RANDOMNESS VS. MEMORY: Prospects and Barriers Omer Reingold, Microsoft Research and Weizmann With insights courtesy of Moni Naor, Ran Raz, Luca Trevisan, Salil Vadhan, Avi Wigderson, many more …

Randomness In Computation (1)  Distributed computing (breaking symmetry)  Cryptography: Secrets, Semantic Security, …  Sampling, Simulations, …

Randomness In Computation (2)  Communication Complexity (e.g., equality)  Routing (on the cube [Valiant]) - drastically reduces congestion

Randomness In Computation (3)  In algorithms – useful design tool, but many times can derandomize (e.g., PRIMES in P). Is it always the case?  RL=L means that every randomized algorithm can be derandomized with only a constant factor increase in memory

Talk’s Premise: Many Frontiers of RL=L RL=L RL in L 3/2 And Beyond Barriers of previous proofs  wealth of excellent research problems.

RL  (NL  ) L 2 [Savitch 70] poly(|x|) configs transitions on current random bit duplicate (running time T) ≤ poly(|x|) times s = start config t = accept config Configuration graph (per RL algorithm for P & input x): x  P  random walk from s ends at t w.p. ≥ ½ x  P  t unreachable from s Enumerating all possible paths – too expensive. Main idea: 1st half of computation only transmits log n bits to 2nd half

Oblivious Derandomiztion of RL  Pseudorandom generators that fool space-bounded algorithms [AKS 87, BNS 89, Nisan 90, NZ 93, INW 94 ]  Nisan’s generator has seed length log 2 n  Proof that RL in L 2 via oblivious derandomization  Major tool in the study of RL vs. L  Applications beyond [Ind 00, Siv 02, KNO 05,…]  Open problem: PRGs with reduced seed length

Randomness Your Service  Basic idea [NZ93] (related to Nisan’s generator):  Let log -space A read a random 100 logn bit string x.  Since A remembers at most logn bits, x still contains (roughly) 99 logn bits of entropy (independent of A ’s state).  Can recycle x : G x,y x, Ext(x,y)

Randomness Your Service  NZ generator:  Possible setting of parameters: x is O(log n) long. Each y i is O(log ½ n) long and have log ½ n y i ’s.  Expand O(log n) bits to O(log 3/2 n) (get any poly)  Error >> 1/n ([AKS87] gets almost log 2 n bits w. error 1/n) G x,y 1,y 2, … x, Ext(x,y 1 ), Ext(x,y 2 ),

Randomness Your Service  NZ generator:  Error >> 1/n ([AKS87] gets almost log 2 n bits w. error 1/n)  Open: get any polynomial expansion w. error 1/n  Open: super polynomial expansion with logarithmic seed and constant error (partial result [RR99]). G x,y 1,y 2, … x, Ext(x,y 1 ), Ext(x,y 2 ),

Nisan,INW Generators via Extractors  Recall basic generator:  Lets flip it … G x,y x, Ext(x,y)

Nisan,INW Generators via Extractors x,y xExt(x,y) Given state of machine in the middle, Ext(x,y) still  -random log n Loss at each level: log n (possible entropy in state). + log 1/ έ for extractor seed, where έ =  /n Altogether: seed length = log 2 n

Nisan,INW + NZ  RL=L  Let M be an RL machine  Using [Nisan] get M’ that uses only log 2 n random bits  Fully derandomize M’ using [NZ]  Or does it?  M’ is not an RL machine (access to seed of [Nisan, INW] not read once)  Still, natural approach – derandomize seed of [Nisan] Can we build PRGs from read once ingredients? Not too promising …

RL  L 3/2 [SZ95] - “derandomized” [Nis]  Nisan’s generator has following properties:  Seed divided into h (length log 2 n ) and x (length logn ).  Given h in input tape, generator runs in L.   M, w.h.p over h, fixing h and ranging over x implies a good generator for M.  h is shorter if we generate less than n bits

[SZ95] - basic idea  Fix h, divide run of M to segments:  Enumerate over x, estimate all transition probs.  Replace each segment with a single transition  Recurse using the same h  Now M’ depends on h M’ close to some t-power of M. [SZ] perturb M’ to eliminate dependency on h

[SZ95] –further progress  Open: Translate [SZ] to a better generator against space bounded algorithms!  Potentially, can then recursively apply [SZ] and get better derandomization of RL (after constant number of iterations may get RL in L 1+  )  Armoni showed an interesting extrapolation between [NZ] and [INW] and as a result got a slight improvement (RL in L 3/2 /(log L) 1/2 )

Thoughts on Improving INW x,y xExt(x,y) Loss at each level: log n (possible entropy in state). + log 1/ έ for extractor seed, where έ =  /n Avoiding loss due to entropy in state: [RR99] Recycle the entropy of the states. Challenge: how to do it when do not know state probabilities? Open: better PRGs against constant width branching programs Even for combinatorial rectangles we do not know “optimal” PRGs

Thoughts on Improving INW x,y xExt(x,y) Loss at each level: log n (possible entropy in state). + log 1/ έ for extractor seed, where έ =  /n Avoiding loss due to extractor seeds: Can we recycle y from previous computation? Challenge: contain dependencies … Do we need a seed at all? Use seedless extractors instead? x Ext(x,y(x))

Thoughts on Improving INW x,y xExt(x,y) Loss at each level: log n (possible entropy in state). + log 1/ έ for extractor seed, where έ =  /n Extractor seed is long because we need to work with small error έ =  /n Error reduction for PRGs? If use error έ =  /(log n) sequence still has some unpredictability property, is it usable? (Yes for SL [R04,RozVad05]!)

Final Comment on Improving INW  Perhaps instead on reducing the loss per level we should reduce the number of levels?  This means that at each level the number of pseudorandom strings we have should increase more rapidly (e.g., quadraticaly).  Specific approach based on ideas from Cryptography (constructions of PRFs based on PR Synthesizers [NR]), more complicated to apply here.

Its all About Graph Connectivity  Directed Connectivity captures NL  Undirected Connectivity is in L [R04].  Oblivious derandomization: pseudo-converging walks for consistently labelled regular digraphs [R04,RTV05]  Where is RL on this scale?  Connectivity for digraphs w/polynomial mixing time [RTV05] Outgoing edges have labels. Consistent labelling means that each label forms a permutation on vertices A walk on consistently labelled graph cannot lose entropy

Connectivity for undirected graphs [R04] Connectivity for regular digraphs [RTV05], Pseudo-converging walks for consistently-labelled, regular digraphs [R04, RTV05] Pseudo-converging walks for regular digraphs [RTV05] Connectivity for digraphs w/polynomial mixing time [RTV05] RL in L Suffice to prove RL=L Towards RL vs. L? It is not about reversibility but about regularity In fact it is about having estimates on stationary probabilities [CRV07]

Some More Open Problems  Pseudo-converging walks on an (inconsistently labelled) clique. (Similarly, universal traversal sequence).  Undirected Dirichlet Problem:  Input: undirected graph G, a vertex s, a set B of vertices, a function f: B → [0, 1].  Output: estimation of f(b) where b is the entry point of the random walk into B.

Conclusions  Richness of research directions and open problems towards RL=L and beyond:  PRGs against space bounded computations  Directed connectivity. Even if you think that NL=L is plain crazy, many interesting questions and some beautiful research …

Widescreen Test Pattern (16:9) Aspect Ratio Test (Should appear circular) 16x9 4x3