Almost SL=L , and Near-Perfect Derandomization

1. Almost SL=L, andNear-Perfect Derandomization Oded Goldreich The Weizmann Institute Avi Wigderson IAS, Princeton Hebrew University

2. SL vs. L Theseus Ariadne Crete, ~1000 BC

3. 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 “?”

4. “God does not play dice with the universe” Derandomization

5. 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.

6. Alg random bits (|r|=m) r Alg(x,r) correct for most r G Wx = good r’s for x |Wx|/ 2m > 3/4 input x(|x|=n) (|s|=d)sseed The Old Paradigm 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

7. Alg random bits (|r|=m) r Alg(x,r) correct for most r E Wx = good r’s for x |Wx|/2m > 3/4 input x(|x|=n) (|s|=d) sseed The New Idea 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

8. 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. Atmost 2k x{0,1}n satisfy |Pr[ E(x,Ud)  W] – |W|/2m |>  Thm [Z,NZ,T,ISW,SU]: Explicit efficient extractors exist

9. NLnon-deterministic space O(log n) st-connectivity in directed graphs L deterministic space O(log n) st-conn. In directed outdegree 1 graphs SLsymmetric non-deterministic space O(log n) st-connectivity in undirected graphs RLprobabilistic space O(log n) Lbdeterministic space O((log n)b) L  SL  RL  NL  L2 The LogSpace Arena

10. Theorems Open Problems Thm [S]: NL L2 NL = L ? RL = L ? SL = L ? Thm [IS] NL = coNL Thm [ALLKR] SLRL Thm [NSW] SLL3/2 Thm [SZ] RLL3/2 Thm [ASTW] SLL4/3 New ThmSL = L except on rare instances

11. 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 1stk bits ’ ← - first k bits v’ ←ith neighbour of v in G w←w,v’ walk(G,v’,’)

12. 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

13. 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

14. 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

15. random bits (|r|=m>n) r Alg Alg(x,r) correct for most r n |Wx|/ 2m > 1-2-2n W= x Wx |W|/ 2m > 3/4 E input x(|x|=n) (|s|=d)sseed General Derandomization G 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

16. Assumptions vs. Conclusions Thm[IW]: IfDTIME(2O(n))  SIZE(2n) for all >0 Then BPP=P NewThm: IfPis not approx by SIZESAT(nk) for all integers k Then BPP=P for all but exp(n) n-bit inputs Proof: nkrunning 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]

17. 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