Ramanujan graphs by 8ZOi0BQ4


									      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]

    s     … t                               …

Assume wlog
•  Ĝ has constant degree.
G Each connected component of Ĝ an expander.
• regular and
•  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).
             What about PR Walks?
         u     v                           Cv
        s     … t                           …

•       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
• 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

           about reversibility for consistently-labelled,
  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]

But How to Construct an Expander?
•   Goal in explicit constructions: minimize
    degree, maximize expansion.
•   Celebrated sequence of algebraic
    constructions [Mar73,G80,JM85,LPS86,
•   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
•   [RVW 00]: a method to reduce the degree of
    a graph while not harming its expansion by
•   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                               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,
  degree: increases
• #vertices: unchanged family {Gi} of d2-regular
    Construct [RVW00]:
    graphs s.t. Gi has d4i vertices and (Gi)  ½

    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,

   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
• 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
                                 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
      Open: More Unbalanced


• 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

   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) )

To top