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 42 4 4 24 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 42 4 4 24 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
UG
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
UG
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
UG
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
UG
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
UG
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!
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