Walk the Walk: On Pseudorandomness, Expansion,

and Connectivity

Omer ReingoldOmer ReingoldWeizmannWeizmann InstituteInstitute

Based on join works with Michael Capalbo, Kai-Min Based on join works with Michael Capalbo, Kai-Min Chung, Chung, Chi-Jen Lu, Luca Trevisan, Salil Vadhan, and Avi Chi-Jen Lu, Luca Trevisan, Salil Vadhan, and Avi Wigderson Wigderson

(Undirected) Connectivity (Undirected) Connectivity

How to Walk an Undirected Graph?How to Walk an Undirected Graph?

• Random walk - when in doubt, flip a coin:• At each step, follow a uniformly selected edge. • If there is a path between s and t, a random

walk will find it (polynomial number of steps).• Algorithm uses logarithmic memory (minimal).

Pseudorandom Walks?Pseudorandom Walks?• Can we invest less randomness in the walk?• Can we escape a maze deterministically? • (N,D)-Universal Traversal Sequence [Cook]:

sequence of edge labels which guides a walk through all of the vertices of any D-regular graph on N vertices.

• [AKLLR79] poly-long UTS exist (probabilistic).• What about explicit (efficient) poly-long UTS?• Can connectivity in undirected graphs be

solved deterministically using logarithmic memory?

• Yes! & partial positive answers for the above …

• Exploits Expander Graphs …

Log-Space Algorithm [R04]Log-Space Algorithm [R04]

• ĜĜ has constant degree. • Each connected component of ĜĜ an expander. v in GG define the set Cv={<v,*>} in ĜĜ. • u and v are connected Cu and Cv are in the

same connected component. Enough to verify the existence of a path between

<s,00…0> and <t,00…0> (easy in log-space).

……GG

sstt

Assume wlog G regular and non-bipartite

……ĜĜ

Log-space transformation

highly connected; logarithmic diameter; random walk converges to uniform in logarithmic number of steps

What about PR Walks? What about PR Walks?

• An edge between Cu and Cv in ĜĜ “projects” to a polynomial path between u and v in GG

• GG is connected ĜĜ an expander log path in ĜĜ converges to uniform projects to a poly path in GG that converges to uniform

• The projection is logspace • “Oblivious of G”G”, if GG is consistently labelled

……GG

sstt ……

ĜĜCu

Cvvu

Labellings of Regular Labellings of Regular DigraphsDigraphs

• Denote by i(v) the ith neighbor of v

• Inconsistently labelled: u,v,i s.t. i(u)=i(v)

• Consistently labelled: i i is a permutation

(Every regular digraph has a consistent labelling)

32 1 1

243

4u v

More Results [R04,RTV05]More Results [R04,RTV05]

• For consistently-labelled digraphs:• Universal-Traversal Sequence (poly long,

log-space constructible). • Psedorandom Walk Generator:

log-long uniform seed poly-long sequence of edge labels s.t. the walk (on any appropriate-size graph) converges to the stationary distribution.

• In general:• Universal Exploration Sequence

Some Open ProblemsSome Open Problems

• Pseudorandom-Walk Generator for inconsistently-labelled digraphs• Far reaching implication [RTV 05]: Every

randomized algorithm can be derandomized with small penalty in space (RL=L).

• A walk that is pseudorandom all the way (not just in the limit): every node of the walk should be distributed “correctly”.• A very powerful derandomization tool

(generalizes eps-bias, expander walks, etc.)

Connectivity for undirected graphs [R04]

Connectivity for regular digraphs [RTV05]

Pseudorandom walks for consistently-labelled, regular digraphs [R04, , RTV05]

Pseudorandom walks for regular digraphs [RTV05]

Connectivity for digraphs w/polynomial mixing time [RTV05]

RL

in L

Suffice toprove RL=L

Summary on RL vs. LSummary on RL vs. L

It is not about reversibility but about regularity In fact it is about having estimates on stationary probabilities [CRV07]

But How to Construct an But How to Construct an Expander?Expander?

• Goal in explicit constructions: minimize degree, maximize expansion.

• Celebrated sequence of algebraic constructions [Mar73,G80,JM85,LPS86, AGM87,Mar88,Mor94,...].

• Ramanujan graphs: Optimal 2nd eigenvalue (as a function of degree).

