# Ramanujan graphs by 8ZOi0BQ4

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```									      Walk the Walk: On
Pseudorandomness, Expansion,
and Connectivity
Omer Reingold
Weizmann Institute

Based on join works with Michael Capalbo, Kai-Min Chung,
Chi-Jen Lu, Luca Trevisan, Salil Vadhan, and Avi Wigderson
(Undirected) Connectivity
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?
•       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]

Ĝ
G
s     … t                               …

Log-space
Assume wlog
•  Ĝ has constant degree.
G Each connected component of Ĝ an expander.
transformation
• regular and
non-bipartite
•  v in G define the set Cv={<v,*>} in Ĝ.
•   u and v are connected  Cu and Cv are in the same
highly connected; logarithmic diameter;
connected component.
random walk converges to uniform in
Enough to verify the existence of a path between
<s,00…0> number of (easy
logarithmic and <t,00…0> stepsin log-space).
u     v                           Cv
Ĝ
G
s     … t                           …
Cu

•       An edge between Cu and Cv in Ĝ “projects” to a
polynomial path between u and v in G
•       G is connected  Ĝ an expander  log path in
Ĝ converges to uniform  projects to a poly
path in G that converges to uniform
•       The projection is logspace
•       “Oblivious of G”, if G is consistently labelled
Labellings of Regular Digraphs

u                v
3         4      4         2
2 1              1 3

• 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)
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 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.)
Summary on RL vs. L
Connectivity for undirected graphs [R04]

Connectivity for regular digraphs [RTV05]
in L

It is notPseudorandom walksbut about regularity
regular digraphs [R04, RTV05]
 In fact it is about having estimates on stationary
probabilities [CRV07]
Pseudorandom walks for
regular digraphs [RTV05]
Suffice to
prove RL=L
Connectivity for digraphs w/polynomial
mixing time [RTV05]

RL
But How to Construct an 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
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 Product
Somewhat easier to describe. Somewhat weaker
expansion properties [RVW00,MR00]

5                               (u,5)
4                               (u,4)
6                           (u,6)
3                  H
(u,3)
u                               u
7                       2       (u,7)            (u,2)
1                               (u,1)
8                               (u,8)
Zig-Zag Construction of Expanders
• Squaring: Block: H degree d on d4 vertices,
Building
reduces
:(H)1/4.
degree: increases
• #vertices: unchanged family {Gi} of d2-regular
Construct [RVW00]:
graphs s.t. Gi has d4i vertices and (Gi)  ½

Zig-Zag:
G1 =H2
Gi+1 = (Gi)2ⓏH : increases
degree: reduces
•   Iteratively pulling the blanket from both
#vertices: increases
sizes, stretches the blanket
Usefulness for Connectivity
•   Building Block: H degree d on d10 vertices,
(H)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-
Zag 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 Expansion
N           N

S, |S| K         D         |(S)|  A |S|
(A > 1)

Every (not too large) set expands.
• Goal: maximize expansion parameter A
• In random graphs AD-1
Explicit constructions – Vertex
Expansion
• Optimal 2nd eigenvalue expansion does not imply
optimal vertex expansion
• Exist Ramanujan graphs with vertex expansion 
D/2 [Kah95].
• Lossless Expander – Expansion > (1-) D
• Why should we care?
• Limitation of previous techniques
• Many beautiful applications
Strong Unique Neighbor Property

S, |S| K, |(S)|  0.9 D |S|

Unique neighbor of S
S
Non 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
Explicit Vertex Expansion
• Current state of knowledge – extremely far
from optimal.
• Open: lossless undirected expanders.
• Unique neighbor expanders are known [AC02]
• Based on the zig-zag product: lossless directed
expanders [CRVW02]. Expansion D-O(D).
• Works even if right-hand side is smaller by a
constant factor.
• Open: expansion D-O(1) (even with non-constant
degree).
Open: More Unbalanced
N
M

D

• 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 Degree
N
M

S, |S| K
D         |(S)|  ¾ D |S|

• 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) )

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