Transcript Document

Almost SL=L, and
Near-Perfect Derandomization
Oded Goldreich
The Weizmann Institute
Avi Wigderson
IAS, Princeton
Hebrew University
SL vs. L
Theseus
Ariadne
Crete, ~1000 BC
SL vs L
Thm (informal): SL=L except on rare inputs
Thm (formal): For every >0 there is a deterministic
logspace algorithm, which correctly determines
undirected st-connectivity, except on at most
exp(n) graphs on n vertices on which it answers “?”
Thm: Fix any language A in SL. Then
for every  >0 there is a deterministic
logspace algorithm, which correctly determines
membership in A, except on at most exp(n)
inputs of length n, on which it answers “?”
Derandomization
“God does not play dice
with the universe”
General Derandomization
Thm (informal): BPP=P except on rare inputs
under some natural complexity assumption.
Thm (formal):
Assumption: There is a function in P, which has no
approx nk–size circuits with SAT oracle for any k.
Conclusion: Fix any language A in BPP. Then
for every  >0 there is a deterministic
polyime algorithm, which for every n errs
on at most exp(n) inputs of length n.
The Old Paradigm
random bits
(|r|=m) r
G
Alg
input x (|x|=n)
Alg(x,r)
correct
for most r
Wx = good r’s for x
|Wx|/ 2m > 3/4
(|s|=d) s seed
Alg’(x): Majority {Alg(x,G(1)),…,Alg(x,G(2d)}
If: G efficient, pseudo-random generator for Alg., d=O(log n)
Then: Alg’ is deterministic, efficient, correct for every x
The New Idea
random bits
(|r|=m) r
E
Alg
input x (|x|=n)
Alg(x,r)
correct
for most r
Wx = good r’s for x
|Wx|/2m > 3/4
(|s|=d) s seed
Alg’(x): Majority {Alg(x,E(x,1)),…,Alg(x,E(x,2d)}
If: E efficient extractor, d=O(log n)
Then: Alg’ is deterministic, efficient, correct for all but few x
(*) m<n
(**) Wx independent of x
Extractors
Def (informal): Extractors “smooth out” every probability
distribution of sufficient “entropy” with the
aid of “few” truly random bits.
Def [NZ]: A probability distribution X on {0,1}n
is a k-source if for every x Pr[X=x]2-k
Def [NZ]: A function E:{0,1}n  {0,1}d→ {0,1}m
is a (k,)-extractor if for every k-source X
|E(X,Ud) – Um|1 < 
Lemma [NZ]: Fix any event W{0,1}m. At most 2k x{0,1}n
satisfy |Pr[ E(x,Ud)  W] – |W|/2m |> 
Thm [Z,NZ,T,ISW,SU]: Explicit efficient extractors exist
The LogSpace Arena
NL non-deterministic space O(log n)
st-connectivity in directed graphs
L
deterministic space O(log n)
st-conn. In directed outdegree 1 graphs
SL symmetric non-deterministic space O(log n)
st-connectivity in undirected graphs
RL probabilistic space O(log n)
Lb deterministic space O((log n)b)
L  SL  RL  NL  L2
Theorems
Open Problems
Thm [S]:
NL  L2
Thm [IS]
NL = coNL
Thm [ALLKR]
SL  RL
Thm [NSW]
SL  L3/2
Thm [SZ]
RL  L3/2
RL = L ?
Thm [ASTW]
SL  L4/3
SL = L ?
New Thm
SL = L except on rare instances
NL = L ?
Traversal Sequences
G undirected graph on n vertices
 = (1, 2,…, p) in {0,1}p
A walk w on G starting at v using  :
w ← v, set k = log deg(v)
walk(G,v,):
(1) If ||<k output w
(2) If not, let i be the value of the 1st k bits
’ ←  - first k bits
v’ ← ith neighbour of v in G
w ← w,v’
walk(G,v’,’)
Universal Traversal Sequences
Def [C]: A sequence  is n-universal (n-uts)if
for every graph on G on n vertices, and
for every vertex v of G, walk(G,v,)
visits all vertices in v’s connected component.
Conj [C]: Computing n-uts is in L
Thm [AKLLR]: A random walk of length n4 visits
all vertices of a connected n-vertex graph
Cor [AKLLR]: Most sequences of length n6 are n-uts
Thm [N]: There is a pseudo-random generator for RL which
uses only O((log n)2) random bits and space.
Cor [N]: Computing n-uts is in L2
The NSW Connectivity
Algorithm
Main subroutine (in L):
Input: an n-vertex graph G, any k-uts 
Output: an n/k-vertex graph G’,
such that G is connected iff G’ is.
The algorithm:
Repeat main subroutine (log n)/(log k) steps
Total space complexity: (log n)2/(log k)
In [NSW]: k = exp ((log n)1/2)  SL in L3/2
(since by [N] k-uts can be found in L)
Here: k=n/6 for any  >0,  random m-bit string, m=k6
Most  of length m=n are k-uts
>3/4
<n
independent of G
The New Connectivity
Algorithm
Fix >0, set m=l6 =n , d=O(log n)
Fix a logspace (m2, 1/8)-extractor E:{0,1}n  {0,1}d→ {0,1}m
Set E(G)= E(G,1),E(G,2),…,E(G,2d)
Main subroutine (in L):
Input: an n-vertex graph G, =E(G)
Output: an graph G’, connected iff G is,
with n/k vertices if  is an k-uts
The algorithm:
Repeat main subroutine (log n)/(log k) steps
Total space complexity: (log n)2/(log k) =O(log n)
Whenever E(G) is an k-uts
This fails for at most exp(m2) = exp(n2) graphs
General Derandomization
random bits
(|r|=m>n) r
G
Alg
n
E
(|s|=d) s seed
input x (|x|=n)
Alg(x,r)
correct
for most r
|Wx|/ 2m > 1-2-2n
W= x Wx
|W|/ 2m > 3/4
Alg’(x): Majority {Alg(x,G(E(x,1))),…,Alg(x,G(E(x,2d)))}
If: E efficient extractor, G pseudorandom generator
Then: Alg’ is deterministic, efficient, correct for all but few x
Assumptions vs. Conclusions
Thm[IW]: If DTIME(2O(n))  SIZE(2n) for all >0
Then BPP=P
New Thm: If P is not approx by SIZESAT(nk) for all integers k
Then BPP=P for all but exp(n) n-bit inputs
Proof:
nk running time of Alg on length n inputs
W can be recognized in SIZESAT(nk)
f P cannot be approx by SIZESAT(nk/)
G=NWf fools W [NW,KvM]
Discussion & Problems
Efficient deterministic algorithms which are
correct on all but exp(cn) length n inputs (c<1)
correct (whp) on dist with high enough (min) entropy
Generalize some known classes of algorithms
(1) Derandomizations under uniform assumptions
correct (whp) on efficiently samplable distributions
(2) Average case analysis
correct for specific structured distributions
OPEN
 Find other examples of such algorithms
 Prove: SL = L