1. The Zig-Zag Product and Expansion Close to the Degree
Salil Vadhan
Harvard University
joint work with Michael Capalbo (DIMACS),
Omer Reingold* (Weizmann),
and Avi Wigderson (IAS)

2. Measures of Expansion Combinatorial: vertex expansion, large cuts
Algebraic: second eigenvalue
Probabilistic: like randomness extractors
Thm [...]: All equivalent.
This talk:
Optimize vertex expansion.
Analysis of zig-zag with extractor-like measure.
Unified entropy-theoretic view of all measures.
almost

3. Vertex Expansion N S, |S| K D |(S)|  A |S|
Goals: D small (even constant), A large
Many applications: network design, sorting, complexity theory, cryptography, coding theory, proof complexity

4. Expander Graphs N S, |S| K D |(S)|  A |S| Q: How large can A be?
Trivial upper bound: A  D.
Random graphs: A=D-1.01
But many applications need explicit (deterministic & efficient) constructions.
Previously, best explicit expanders: A =D/2 (for constant D and K=(N)).

5. This Work: Constant-Degree “Lossless” Expanders
… which may even be unbalanced: N
M= N
S, |S| K
|(S)| (1-) D |S| D
0<, 1 constants  D constant & K= (N)
For the experts: K= ( M/D) & D= poly(1/ , log (1/ )) (fully explicit: D= quasipoly(1/ , log (1/ )))

6. Previous explicit constructions
Celebrated sequence of algebraic constructions [Mar73,GG80,JM85,LPS86,AGM87,Mar88,Mor94,...].
Achieved optimal 2nd eigenvalue (Ramanujan graphs), but this only implies vertex expansion  D/2 [Kah95].
“Combinatorial” constructions: Ajtai [Ajt87], more explicit and simple [RVW00].
“Lossless objects”: [Alo95,RR99,TUZ01]
Unique neighbor, constant-degree expanders [Cap01].

7. Why Bother with the Deg./2 Barrier?
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 [BMRS00].
Distributed storage schemes [UW87].
Hard tautologies for various proof systems [BW99,ABRW00,AR01].

8. Properties of Lossless Expanders
At least (1-2) D |S| elements of (S) are unique neighbors: touch exactly one edge from S N
M= N
S, |S| K
|(S)| (1-) D |S| x D
Fault tolerance: Even if an adversary removes most (say ¾) edges from each vertex, lossless expansion maintained (with =4)

9. Outline Overview & results Entropy view of expansion
The original zig-zag product & its limitation
The new zig-zag product for conductors

10. Entropy View of Expansion
M= N
prob. dist. S H(S)  k
(A=2a, B=2b, ...)
H(induced dist)  H(S)+a
S, |S| K
|(S)| A |S| D
Think of expanders as “entropy increasers”
H(X) = number of bits of “randomness” in X
H(X)log2|Support(X)| w/equality if X uniform on Support(X)
H(XY)= H(X) + H(Y) if X,Y independent

11. Entropy View of Vertex Expansion
M= N
prob. dist. S H(S)  k
(A=2a, B=2b, ...)
H(S )  H(S)+a
S, |S| K
|(S)| A |S| D
Use: H(X)=log2|Support(X)|

12. Entropy View of 2nd E-Value
M= N
prob. dist. S H(S)  k
S, |S| K
|(S)| A |S|
H(S )  H(S)+a D
(A=2a, B=2b, ...)
Use:
Fact: 

13. Randomness Conductors [CRVW02]
prob. dist. S H(S)  k
(A=2a, B=2b, ...)
H(S )  H(S)+a
S, |S| K
|(S)| A |S| D
Allow statistical distance  on output
Use:

14. Randomness Conductors [CRVW02]
prob. dist. S H(S)  k
(A=2a, B=2b, ...)
H(S )  H(S)+a
S, |S| K
|(S)| A |S| D
Special Cases:
k+a = m  randomness extractors [NZ93]
a= d  lossless expanders [RR99,TUZ01]

