Testing Expansion in Bounded Degree Graphs

1
Testing Expansion in Bounded Degree Graphs

• Christian Sohler
• University of Dortmund
• (joint work with Artur Czumaj, University of
  Warwick)

2
Testing Expansion in Bounded Degree
GraphsIntroduction

• Property TestingRubinfeld, Sudan
• Formal framework to analyze Sampling-algorithm
  s for decision problems
• Decide with help of a random sample whether a
  given object has a property or is far away from
  it

Close to property
Far away from property
Property
3
Testing Expansion in Bounded Degree
GraphsIntroduction

• Property TestingRubinfeld, Sudan
• Formal framework to analyze Sampling-algorithm
  s 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 e-far from a property P, if it
  differs in more than an e-fraction of ist formal
  description from any object with property P.

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Testing Expansion in Bounded Degree
GraphsIntroduction

• Bounded degree graphs
• Graph (V,E) with degree bound d
• V1,,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 3 4
2 4 4 2
4 1 3
1
3
1
d
4
2
n
5
Testing Expansion in Bounded Degree
GraphsIntroduction

• Definition
• A graph (V,E) with degree bound d and n vertices
  is e-far from a property P, if more than edn
  entries in the adjacency lists have to be
  modified to obtain a graph with property P.
• Example (Bipartiteness)

1 2 3 4
2 4 4 2
4 1 3
1
3
1
d
4
2
1/7-far from bipartite
n
6
Testing Expansion in Bounded Degree
GraphsIntroduction

• Goal
• Accept graphs that have property P with
  probability at least 2/3
• Reject graphs that are e-far from P with
  probability at least 2/3
• Complexity Measure
• Query (sample) complexity
• Running time

7
Testing Expansion in Bounded Degree
GraphsIntroduction

• Definition Neighborhood
• N(U) denotes the neighborhood of U, i.e. N(U)
  v?V-U ? u?U such that (v,u)?E
• Definition Expander
• A Graph is an a-Expander, if N(U)? a?U for each
  U?V with U?V/2.

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Testing Expansion in Bounded Degree
GraphsIntroduction

• Testing Expansion
• Accept every graph that is an a-expander
• Reject every graph that is e-far from an
  a-expander
• If not an a-expander and not e-far then we can
  accept or reject
• Look at as few entries in the graph
  representation as possible

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Testing 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 FOCS00
• Parallel / follow-up work
• An expansion tester for bounded degree graphs
• Kale, Seshadhri, ICALP08
• Testing the Expansion of a Graph
• Nachmias, Shapira, ECCC07

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Testing Expansion in Bounded Degree
GraphsIntroduction

• Difficulty
• Expansion is a rather global property

Expander with n/2 vertices
Expander with n/2 vertices
Case 1 A good expander
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Testing Expansion in Bounded Degree
GraphsIntroduction

• Difficulty
• Expansion is a rather global property

Expander with n/2 vertices
Expander with n/2 vertices
Case 2 e-far from expander
12
Testing 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/e)
  steps
• Case 1 Distribution of end points is essentially
  uniform
• Case 2 Random walk will typically not cross cut
  -gt distribution differs significantly from
  uniform

Expander with n/2 vertices
Expander with n/2 vertices
Expander with n/2 vertices
Expander with n/2 vertices
Case 1 A good expander
Case 2 e-far from expander
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Testing Expansion in Bounded Degree GraphsThe
algorithm of Goldreich and Ron
Idea Count the number of collisions among end
points of random walks

• How to distinguish these two cases?
• Perform a random walk for L poly(log n, 1/e)
  steps
• Case 1 Distribution of end points is essentially
  uniform
• Case 2 Random walk will typically not cross cut
  -gt distribution differs significantly from
  uniform

Expander with n/2 vertices
Expander with n/2 vertices
Expander with n/2 vertices
Expander with n/2 vertices
Case 1 A good expander
Case 2 e-far from expander
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Testing Expansion in Bounded Degree GraphsThe
algorithm of Goldreich and Ron

• ExpansionTester(G,e,l,m,s)
• 1. repeat s times
• 2. choose vertex v uniformly at random from V
• 3. do m random walks of length L starting from
  v
• 4. count the number of collisions among
  endpoints
• if collisionsgt (1e) Ecollisions in uniform
  distr. then reject
• 6. accept

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Testing Expansion in Bounded Degree GraphsMain
result

• ExpansionTester(G,e,l,m,s)
• 1. repeat s times
• 2. choose vertex v uniformly at random from V
• 3. do m random walks of length L starting from
  v
• 4. count the number of collisions among
  endpoints
• if collisionsgt (1e) Ecollisions in uniform
  distr. then reject
• 6. accept
• TheoremThis work
• Algorithm ExpansionTester with sQ(1/e),
  mQ(?n/poly(e)) and L poly(log n, d, 1/a, 1/e)
  accepts every a-expander with probability at
  least 2/3 and rejects every graph, that is e-far
  from every a-expander with probability 2/3,
  where a Q(a²/(d² log (n/e)).

