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

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

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, …

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

Circuits y1 y2 … yn-1 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.

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

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

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

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

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

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

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

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.

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!

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!

Bounds on relaxed superconcetrators [Dolev, Dwork, Pippinger, and Wigderson ’83, Pudlák’92] depth d circuits size Ω(…) d=2 n log n d=3 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]

Depth-1 circuits for Prefix-XOR y1 y2 … yn-1 yn      x1 … x2 xn → total size Θ(n2) Prefix-XOR: yk = x1  x2  …  xk-1  xk

Depth-2 circuits for Prefix-XOR y1 … 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)

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

Comparison of depth-d superconcentrators d=2 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)

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

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]

Comparison of depth-d superconcentrators d=2 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)

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

Recent improvements Explicit functions (matrix multiplication) [ Cherukhin ‘08, Jukna ’10, Drucker ‘12] depth d circuits size Ω(…) d=2 n3/2 d=3 n log n d=4 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)

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