Constant Degree, Lossless Expanders

1. Constant Degree, Lossless Expanders Omer Reingold AT&T joint work with Michael Capalbo (IAS), Salil Vadhan (Harvard), and Avi Wigderson (Hebrew U., IAS)

2. N N |(S)| >A |S| S, |S| K D 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, ...)

3. N N |(S)| >A |S| S, |S| K D 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).

4. N M= N D |(S)| >(1-) D |S| S, |S| K This Work: Const. Degree, Lossless Expanders … … that may even be slightly unbalanced: 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/( ) )).

5. 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].

6. Why Bother with the Deg./2 Barrier? • Because it’s 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].

7. 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.

8. |S| K Step 1: Match to “unique neighbors” of S Then, continue with (≤ |S|/10) unmatched vertices in S Distributed routing in networks Simplified scenario: Vertex disjoint paths in a layered graph. Expansion factor from left to right 9D/10. Inputs Outputs ... ...

9. |S| K Distributed routing in networks Simplified scenario: Vertex disjoint paths in a layered graph. Expansion factor from left to right 9D/10. Inputs Outputs ... Incredibly Fault Tolerant[UW87]: Works even if adversary removes 3/4 of Dedges from each vertex.

10. + + 1 0 0 + 0 D 1 + 1 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. Rate1 - 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/).

11. + Error set B, |B|K/2 + 1 0 |(B)| > (1-) D |B| 0 1 + |(B)Sat|< 2D|B| 1 0 + 0 1 1 1 0 0 Simple Decoding Algorithm in Linear Time (& log n parallel phases) [SS 96] • Algorithm: At each phase, flip every variable that “sees” a majority of 1’s (i.e, unsatisfied constraints). N (Variables) M= N (Constraints) |Flip\B| |B|/4.|B\Flip| |B|/4. |Bnew||B|/2.

12. 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.

13. z z • Thm. If G1 is a “good” expander, then Expansion (G1 G2)  Expansion (G2) The Zig-Zag Product [RVW00]

14. 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.

15. Trying to improve ??? ???

16. 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.

17. Randomness conductors: • As in extractors. • Allows the entire spectrum. 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? • …

18. 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.

19. 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.

20. Uniform f S’ =f (S,Uniform) S, of min entropy k 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, Pr[x]  2-k)

21. N N S, |S| K D Lossless Expanders are Incredibly Fault Tolerant [UW87] Let an adversary remove (1-) Dedges for each vertex. • Still expands by a factor (1-  / ) D’ !! |(S)| >(1-)|S|