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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• Overview of the proof
• Algorithm ExpansionTester accepts every
  a-expander with probability at least 2/3

17
Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• Overview of the proof
• Algorithm ExpansionTester accepts every
  a-expander with probability at least 2/3 ?
  (Chebyshev inequality)

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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• Overview of the proof
• Algorithm ExpansionTester accepts every
  a-expander with probability at least 2/3 ?
  (Chebyshev inequality)
• If G is e-far from an a-expander, then it
  contains a set U of dn vertices such that N(U) is
  small

U
G
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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• Overview of the proof
• Algorithm ExpansionTester accepts every
  a-expander with probability at least 2/3 ?
  (Chebyshev inequality)
• If G is e-far from an a-expander, then it
  contains a set U of dn vertices such that N(U) is
  small
• If G has a set U of dn vertices such that N(U) is
  small, thenExpansionTester rejects

U
G
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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• Overview of the proof
• Algorithm ExpansionTester accepts every
  a-expander with probability at least 2/3 ?
  (Chebyshev inequality)
• If G is e-far from an a-expander, then it
  contains a set U of dn vertices such that N(U) is
  small
• If G has a set U of dn vertices such that N(U) is
  small, thenExpansionTester rejects ? Random
  walk is unlikely to cross cut -gt more collisions

U
G
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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

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

U
G
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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• If G is e-far from an a-expander, then it
  contains a set U of dn vertices such that N(U) is
  small
• Lemma
• If G is e-far from an a-expander, then for every
  A?V of size at most en/4 we have that GV-A is
  not a (ca)-expander

U
G
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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• If G is e-far from an a-expander, then it
  contains a set U of dn vertices such that N(U) is
  small
• Lemma
• If G is e-far from an a-expander, then for every
  A?V of size at most en/4 we have that GV-A is
  not a (ca)-expander
• Procedure to construct U
• As long as U is too small apply lemma with AU
• Since GV-A is not an expander, we have a set B
  of vertices that is badly connected to the rest
  of GV-A
• Add B to U

U
G
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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• Lemma
• If G is e-far from an a-expander, then for every
  A?V of size at most en/4 we have that GV-A is
  not a (ca)-expander
• Proof (by contradiction)
• Assume A as in lemma exists with GV-A is
  (ca)-expander
• Construct from G an a-expanderby changing at
  most edn edges
• Contradiction G is not e-far from a-expander

G
A
(ca)-Expander
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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• Lemma
• If G is e-far from an a-expander, then for every
  A?V of size at most en/4 we have that GV-A is
  not a (ca)-expander
• Proof (by contradiction)

G
Construction of a-expander 1. Remove edges
incident to A 2. Add (d-1)-regular
c-expander to A 3. Remove arbitrary matching M
of size A/2 from GV-A 4. Match endpoints
of M with points from A
A
(ca)-Expander
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Testing Expansion in Bounded Degree
GraphsAnalysis of the algorithm

• Lemma
• If G is e-far from an a-expander, then for every
  A?V of size at most en/4 we have that GV-A is
  not a (ca)-expander
• Proof (by contradiction)

Show that every set X has large neighborhood by
case distinction
G
Construction of a-expander 1. Remove edges
incident to A 2. Add (d-1)-regular
c-expander to A 3. Remove arbitrary matching M
of size A/2 from GV-A 4. Match endpoints
of M with points from A
A
X
(ca)-Expander
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Testing Expansion in Bounded Degree GraphsMain
result

• ExpansionTester(G,e,l,m,s)
• 1. repeat s times
• 2. choose vertex v uniformly at random from V
• 3. do m random walks of length L starting from
  v
• 4. count the number of collisions among
  endpoints
• if collisionsgt (1e) Ecollisions in unif.
  Distr. then reject
• 6. accept
• TheoremThis work
• Algorithm ExpansionTester with sQ(1/e),
  mQ(?n/poly(e)) and L poly(log n, d, 1/a, 1/e)
  accepts every a-expander with probability at
  least 2/3 and rejects every graph, that is e-far
  from every a-expander with probability 2/3,
  where a poly(1/log n, 1/d, a, e).

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Thank you!