University of Dortmund
Testing Expansion in Bounded
Degree Graphs
Christian Sohler
University of Dortmund
(joint work with Artur Czumaj,
University of Warwick)
Christian Sohler
1
University of Dortmund
Testing Expansion in Bounded Degree Graphs
Introduction
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
Close to
property
Property
Far away from
property
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Introduction
University of Dortmund
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 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.
Christian Sohler
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University of Dortmund
Testing Expansion in Bounded Degree Graphs
Introduction
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
d
1
2
4
2
4
1
3
4
■
4
2
3
■
■
■
1
1
3
2
4
n
Christian Sohler
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University of Dortmund
Testing Expansion in Bounded Degree Graphs
Introduction
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):
d
1
2
4
2
4
1
3
4
■
4
2
3
■
■
■
1
n
1
3
2
4
1/7-far from bipartite
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Introduction
University of Dortmund
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
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Introduction
University of Dortmund
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.
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Introduction
University of Dortmund
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
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Introduction
University of Dortmund
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
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Testing Expansion in Bounded Degree Graphs
Introduction
University of Dortmund
Difficulty:
• Expansion is a rather global property
Expander
with n/2
vertices
Expander
with n/2
vertices
Case 1: A good expander
Christian Sohler
10
Testing Expansion in Bounded Degree Graphs
Introduction
University of Dortmund
Difficulty:
• Expansion is a rather global property
Expander
with n/2
vertices
Expander
with n/2
vertices
Case 2: e-far from expander
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
The algorithm of Goldreich and Ron
University of Dortmund
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
-> distribution differs significantly from uniform
Expander
with n/2
vertices
Expander
with n/2
vertices
Case 1: A good expander
Expander
with n/2
vertices
Expander
with n/2
vertices
Case 2: e-far from expander
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
The algorithm of Goldreich and Ron
University of Dortmund
How to distinguish these
Idea:two cases?
Count
the number
of collisions
• Perform a random walk for
L= poly(log
n, 1/e)
steps
among
end
of random
walks
• Case 1: Distribution of end
points
is points
essentially
uniform
• Case 2: Random walk will typically not cross cut
-> distribution differs significantly from uniform
Expander
with n/2
vertices
Expander
with n/2
vertices
Case 1: A good expander
Expander
with n/2
vertices
Expander
with n/2
vertices
Case 2: e-far from expander
Christian Sohler
13
Testing Expansion in Bounded Degree Graphs
The algorithm of Goldreich and Ron
University of Dortmund
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
5. if #collisions> (1+e) E[#collisions in uniform distr.] then
reject
6. accept
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Main result
University of Dortmund
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
5. if #collisions> (1+e) E[#collisions in uniform distr.] then
reject
6. accept
Theorem:[This work]
Algorithm ExpansionTester with s=Q(1/e), m=Q(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)).
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
Overview of the proof:
• Algorithm ExpansionTester accepts every a-expander with
probability at least 2/3
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
Overview of the proof:
• Algorithm ExpansionTester accepts every a-expander with
probability at least 2/3  (Chebyshev inequality)
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
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
Christian Sohler
18
Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
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
• If G has a set U of dn vertices such that N(U) is small, then
ExpansionTester rejects
Christian Sohler
19
Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
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
• If G has a set U of dn vertices such that N(U) is small, then
ExpansionTester rejects
 Random walk is unlikely to cross cut -> more collisions
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
• 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
Christian Sohler
21
Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
• 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
Lemma:
If G is e-far from an a*-expander, then for every AV of size at
most en/4 we have that G[V-A] is not a (ca*)-expander
Christian Sohler
22
Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
• 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
Lemma:
If G is e-far from an a*-expander, then for every AV of size at
most en/4 we have that G[V-A] is not a (ca*)-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
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
Lemma:
If G is e-far from an a*-expander, then for every AV of size at
most en/4 we have that G[V-A] is not a (ca*)-expander
Proof (by contradiction):
• Assume A as in lemma exists with G[V-A] is (ca*)-expander
• Construct from G an a*-expander
G
by changing at most edn
edges
• Contradiction: G is not
A
e-far from a*-expander
(ca*)-Expander
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Analysis of the algorithm
University of Dortmund
Lemma:
If G is e-far from an a*-expander, then for every AV of size at
most en/4 we have that G[V-A] is not a (ca*)-expander
Proof (by contradiction):
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 G[V-A]
4. Match endpoints of M with
points from A
G
A
(ca*)-Expander
Christian Sohler
25
University of Dortmund
Testing Expansion in Bounded Degree Graphs
Analysis 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 G[V-A] is not a (ca*)-expander
Proof (by contradiction):
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 G[V-A]
4. Match endpoints of M with
points from A
Show that every set X
G
has large neighborhood
by case distinction
A
X
(ca*)-Expander
Christian Sohler
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Testing Expansion in Bounded Degree Graphs
Main result
University of Dortmund
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
5. if #collisions> (1+e) E[#collisions in unif. Distr.] then reject
6. accept
Theorem:[This work]
Algorithm ExpansionTester with s=Q(1/e), m=Q(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).
Christian Sohler
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University of Dortmund
Thank you!
Christian Sohler
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