The Zigzag Graph Product and ConstantDegree Lossless Expanders

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Expander Graphs The Unbalanced Case
Omer Reingold The Weizmann Institute
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What's in This Talk?

• Expander Graphs an array of definitions.
• Focus on most established notions, and open
  problems on explicit constructions. Mainly in the
  unbalanced case since this is
• What applications often require
• Where constructions are very far from optimal
• Will flash one construction (no details) -
  Unbalanced expanders based on Parvaresh-Vardy
  Codes Guruswami,Umans,Vadhan 06

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Bipartite Graphs

• As a preparation for the unbalanced case we will
  talk of bipartite expanders.
• Can also capture undirected expanders

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Vertex Expansion

• Every (not too large) set expands.

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Vertex Expansion

• Goal minimize D (i.e. constant D)
• Degree 3 random graphs are expanders Pin73

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Vertex Expansion

• Also maximize A.
• Trivial upper bound A ? D
• even A ? D-1
• Random graphs A?D-1

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2nd Eigenvalue Expansion

• 2nd eigenvalue (in absolute value) of
  (normalized) adjacency matrix is bounded away
  from 1
• Can be interpreted in terms of Renyi (l2)
  entropy

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Expanders Add Entropy
N
Prob. dist. X
Induced dist. X
D
x

• Vertex expansion Support(X) ? A
  Support(X)
• Some applications rely on less naïve measures
  of entropy.
• Col(X) PrX(1)X(2) X2

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2nd Eigenvalue Expansion
X
X

• Col(X) 1/N ? ?2 (Col(X) 1/N)
• Renyi entropy (log 1/Col(X)) increases as long
  as ? lt 1 and Col(X) is not too small

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2nd Eigenvalue Expansion
X
X

• Interestingly, vertex expansion and
  2nd-eigenvalue expansion are essentially
  equivalent for constant degree graphs Tan84,
  AM84, Alo86

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Explicit Constructions

• Applications need explicit constructions
• Weakly explicit easy to build the entire graph
  (in time poly N).
• Strongly explicit
• Given vertex name x and edge label i easy to find
  the ith neighbor of x (in time poly log N).

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Explicit constructions 2nd Eigenvalue

• Celebrated sequence of algebraic constructions
  Mar73, GG80,JM85,LPS86,AGM87,Mar88,Mor94,....
• Optimal 2nd eigenvalue (Ramanujan graphs)
• Combinatorial constructions Ajt87, RVW00,
  BL04.
• Open Combinatorial constructions of strongly
  explicit Ramanujan (or almost Ramanujan) graphs.
• Getting close Ben-Aroya,Ta-Shma 08

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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 gt (1-??) D
• Why should we care?
• Limitation of previous techniques
• Many applications

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Property 1 A Very Strong Unique Neighbor Property
?S, S? K, ?(S) ? 0.9 D S
S

• S has ? 0.8 D S unique neighbors !
• We call graphs where every such S has even a
  single unique neighbor unique neighbor
  expanders

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Property 2 Incredibly Fault Tolerant
?S, S? K, ?(S) ? 0.9 D S
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Explicit constructions Vertex Expansion

• Open lossless expanders for the undirected case.
• Unique neighbor expanders are known AC02
• For the directed case (expansion only from left
  side), lossless expanders are known CRVW02.
  Expansion D-O(D?).
• Open expansion D-O(1) (even with non-constant
  degree).

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Unbalanced Expanders

• Many applications need

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Unbalanced Expanders

• Many applications need unbalanced expanders

M
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Array of Definitions
M
X
X

• Many flavors
• How unbalanced.
• Measure of entropy.
• Lossless vs. lossy.
• Is X close to full entropy?
• Lower vs. upper bound on entropy of X.

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Vertex Expansion Revisited
M

• Even previously trivial tasks
• require D ??(log N/log M)
• M ltlt N ? Farewell constant degree

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Slightly-Unbalanced Constant-Degree Lossless
Expanders
M? N
?(S) ?(1-?) D S
CRVW02 0lt?,?? 1 constants ? D constant K? (N)
In case someone asks K? (? M/D) D poly(1/?
, log (1/? )) (fully explicit D quasipoly(1/?
, log (1/? )))
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Open More Unbalanced
M

• E.g. MN0.5 and sets of size at most KN0.2
  expand. While being greedy
• Unique neighbor expanders
• Lossless expanders
• Minimal Degree

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Super-Constant Degree
M
?S, S? K
?(S) ? (1-?)D S

• State of the art GUV06 DPoly(LogN),
  MPoly(KD) (w. some tradeoff).
• Open MO(KD) (known w. DQuasiPoly(LogN))
• Open D O(LogN)

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Dispersers Sipser 88
N
M

• Bounds
• D 1/? log(N/K)
• DK/M log 1/? -- must be lossy
• Explicit constructions are (comparably) good but
  still not optimal

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Increasing Entropy?
M
Prob. dist. X
Induced dist. X
D
x

• Can Renyi entropy increase ?
• Col(X) lt Col(X) ? (essentially) Dgt
  minM0.5, N/M

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Extractors NZ 93
M N
X
X

• (k,?)-extractor if Min-entropy(X) ? k ? X
  ?-close to uniform
• Min-entropy(X) ? k if ?x, Prx ? 2-k
• X and Y are ?-close if maxT PrX?T - PrY?T
  ½ X-Y1 ? ?

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Equivalently Extractors Mixing
M

• Vertex Expansion Sets on the left have many
  neighbors.
• Mixing Lemma the neighborhood of S hits any T
  with roughly the right proportion.

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2-Source Extractors
source of biased correlated bits
EXT
almost uniform output
another independent weak source

• Recently lots of attention and results
• Randomness Extractors are a special case, where
  the 2nd source is truly random.

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Explicit Constructs. of Extractors

• Extractors are highly motivated in applications.
  As a general rule of thumb Anything expanders
  can do, extractors can do better
• Lots of progress. Still very far from optimal.
  Best in one direction LRVW03, GUV06
  DPoly(LogN / ?), M2k(1-?)
• Selected open problem M2k with DPoly(LogN / ?)
  

Interpretation extracting an arbitrary constant
fraction of entropy
Interpretation extracting all the entropy
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A Word About Techniques

• Research on randomness extractors was invigorated
  with the discovery of a beautiful and surprising
  connection to pseudorandom generators Tre99.
• This further led to discoveries of connections
  between extractors and error correcting codes
  Tre99, RRV99, TZ01, TZS01, SU01.
• In particular, GUV06 relies on Parvaresh-Vardy
  list-decodable codes

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GUV06 - Basic Construction

• Left vertex f ?? Fqn (poly. of degree n-1 over
  Fq)
• Edge Label y ? F
• Right vertices Fqm1
• yth neighbor of f
• (y, f(y), (f h mod E)(y), (f h2 mod E)(y), , (f
  hm-1 mod E)(y))
• where E(Y) irreducible poly of degree n h
  a parameter
• Thm This is a (K,A) expander with Khm, A
  q-hnm.

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Conclusions

• Many interesting variants of expander graphs
• Constructions in general very far from optimal
• Any clean and useful algebraic characterization?