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. 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 NN

3. Expander Graphs (Balanced Case) |  (S)| >A |S|  S, |S|  K 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 ). D NN

4. This Work: Const. Degree, Lossless 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/(   ) ) ).

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 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 ].

7. Distributed routing in networks [ PU89,ALM96,BFU99 ] 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. Distributed 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...

9. Distributed 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 Tolerant [ UW87 ] Incredibly Fault Tolerant [ UW87 ]: Works even if adversary removes 3/4 of D edges from each vertex....

10. 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 9 D/ 10. D 1 1 0 0 1 + + + + 0

11. 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 Time Simple Decoding Algorithm in Linear Time (& log n parallel phases) [ SS 96 ] M=  N (Constraints) N (Variables) + + + + 1 1 0 0 1 |Flip\B|  |B|/4. |B\Flip|  |B|/4.  |B new |  |B|/2. |  (B)| > (1-  ) D |B| |  (B)  Sat| < 2  D |B| 0 1 0 0 1 1 0

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. The Zig-Zag Product [ RVW00 ] z Thm. If G 1 is a “good” expander, then Expansion ( G 1 G 2 )  Expansion ( G 2 ) z

14. 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.

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 Expanders, extractors, condensers & hash functions are all functions, f : [N]  [D]  [M], that transform: S “of entropy” k  S’ = f (S,U niform ) “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.

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 (Slightly Unbalanced), Constant Degree, Lossless ExpandersOur 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. 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, of min entropy k f U niform S’ = f (S,U niform ) S’ is  - close to min entropy” k’ ( min entropy k   x, Pr[x]  2 -k )

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