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The story of superconcentrators The missing link

The story of superconcentrators The missing link. Michal Ko u ck ý 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.

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The story of superconcentrators The missing link

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  1. The story of superconcentratorsThe 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 y1 y2 … yn-1 yn • gates are of arbitrary fan-in and may compute arbitrary Boolean functions. • size of circuit= number of wires. Output     depth d    xi x1 … … xm Input

  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 0000000000000000 b 000000000000000 a 000000000000000 n bits

  8. Integer Addition n+1 bits c=a+b 0100101110101100 b 000000000000000 a 100101110101100 n bits

  9. Integer Addition n+1 bits c=a+b 0010110100010110 b 000000000000000 a 010110100010110 n bits

  10. Integer Addition n+1 bits c=a+b 0100001110010000 b 011001110001111 a 001000000000001 n bits

  11. Integer Addition n+1 bits c=a+b 0100010000010000 b 011001110001111 a 001000010000001 n bits

  12. Integer Addition n+1 bits c=a+b 0011010000010000 a 011001110001111 b 000000010000001 n bits

  13. Connectivity property Y • 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. c=a+b b a X

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

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

  16. Bounds on relaxed superconcetrators[Dolev, Dwork, Pippinger, and Wigderson ’83,Pudlák’92] depth d circuits size Ω(…) d=2 nlog n d=3 nlog log n d=2k or d=2k+1 nλk(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 → total size Θ(n2) Prefix-XOR: yk= x1x2 … xk-1xk y1 y2 … yn-1 yn      x1 … x2 xn

  18. Depth-2 circuits for Prefix-XOR y1 … yj … yn • 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)    Output n n/2i 1 xi xn x1 … … Input

  19. Variants of superconcetrators For any k, sets X, Y where |X|=|Y|=k any X and any Yf(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=2 size Θ(…) superconcentrators n (log n)2/log log n middle ground n (log n/log log n)2 relaxed superconcentratorsnlog n d=2k or d=2k+1 all variants nλk(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 Y • For any k, any X, and randomly chosen Y, |X|=|Y|=k EY[f(X,Y)] ≥δk [Gál, Hansen, K., Pudlák, Viola ‘12] Out = f(X,Y) In X

  23. 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 superconcentratorsnlog n d=2k or d=2k+1 all variants nλk(n) where λ1(n) = log n and λk+1(n) = λk*(n)

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

  25. Recent improvements Explicit functions (matrix multiplication) [ Cherukhin ‘08, Jukna ’10, Drucker ‘12] depth d circuits size Ω(…) d=2 n3/2 d=3 nlog n d=4 nlog log n d=2k+1 or d=2k+2 nλk(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).

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