15. Entropy View of Expansion
M= N
prob. dist. S H(S)  k
(A=2a, B=2b, ...)
H(induced dist)  H(S)+a
S, |S| K
|(S)| A |S| D
Think of expanders as “entropy increasers”
H(X) = number of bits of “randomness” in X
H(X)log2|Support(X)| w/equality iff X uniform on Support(X)
H(XY)= H(X) + H(Y) if X,Y independent

16. Our Expander Construction
Starting Point: Zig-Zag Graph Product [RVW00] 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.
Original motivation: A combinatorial construction of constant-degree expanders with simple analysis.

17. Outline Overview & results Entropy view of expansion
The original zig-zag product & its limitation
The new zig-zag product for conductors

18. The Zig-Zag Product [RVW00]
“Theorem”: Expansion (G1 G2)  min(Expansion (G2), Expansion (G1))
formal versions for 2nd e-value, extractors [RVW00] z

19. Zig-Zag Analysis (Case I) [RVW00]
First step on small graph adds entropy.
Case I: Conditional distributions within “clouds” far fr. uniform.
Next two steps can’t lose entropy.

20. Zig-Zag Analysis (Case II) [RVW00]
First small step does nothing.
Step on big graph “scatters” among clouds (shifts entropy)
Second small step adds entropy.
Case II: Conditional distributions within clouds uniform.

21. Inherent Entropy Loss
In each case, only one of two small steps “works”
But paid for both in degree.

22. Trying to improve ??? ??? Idea: Use two “optimal” small graphs.
How to measure?
???

23. Zig-Zag for Unbalanced Graphs
Second eigenvalue probably not useful.
Extractors [NZ93] and condensers [RR99,RSW00,TUZ01] work well in unbalanced case.
In fact, [RVW00] analyzed a zig-zag product for extractors (with an “easier goal”).
 try to analyze zig-zag for randomness conductors

24. Randomness Conductors [CRVW02]
prob. dist. S H(S)  k
(A=2a, B=2b, ...)
H(S )  H(S)+a
S, |S| K
|(S)| A |S| D
Special Cases:
k+a = m  randomness extractors [NZ93]
a= d  lossless expanders [RR99,TUZ01]

25. Outline Overview & results Entropy view of expansion
The original zig-zag product & its limitation
The new zig-zag product for conductors

26. Zig-Zag for Conductorsx
Start w/lossy conductor G1 d1 G1 y
Goal: make it lossless
y’th nbr of x n1

27. Zig-Zag for Conductors
n2=O(d1)
d2=O(log d1) G2 d1 G1 n1

28. Zig-Zag for Conductors
n2=O(d1)
d2=O(log d1) G2 d1 G1
b2=O(d1) d1
Keep “buffers” to retain lost entropy
(remember which edge used) n1

29. Zig-Zag for Conductors
n2=O(d1)
d2=O(log d1) G2 d1 G1
b2=O(d1) d1
d3=O(log d1) G3
E3: 2nd small conductor
lossless n1
m3<n2

30. Analysis of Entropy Flow
Case I:
Conditional
entropy within
clouds large. n1
n2=O(d1)
d2=O(log d1) G2 d1 G1
b2=O(d1) d1
d3=O(log d1) G3 n1
m3<n2

31. Analysis of Entropy Flow
Case II:
Conditional
entropy within
clouds small. n1
n2=O(d1)
d2=O(log d1) G2 d1 G1
b2=O(d1) d1
d3=O(log d1) G3 n1
m3<n2

32. Summary and Open Problems
Our Result: Constant-Degree Lossless Expanders.
Main tools: randomness conductors, zig-zag product
Further Research:
Analyze algebraic expanders wrt conductor measure.
The undirected case (being lossless from both sides).
Better expansion yet? D-O(1)
Continue the study of randomness conductors.