Constant Degree, Lossless Expanders

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Constant Degree, Lossless Expanders

• Omer Reingold
• ATT

joint work with Michael Capalbo (IAS), Salil
Vadhan (Harvard), and Avi Wigderson (Hebrew U.,
IAS)
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Expander Graphs (Balanced Case)

• An innocent looking object but intimately
  related to various fundamental problems (Network
  Design, Complexity and Proof Theory,
  Derandomization, Coding Theory, Cryptography, ...)

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Expander Graphs (Balanced Case)

• How large can A be?
• Trivial upper bound A ? D.
• Random graphs A?D.
• Previously, best explicit expanders A D/2
  (for constant D and large K).

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This Work Const. Degree, Lossless Expanders
that may even be slightly unbalanced
0lt?,?? 1 are constants ? D is constant K? (N)
For the very curious onlyK? (? M/D) D poly
log (1/(? ?)) (fully explicit D quasi poly
log(1/(? ?) )).
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A Bit of Context

• Explicit construction of constant degree
  expanders is difficult.
• Celebrated sequence of algebraic constructions
  Mar73 ,GG80,JM85,LPS86,AGM87,Mar88,Mor94.
• Ramanujan graphs with expansion ? D/2
  Kahale95.
• Combinatorial constructions Ajtai Ajt87,
  more explicit and very simple RVW00.
• Lossless objects Alo95,RR99,TUZ01
• Unique neighbor, constant degree expanders
  Cap01.

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Why Bother with the Deg./2 Barrier?

• Because its there ???
• For most applications of expanders the more
  expansion the better.
• Specific applications for lossless expanders
• Distributed routing in networks
  PU89,ALM96,BFU99.
• Expander codes Gal63,Tan81,SS96,Spi96,LMSS01.
• Bitprobe complexity of storing subsets
  BMRRS00.
• Various storage schemes UW88,BMRS00.
• Hard tautologies for various proof
  systemsBW99,ABRW00,AR01.

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Distributed routing in networks

• The Task PU89,ALM96,BFU99 Finding vertex/edge
  disjoint paths in a distributed manner. Much
  easier if the network is composed of lossless
  expanders.

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Distributed routing in networks

• Simplified scenario Vertex disjoint paths in a
  layered graph. Expansion factor from left to
  right ? 9D/10.

Outputs
Inputs
...
...
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Distributed routing in networks

• Simplified scenario Vertex disjoint paths in a
  layered graph. Expansion factor from left to
  right ? 9D/10.

Outputs
Inputs
...
Incredibly Fault Tolerant UW87 Works even if
adversary removes 3/4 of D edges from each vertex.
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Simple Expander Codes G63,Z71,ZP76,T81,SS96
N (Variables)
M? N (Parity Checks)
Fix ? 1/10 Sets of size ? K? (? N/D)
expand by a factor 9D/10.
Linear code. Rate 1 - M/N 1 - ? Minimum
distance ? K. Relative distance ? K/N? (? / D)
? / poly log (1/?). For small ? beats the
Zyablov bound and is quite close to the
Gilbert-Varshamov bound of ? log (1/?).
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Simple Decoding Algorithm in Linear Time ( log n
parallel phases) SS 96
N (Variables)
M? N (Constraints)
Flip\B ? B/4.B\Flip ? B/4.? Bnew?
B/2.

• Algorithm At each phase, flip every variable
  that sees a majority of 1s (i.e, unsatisfied
  constraints).

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Hints Into the Expander Construction

• Starting point RVW00 A simple combinatorial
  construction of constant-degree expanders with
  simple analysis.
• The heart of the construction New Graph
  Product Compose large graph w/ small graph to
  obtain a new graph which (roughly) inherits
• Size of large graph.
• Degree from the small graph.
• Expansion from both.

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The Zig-Zag Product RVW00
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Zig-Zag Analysis (Case I) RVW00
In Case I, the second small step is guaranteed
to expand. The first may be lost. In Case II,
the reversed picture ? Need both small steps.
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Trying to improve
???
???
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Zig-Zag for Unbalanced Graphs

• Second eigenvalue analysis for expanders
  probably not useful in the unbalanced case.
• Extractors NZ93 and condensers (under various
  formalizations RR99,RSW00,TUZ01), work well in
  the unbalanced case.
• In fact, RVW00 analyzed a zig-zag product for
  extractors (with an easier goal).
• We introduce randomness conductors that
  interpolate expanders, extractors, condensers
  hash functions, and analyze the zig-zag product
  for conductors.

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Randomness Conductors

• Expanders, extractors, condensers hash
  functions are all functions, f N ? D ?
  M, that transform S of entropy k ? S
  f (S,Uniform) of entropy k
• Many flavors
• Measure of entropy.
• Balanced vs. unbalanced.
• Lossless vs. lossy.
• Lower vs. upper bound on k.
• Is S close to uniform?

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On the Board ?

• Randomness conductors -- a space of
  combinatorial objects
• From Expanders to Extractors in a few easy steps.
• On measures of entropy.
• The definition of randomness conductors.
• Previous constructions and composition techniques
  from the extractor literature extend to (useful)
  explicit constructions of conductors.
• The zig-zag product for conductors can produce
  constant degree, lossless expanders.

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Summary and Open Problems

• Our Result (Slightly Unbalanced), Constant
  Degree, Lossless Expanders.
• Seen some applications, hints into the
  construction, and a short encounter with
  randomness conductors.
• Further Research
• The undirected case (being lossless from both
  sides).
• Better expansion yet?
• Continue the study of randomness condensers.

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Definition Randomness Conductors

• For any function ? 0, log N ? 0, log D ?
  0,1, the function f N ? D ? M, is an ?
  - conductor if ? k, k,

S is ? - close to min entropy k
(min entropy k ? ? x, Prx ? 2-k)
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Lossless Expanders are Incredibly Fault Tolerant
UW87
?(S) gt(1-?) S

• Let an adversary remove (1-?) D edges for each
  vertex.
• Still expands by a factor (1- ? / ?) D !!