• More relevant to us: a simple combinatorial construction w/simple analysis of constant degree expanders [RVW00]

Reducing Degree, Preserving Reducing Degree, Preserving Expansion Expansion

• [RVW 00]: a method to reduce the degree of a graph while not harming its expansion by much.

• For that, introduced a new graph product -the zig-zag product:

H: degree d on D vertices,G: degree D on N vertices

GⓏH: degree d2 on ND vertices

• If H & G are good expanders so is GⓏH

Replacement ProductReplacement ProductSomewhat easier to describe. Somewhat weaker expansion properties [RVW00,MR00]

uu

88

77 22

33

66

5544

11

uu

(u,(u,88))

(u,(u,77)) (u,(u,22))

(u,(u,33))(u,(u,66))

(u,(u,55))(u,(u,44))

(u,1)(u,1)

HH

Zig-Zag Construction of ExpandersZig-Zag Construction of Expanders

• Building Block: H degree d on d4 vertices, ((H))1/4. 1/4.

• Construct [RVW00]: family {Gi} of d2-regular graphs s.t. Gi has d4i vertices and (Gi) ½

G1 = H2

Gi+1 = (Gi)2ⓏH

• Iteratively pulling the blanket from both sizes, stretches the blanket

Squaring: : reducesdegree: increases#vertices: unchanged

Zig-Zag:: increasesdegree: reduces#vertices: increases

Usefulness for ConnectivityUsefulness for Connectivity

• Building Block: H degree d on d10 vertices, ((H))1/4. 1/4.

G1 = G non-bipartite, d2-regular on n vertices

Gi+1 = (Gi)5ⓏH

• Thm [R04]: If G connected then for L=c logn (GL) ½

• Transformation G GL is log-space.

• Zig-Zag product applied to non-expanders!

More Consequences of the Zig-More Consequences of the Zig-Zag ConstructionZag Construction

• Connection with semi-direct product in groups [ALW01]

• New expanding Cayley graphs for non-simple groups [MW02, RSW04]

• Vertex Expansion beating eigenvalue bounds [RVW00, CRVW01]

Vertex ExpansionVertex Expansion

|(S)| A |S|

(A > 1)

S, |S| K

Every (not too large) set expands.• Goal: maximize expansion parameter A

• In random graphs AD-1

D

N N

Explicit constructions – Vertex Explicit constructions – Vertex ExpansionExpansion

• Optimal 2Optimal 2nd nd eigenvalue expansion does eigenvalue expansion does notnot imply optimal vertex expansion imply optimal vertex expansion

• Exist Ramanujan graphs with vertex Exist Ramanujan graphs with vertex expansion expansion D/2 D/2 [Kah95].[Kah95].

• Lossless ExpanderLossless Expander – Expansion > – Expansion > (1-(1-) D) D• Why should we care?Why should we care?

• Limitation of previous techniquesLimitation of previous techniques• Many beautiful applicationsMany beautiful applications

Strong Unique Neighbor Strong Unique Neighbor PropertyProperty

S, |S| K, |(S)| 0.9 D |S|

SNon Unique neighbor

S has 0.8 D |S| unique neighbors !

• We call graphs where every such S has even a single unique neighbor – unique neighbor expanders

Unique neighbor of S

Explicit Vertex ExpansionExplicit Vertex Expansion

• Current state of knowledge – extremely far Current state of knowledge – extremely far from optimal. from optimal.

• Open:Open: lossless lossless undirectedundirected expanders. expanders.• Unique neighbor expanders are known Unique neighbor expanders are known

[AC02][AC02]• Based on the zig-zag product: lossless Based on the zig-zag product: lossless directed directed

expanders [CRVW02]. Expansion expanders [CRVW02]. Expansion D-O(DD-O(D)). . • Works even if right-hand side is smaller by a Works even if right-hand side is smaller by a

constant factor.constant factor.• Open:Open: expansion expansion D-O(1)D-O(1) (even with non- (even with non-

constant degree). constant degree).

Open: More UnbalancedOpen: More Unbalanced

D

NM

• Open: D constant, M=N0.5, and sets of size at most K=N0.2 expand. More ambitious:• Unique neighbor expanders• Lossless expanders• Minimal Degree

Super-Constant DegreeSuper-Constant Degree

D

NM

• State of the art [GUV06]: D=Poly(Log N), M=Poly(KD)

• Open: M=O(KD) (D=Poly(Log N) )• Open: D= O(Log N) (M=Poly(KD) )

S, |S| K |(S)| ¾ D |S|