Constant Degree, Lossless Expanders Omer Reingold AT&T joint work with Michael Capalbo (IAS), Salil...
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Transcript of Constant Degree, Lossless Expanders Omer Reingold AT&T joint work with Michael Capalbo (IAS), Salil...
Constant Degree, Lossless Constant Degree, Lossless ExpandersExpanders
Omer Reingold
AT&T
joint work with Michael Capalbo (IAS),
Salil Vadhan (Harvard),
and Avi Wigderson (Hebrew U., IAS)
Expander Graphs (Balanced Case)Expander Graphs (Balanced Case)
|(S)| >A |S|S, |S| K
An innocent looking object … but intimately related to various fundamental problems (Network Design, Complexity and Proof Theory, Derandomization, Coding Theory, Cryptography, ...)
D
N N
Expander Graphs (Balanced Case)Expander Graphs (Balanced Case)
|(S)| >A |S|S, |S| K
How large can A be? • Trivial upper bound: A D. • Random graphs: AD.• Previously, best explicit expanders: A =D/2
(for constant D and “large” K).
D
N N
This Work: Const. Degree, Lossless This Work: Const. Degree, Lossless Expanders …Expanders …
… that may even be slightly unbalanced:
|(S)| >(1-) D |S|D
N M= N
S, |S| K
0<, 1 are constants D is constant & K= (N)
For the very curious only:K= ( M/D) & D= poly log (1/( )) (fully explicit: D= quasi poly log(1/( ) )).
A Bit of ContextA 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].
Why Bother with the Deg./2 Barrier?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 systems
[BW99,ABRW00,AR01].
Distributed routing in networksDistributed routing in networks
The Task [[PU89,ALM96,BFU99PU89,ALM96,BFU99]]: Finding vertex/edge disjoint paths in a distributed manner. Much easier if the network is composed of lossless expanders.
Distributed routing in networksDistributed routing in networks
Simplified scenario: Vertex disjoint paths in a layered graph. Expansion factor from left to right 9D/10.
|S| K
OutputsInputs
...
Step 1: Match to “unique neighbors” of S
Then, continue with (at most |S|/10) unmatched vertices in S
...
Distributed routing in networksDistributed routing in networks
Simplified scenario: Vertex disjoint paths in a layered graph. Expansion factor from left to right 9D/10.
|S| K
OutputsInputs
Incredibly Fault TolerantIncredibly Fault Tolerant [[UW87UW87]]: Works even if adversary removes 3/4 of D edges from each vertex.
...
Simple Expander Codes Simple Expander Codes [G63,Z71,ZP76,T81,SS96]
M= N (Parity Checks)
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/).
N (Variables)
Fix =1/10 :
Sets of size K= ( N/D) expand by a factor 9D/10.
D
1
100
1
++
+
+
0
Error set B, |B| K/2
• Algorithm: At each phase, flip every variable that “sees” a majority of 1’s (i.e, unsatisfied constraints).
Simple Decoding Algorithm in Linear Simple Decoding Algorithm in Linear TimeTime (& log n parallel phases) [SS 96]
M= N (Constraints)N (Variables)
++
+
+1
100
1|Flip\B| |B|/4.|B\Flip| |B|/4. |Bnew| |B|/2.
|(B)| > (1-) D |B|
|(B)Sat| < 2 D |B|0
10
0
11
0
Hints Into the Expander ConstructionHints 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.
The Zig-Zag Product [RVW00]
z
Thm. If G1 is a “good” expander, then Expansion (G1 G2) Expansion (G2)z
Zig-Zag Analysis (Case I)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.
Zig-Zag for Unbalanced GraphsZig-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.
Randomness ConductorsRandomness 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?
– …
Randomness conductors:
As in extractors.
Allows the entire spectrum.
On the Board ?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.
Summary and Open ProblemsSummary and Open Problems• Our Result: (Slightly Unbalanced), Constant Degree, (Slightly Unbalanced), Constant Degree,
Lossless ExpandersLossless 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.
Definition: Randomness ConductorsDefinition: 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, of min entropy k
f
Uniform
S’ = f (S,Uniform)
S’ is - close to min entropy” k’
(min entropy k x, Pr[x] 2-k)