1. The story of superconcentrators The missing link
Michal Koucký
Institute of Mathematics, Prague

2. Computational complexity
How much computational resources do we need to compute various functions. (time, space, etc.)
Upper bounds (algorithms).
Lower bounds.

3. Lower bound techniques
We have very little understanding of actual computation.
Diagonalization.
Gödel, Turing, …
Information theory.
Shannon, Kolmogorov, …
Other special techniques – random restrictions, approximation by polynomials.
Ajtai, Sipser, Razborov, …

4. Integer Addition
n+1 bits
c=a+b b a
n bits

5. Circuits
y y2 … yn yn
Output    
depth d   
Input
x1 … … xi xm
gates are of arbitrary fan-in and may compute arbitrary Boolean functions.
size of circuit = number of wires.

6. Circuits vs Turing machines
polynomial size circuits ~ polynomial time computation
Open: Exponential time computation cannot be simulated by polynomial size circuits.

7. Integer Addition n+1 bits c=a+b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8. Integer Addition n+1 bits c=a+b 0 1 0 0 1 0 1 1 1 0 1 0 1 1 0 0

9. Integer Addition n+1 bits c=a+b 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0

10. Integer Addition n+1 bits c=a+b 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0

11. Integer Addition n+1 bits c=a+b 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0

12. Integer Addition n+1 bits c=a+b 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0

13. Connectivity property
c=a+b b a X
For any two interleaving sets X and Y,
where X are inputs a and Y are outputs c
there are |X|=|Y| vertex disjoint paths between X and Y in any circuit computing integer addition.

14. Superconcentrators [Valiant’75]Y
Out
= f(X,Y) In X
For any k, any X, and any Y, |X|=|Y|=k
f(X,Y) = k
 Can be built using O(n) wires. Oooopss!

15. Relaxed superconcentrators [Dolev et al.’83]Y
Out d
= f(X,Y) In X
For any k, random X, and random Y, |X|=|Y|=k
EX,Y[f(X,Y)] ≥ δk
 Fixed depth requires superlinear number of wires!

16. Bounds on relaxed superconcetrators [Dolev, Dwork, Pippinger, and Wigderson ’83, Pudlák’92]
depth d circuits size Ω(…)
d= n log n
d= n log log n
d=2k or d=2k+1 n λk(n)
where λ1(n) = log n and λk+1(n) = λk*(n)
Applications [Chandra, Fortune, and Lipton ’83]

17. Depth-1 circuits for Prefix-XOR
y y … yn yn
    
x … x2 xn
→ total size Θ(n2)
Prefix-XOR: yk = x1  x2  …  xk-1  xk

18. Depth-2 circuits for Prefix-XOR
y … yj … yn
Output    n
n/2i 1
Input
x1 … … xi xn
Each middle block computes n/2i parities of input blocks of size 2i i=1, …, log n
→ the total size is O(n log n)

19. Variants of superconcetrators
For any k, sets X, Y where |X|=|Y|=k
any X and any Y f(X,Y) = k (≥ δk) superconcetrators
any X and random Y EY[f(X,Y)] ≥ δk middle ground
random X and random Y EX,Y[f(X,Y)] ≥ δk relaxed superconcetrators

20. Comparison of depth-d superconcentrators
d= size Θ(…)
superconcentrators n (log n)2/log log n
middle ground n (log n/log log n)2
relaxed superconcentrators n log n
d=2k or d=2k+1
all variants n λk(n)
where λ1(n) = log n and λk+1(n) = λk*(n)

21. Good error-correcting codes
0<ρ,δ<1 constants, m < n:
enc : {0,1}m → {0,1}n
For any x, x’  {0,1}m, where x  x’ distHam(enc(x),enc(x’)) ≥ δn.
m ≥ ρn.
Applications: zillions

22. Connectivity of circuits computing codes
Out
= f(X,Y) In X
For any k, any X, and randomly chosen Y, |X|=|Y|=k
EY[f(X,Y)] ≥ δk
[Gál, Hansen, K., Pudlák, Viola ‘12]

23. Comparison of depth-d superconcentrators
d= size Θ(…)
superconcentrators n (log n)2/log log n
middle ground n (log n/log log n)2
relaxed superconcentrators n log n
d=2k or d=2k+1
all variants n λk(n)
where λ1(n) = log n and λk+1(n) = λk*(n)

24. Single output functions
X y
(c*ac*b)*c* [K. Pudlák, and Thérien ’05]
 circuits must contain relaxed superconcentrators

25. Recent improvements
Explicit functions (matrix multiplication) [ Cherukhin ‘08, Jukna ’10, Drucker ‘12]
depth d circuits size Ω(…)
d= n3/2
d= n log n
d= n log log n
d=2k+1 or d=2k+2 n λk(n)
where λ1(n) = log n and λk+1(n) = λk*(n)

26. Conclusions
Information theory is the strongest lower bound tool we currently have (unfortunately).