Bounded Degree Graphs

Christian Sohler

University of Dortmund

(joint work with Artur Czumaj, University of Warwick)

Christian Sohler 2

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Property Testing[Rubinfeld, Sudan]:• Formal framework to analyze „Sampling“-algorithms for decision problems• Decide with help of a random sample whether a given object has a property or is far away from it

Property

Far away from property

Close toproperty

Christian Sohler 3

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Property Testing[Rubinfeld, Sudan]:• Formal framework to analyze „Sampling“-algorithms for decision problems• Decide with help of a random sample whether a given object has a property or is far away from it

Definition:• An object is -far from a property , if it differs in more than an -fraction of ist formal description from any object with property .

Christian Sohler 4

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Bounded degree graphs• Graph (V,E) with degree bound d• V={1,…,n}• Edges as adjacency lists through function f: V {1,…,d} V• f(v,i) is i-th neighbor of v or ■, if i-th neighbor does not exist• Query f(v,i) in O(1) time

1

2 4

31 2 3 4

2 4 4 2

4 1 ■ 3■ ■ ■ 1

d

n

Christian Sohler 5

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Definition:• A graph (V,E) with degree bound d and n vertices is -far

from a property , if more than dn entries in the adjacency lists have to be modified to obtain a graph with property .

Example (Bipartiteness):

1

2 4

31 2 3 4

2 4 4 2

4 1 ■ 3■ ■ ■ 1

d

n 1/7-far from bipartite

Christian Sohler 6

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Goal:• Accept graphs that have property with probability

at least 2/3• Reject graphs that are -far from with probability

at least 2/3

Complexity Measure:• Query (sample) complexity• Running time

Christian Sohler 7

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Definition [Neighborhood]• N(U) denotes the neighborhood of U, i.e.

N(U) = {vV-U: uU such that (v,u)E}

Definition [Expander]:• A Graph is an -Expander, if N(U) |U| for each UV

with |U||V|/2.

Christian Sohler 8

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Testing Expansion:• Accept every graph that is an -expander• Reject every graph that is -far from an *-expander• If not an -expander and not -far then we can accept or

reject• Look at as few entries in the graph representation as

possible

Christian Sohler 9

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Related results:Definition of bounded degree graph model; connectivity, k-connectivity, circle

freeness

[Goldreich, Ron; Algorithmica]Conjecture: Expansion can be tested O(n polylog(n)) time

[Goldreich, Ron; ECCC, 2000]Rapidly mixing property of Markov chains

[Batu, Fortnow, Rubinfeld, Smith, White; FOCS‘00]

Parallel / follow-up work:

An expansion tester for bounded degree graphs

[Kale, Seshadhri, ICALP’08]Testing the Expansion of a Graph

[Nachmias, Shapira, ECCC’07]

Christian Sohler 10

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Difficulty:• Expansion is a rather global property

Expander with n/2vertices

Expanderwith n/2vertices

Case 1: A good expander

Christian Sohler 11

University of DortmundTesting Expansion in Bounded Degree GraphsIntroduction

Difficulty:• Expansion is a rather global property

Expander with n/2vertices

Expanderwith n/2vertices

Case 2: -far from expander

Christian Sohler 12

University of DortmundTesting Expansion in Bounded Degree GraphsThe algorithm of Goldreich and Ron

How to distinguish these two cases?• Perform a random walk for L= poly(log n, 1) steps• Case 1: Distribution of end points is essentially uniform• Case 2: Random walk will typically not cross cut

-> distribution differs significantly from uniform

Expander with n/2vertices

Expanderwith n/2vertices

Case 1: A good expander

Expander with n/2vertices

Expanderwith n/2vertices

Case 2: -far from expander

Christian Sohler 13

University of DortmundTesting Expansion in Bounded Degree GraphsThe algorithm of Goldreich and Ron

How to distinguish these two cases?• Perform a random walk for L= poly(log n, 1) steps• Case 1: Distribution of end points is essentially uniform• Case 2: Random walk will typically not cross cut

-> distribution differs significantly from uniform

Expander with n/2vertices

Expanderwith n/2vertices

Case 1: A good expander

Expander with n/2vertices

Expanderwith n/2vertices

Case 2: -far from expander

Idea:Count the number of collisions among end points of random walks

Christian Sohler 14

University of DortmundTesting Expansion in Bounded Degree GraphsThe algorithm of Goldreich and Ron

ExpansionTester(G,,l,m,s)1. repeat s times 2. choose vertex v uniformly at random from V3. do m random walks of length L starting from v4. count the number of collisions among endpoints5. if #collisions> (1+ E[#collisions in uniform distr.] then

reject6. accept

Christian Sohler 15

University of Dortmund

ExpansionTester(G,,l,m,s) 1. repeat s times 2. choose vertex v uniformly at random from V3. do m random walks of length L starting from v4. count the number of collisions among endpoints5. if #collisions> (1+ E[#collisions in uniform distr.] then

reject6. accept

Theorem:[This work]Algorithm ExpansionTester with s=(1/, m=(n/poly() and

L= poly(log n, d, 1/, 1/) accepts every -expander with probability at least 2/3 and rejects every graph, that is -far from every *-expander with probability 2/3, where * = (²/(d² log (n/)).

