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 n k –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 Alg’(x): Majority {Alg(x,G(1)),…,Alg(x,G(2 d )} Alg random bits (|r|=m) r Alg(x,r) correct for most r input x (|x|=n) Wx = good r’s for x |Wx|/ 2 m > 3/4 If: G efficient, pseudo-random generator for Alg., d=O(log n) Then: Alg’ is deterministic, efficient, correct for every x G (|s|=d) s seed

The New Idea Alg’(x): Majority {Alg(x,E(x,1)),…,Alg(x,E(x,2 d )} If: E efficient extractor, d=O(log n) Then: Alg’ is deterministic, efficient, correct for all but few x Alg random bits (|r|=m) r Alg(x,r) correct for most r input x (|x|=n) W x = good r’s for x |W x |/2 m > 3/4 E (|s|=d) s seed (*) m<n(**) W x independent of x

Extractors 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,U d ) – U m | 1 <  Def [NZ]: A probability distribution X on {0,1} n is a k-source if for every x Pr[X=x]  2 -k Def (informal): Extractors “smooth out” every probability distribution of sufficient “entropy” with the aid of “few” truly random bits. Lemma [NZ]: Fix any event W  {0,1} m. At most 2 k x  {0,1} n satisfy |Pr[ E(x,U d )  W] – |W|/2 m |>  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 st-connectivity in directed graphs L deterministic space O(log n) st-conn. In directed outdegree 1 graphs st-conn. In directed outdegree 1 graphs SL symmetric non-deterministic space O(log n) st-connectivity in undirected graphs st-connectivity in undirected graphs RL probabilistic space O(log n) L b deterministic space O((log n) b ) L  SL  RL  NL  L 2 L  SL  RL  NL  L 2

Theorems Thm [S]: NL  L 2 Thm [IS] NL = coNL Thm [ALLKR] SL  RL Thm [NSW] SL  L 3/2 Thm [SZ] RL  L 3/2 Thm [ASTW] SL  L 4/3 Open Problems NL = L ? RL = L ? SL = L ? New Thm SL = L except on rare instances

Traversal Sequences  = (  1,  2,…,  p ) in {0,1} p G undirected graph on n vertices 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 1 st k bits  ’ ←  - first k bits v’ ← i th 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 n 4 visits all vertices of a connected n-vertex graph Cor [AKLLR]: Most sequences of length n 6 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 L 2

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 L 3/2 (since by [N] k-uts can be found in L) Here: k=n  /6 for any  >0,  random m-bit string, m=k 6 Most  of length m=n  are k-uts >3/4 <n independent of G

The New Connectivity Algorithm 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 Fix  >0, set m=l 6 =n , d=O(log n) Fix a logspace (m 2, 1/8)-extractor E:{0,1} n  {0,1} d → {0,1} m Set E(G)= E(G,1),E(G,2),…,E(G,2 d ) Total space complexity: (log n) 2 /(log k) =O(log n) Whenever E(G) is an k-uts This fails for at most exp(m 2 ) = exp(n 2  ) graphs

General Derandomization Alg’(x): Majority {Alg(x,G(E(x,1))),…,Alg(x,G(E(x,2 d )))} If: E efficient extractor, G pseudorandom generator Then: Alg’ is deterministic, efficient, correct for all but few x G Alg random bits (|r|=m>n) r Alg(x,r) correct for most r input x (|x|=n) |W x |/ 2 m > n W=  x W x |W|/ 2 m > 3/4 E (|s|=d) s seed nn

Assumptions vs. Conclusions Thm[IW]: If DTIME(2 O(n) )  SIZE(2  n ) for all  >0 Then BPP=P New Thm: If P is not approx by SIZE SAT (n k ) for all integers k Then BPP=P for all but exp(n  ) n-bit inputs Proof: n k running time of Alg on length n inputs W can be recognized in SIZE SAT (n k ) f  P cannot be approx by SIZE SAT (n k/  ) G=NW f fools W [NW,KvM]

Discussion & Problems OPEN  Find other examples of such algorithms  Prove: SL = L 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