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.
1.
Diagonalization.
–
2.
Gödel, Turing, …
Information theory.
–
Shannon, Kolmogorov, …
Other special techniques – random restrictions,
approximation by polynomials.
3.
–
Ajtai, Sipser, Razborov, …
Integer Addition
n+1 bits
c=a+b
b
a
n bits
Circuits
y1
y2


Output
yn-1
yn


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
b
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
a
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
n bits
Integer Addition
n+1 bits
c=a+b 0 1 0 0 1 0 1 1 1 0 1 0 1 1 0 0
b
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
a
1 0 0 1 0 1 1 1 0 1 0 1 1 0 0
n bits
Integer Addition
n+1 bits
c=a+b 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0
b
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
a
0 1 0 1 1 0 1 0 0 0 1 0 1 1 0
n bits
Integer Addition
n+1 bits
c=a+b 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0
b
0 1 1 0 0 1 1 1 0 0 0 1 1 1 1
a
0 0 1 0 0 0 0 0 0 0 0 0 0 0 1
n bits
Integer Addition
n+1 bits
c=a+b 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0
b
0 1 1 0 0 1 1 1 0 0 0 1 1 1 1
a
0 0 1 0 0 0 0 1 0 0 0 0 0 0 1
n bits
Integer Addition
n+1 bits
c=a+b 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0
a
0 1 1 0 0 1 1 1 0 0 0 1 1 1 1
b
0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
n bits
Connectivity property
Y
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
d=2
d=3
d=2k or d=2k+1
size Ω(…)
n log n
n log log n
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


x1
x2
…

yn-1
yn


…
xn
→ total size Θ(n2)
Prefix-XOR:
yk = x1
 x2 
…
 xk-1  xk
Depth-2 circuits for Prefix-XOR
y1
…

Output
•
…
n/2i
x1
…
yn


n
Input
yj
xi
1
…
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
size Θ(…)
d=2
superconcentrators
middle ground
relaxed superconcentrators
n (log n)2/log log n
n (log n/log log n)2
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
1.
For any x, x’  {0,1}m, where x  x’
distHam(enc(x),enc(x’)) ≥ δn.
2.
m ≥ ρn.
Applications: zillions
Connectivity of circuits computing codes
Y
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
size Θ(…)
d=2
superconcentrators
middle ground
relaxed superconcentrators
n (log n)2/log log n
n (log n/log log n)2
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
(c*ac*b)*c*
y
[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
d=2
d=3
d=4
d=2k+1 or d=2k+2
size Ω(…)
n3/2
n log n
n log log n
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).