christian sohler 1 university of dortmund testing expansion in bounded degree graphs christian...
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![Page 1: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/1.jpg)
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)
![Page 2: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/2.jpg)
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
![Page 3: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/3.jpg)
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 .
![Page 4: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/4.jpg)
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
![Page 5: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/5.jpg)
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
![Page 6: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/6.jpg)
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
![Page 7: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/7.jpg)
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.
![Page 8: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/8.jpg)
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
![Page 9: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/9.jpg)
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]
![Page 10: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/10.jpg)
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
![Page 11: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/11.jpg)
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
![Page 12: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/12.jpg)
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
![Page 13: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/13.jpg)
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
![Page 14: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/14.jpg)
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
![Page 15: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/15.jpg)
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
![Page 16: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/16.jpg)
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
![Page 17: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/17.jpg)
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)
![Page 18: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/18.jpg)
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
![Page 19: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/19.jpg)
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
![Page 20: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/20.jpg)
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
![Page 21: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/21.jpg)
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
![Page 22: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/22.jpg)
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
![Page 23: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/23.jpg)
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
![Page 24: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/24.jpg)
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
![Page 25: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/25.jpg)
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
![Page 26: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/26.jpg)
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
![Page 27: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/27.jpg)
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, , ).
![Page 28: Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,](https://reader036.vdocuments.us/reader036/viewer/2022062714/56649d605503460f94a41d8d/html5/thumbnails/28.jpg)
Christian Sohler 28
University of Dortmund
Thank you!Thank you!