Testing Expansion in Bounded Degree GraphsMain result

Christian Sohler 16

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

Overview of the proof:• Algorithm ExpansionTester accepts every -expander with

probability at least 2/3

Christian Sohler 17

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

Overview of the proof:• Algorithm ExpansionTester accepts every -expander with

probability at least 2/3 (Chebyshev inequality)

Christian Sohler 18

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

Overview of the proof:• Algorithm ExpansionTester accepts every -expander with

probability at least 2/3 (Chebyshev inequality)• If G is -far from an *-expander, then it contains a set U of

n vertices such that N(U) is small U

G

Christian Sohler 19

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

Overview of the proof:• Algorithm ExpansionTester accepts every -expander with

probability at least 2/3 (Chebyshev inequality)• If G is -far from an *-expander, then it contains a set U of

n vertices such that N(U) is small

• If G has a set U of n vertices such that N(U) is small, thenExpansionTester rejects

U

G

Christian Sohler 20

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

Overview of the proof:• Algorithm ExpansionTester accepts every -expander with

probability at least 2/3 (Chebyshev inequality)• If G is -far from an *-expander, then it contains a set U of

n vertices such that N(U) is small

• If G has a set U of n vertices such that N(U) is small, thenExpansionTester rejects Random walk is unlikely to cross cut -> more collisions

U

G

Christian Sohler 21

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

• If G is -far from an *-expander, then it contains a set U of n vertices such that N(U) is small

U

G

Christian Sohler 22

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

• If G is -far from an *-expander, then it contains a set U of n vertices such that N(U) is small

Lemma:If G is -far from an*-expander, then for every AV of size

at most n/4 we have that G[V-A] is not a (c*)-expander

U

G

Christian Sohler 23

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

• If G is -far from an *-expander, then it contains a set U of n vertices such that N(U) is small

Lemma:If G is -far from an*-expander, then for every AV of size

at most n/4 we have that G[V-A] is not a (c*)-expander

Procedure to construct U:• As long as U is too small apply lemma with A=U

• Since G[V-A] is not an expander, we have a set B of vertices that is badly connected to the rest of G[V-A]

• Add B to U

U

G

Christian Sohler 24

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

Lemma:If G is -far from an*-expander, then for every AV of size

at most n/4 we have that G[V-A] is not a (c*)-expander

Proof (by contradiction):• Assume A as in lemma exists with G[V-A] is (c*)-expander• Construct from G an *-expander

by changing at most dn edges

• Contradiction: G is not -far from *-expander

A

G

(c*)-Expander

Christian Sohler 25

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

Lemma:If G is -far from an*-expander, then for every AV of size

at most n/4 we have that G[V-A] is not a (c*)-expander

Proof (by contradiction):

A

G

(c*)-Expander

Construction of *-expander:1. Remove edges incident to A2. Add (d-1)-regular c‘-expander to A 3. Remove arbitrary matching M of size |A|/2 from G[V-A]4. Match endpoints of M with points from A

Christian Sohler 26

University of DortmundTesting Expansion in Bounded Degree GraphsAnalysis of the algorithm

Lemma:If G is -far from an*-expander, then for every AV of size

at most n/4 we have that G[V-A] is not a (c*)-expander

Proof (by contradiction):

A

G

(c*)-Expander

Construction of *-expander:1. Remove edges incident to A2. Add (d-1)-regular c‘-expander to A 3. Remove arbitrary matching M of size |A|/2 from G[V-A]4. Match endpoints of M with points from A

X

Show that every set X has large neighborhood

by case distinction

Christian Sohler 27

University of DortmundTesting Expansion in Bounded Degree GraphsMain result

ExpansionTester(G,,l,m,s) 1. repeat s times 2. choose vertex v uniformly at random from V3. do m random walks of length L starting from v4. count the number of collisions among endpoints5. if #collisions> (1+ E[#collisions in unif. Distr.] then reject6. accept

Theorem:[This work]Algorithm ExpansionTester with s=(1/, m=(n/poly() and

L= poly(log n, d, 1/, 1/) accepts every -expander with probability at least 2/3 and rejects every graph, that is -far from every *-expander with probability 2/3, where * = poly(1/log n, 1/d, , ).

Christian Sohler 28

University of Dortmund

Thank you!Thank you!