Pseudorandom Walks: Looking Random in The Long Run or All The Way? Omer Reingold Weizmann Institute

Graph Walks G - a (possibly directed) graph. Assume G is out- regular (all out degrees are D). s - a vertex. Assume s sink connected component (all vertices reachable from s can also reach s). A walk in G from s: a sequence v 0, v 1,…v i,… s.t v 0 =s, and (v i,v i+1 ) is an edge in G. … G s v0v0 v4v4 v3v3 v2v2 v1v1 Alternatively a walk is a sequence of edge-labels a 1, a 2,…a i (each a i [D]). For a given G and s, labels ā v 0 =s,v 1,…,v i,… with (v i-1,v i ) beings the edge labeled a i out of v i-1.

How to Walk a Graph? Random walk - when in doubt, flip a coin: At step i, follow edge labeled a i uniformly and independently of all previous labels. If G is expanding enough, walk converges quickly to unique stationary distribution.

What is a Pseudorandom-Walk Generator? Short random seed Efficient Deterministic Generator a 1, a 2,…a i,… N,D, pseudorandom walk But what is a pseudorandom walk? First option: v 0, v 1,…v i looks like a random walk to a bounded distinguisher Same as a pseudorandom bit generator … More interesting: think of the walk as the distinguisher: The output of a length-i walk is v i A distribution W on labels a 1, a 2,…,a i implies a distribution on vertices v 0, v 1 [ W ],…v i [ W ]. W is pseudorandom if for every G (size N and degree D), s and i, the distributions v i [ W ] and v i [ U ] are statistically close. - -

Walks & Space-Bounded Computations Walking a graph requires log N memory – ith state: (v i,i)..Already know pseudorandom generators that fool space-bounded algorithms [AKS87, BNS89, Nisan90, INW94]. Shortest seed – log 2 N bits. In fact, pseudorandom walks PRGs that fool space-bounded algorithms … Seems like this talk is reaching a dead end … Lets shift topic and see if they notice …

Walks & Space-Bounded Computations Walking a graph requires log N memory – ith state: (v i,i)..Already know pseudorandom generators that fool space-bounded algorithms [AKS87, BNS89, Nisan90, INW94]. Shortest seed – log 2 N bits. In fact, pseudorandom walks PRGs that fool space-bounded algorithms …

R/W Space Bounded Algorithms.. read only... read only M.. write only write only.. Input Work Output: One Way Randomcoins 0/1 Space L : deterministic logspace L k : deterministic space O(log k n) RL : randomized logspace, poly-time one-sided error ( BPL : randomized logspace, poly-time two-sided error) RL (BPL) vs. L Does randomness help space bounded computation? RL (BPL) vs. L Does randomness help space bounded computation?

Given an RL algorithm for language P & input x configuration graph: poly(|x|) configs transitions on current random bit T (running time) poly(|x|) times s = start config t = accept config PR Walks Derandomize RL Walking the configuration graph with labels ā Running the algorithm with randomness ā a PR walk generator fools the RL algorithm Run your algorithm on the labels of PR walk: x P algorithm accepts w.p. ½ - x P algorithm never accepts == A PR walk generator that runs in logspace with seed length O(log N) RL = BPL = L By defn of RL: x P random walk from s ends at t w.p. ½ x P t unreachable from s

Where are we at? PR walks on directed graphs PRGs that fool space-bounded algorithms. Sufficiently efficient such PRGs imply RL = L viaOblivious Derandomization Nisans generator [Nisan 90] - O(log 2 N)-long seed RL L 2 ([Saks,Zhou 95] RL L 3/2 Oblivious derandomiztion is not the most powerful) Pseudorandom generators for space bounded computations – powerful derandomization tool

Undirected Connectivity Basic graph problem. Extensively studied. Time complexity – well understood: Two linear time algorithms, BFS and DFS, are known and used at least since the 1960s (context of AI, mazes, wiring of circuits, …). Work also for the directed case. Require linear space … G st

Undirected Connectivity in RL [Aleliunas, Karp, Lipton, Lovasz, Rackoff 79] Randomized space O(log N) for USTCON. The algorithm: take a, polynomially long, random walk from s and see if you reach t. Works because the walk converges in poly number of steps to uniform distribution on connected component. Undirected connectivity can be solved also in deterministic logspace [ Savitch 70, Nisan,Szemerédi,Wigderson 92, Armoni,Ta- Shma,Wigderson,Zhou 97, Trifonov 04, R 04 ] Undirected Connectivity in L Assume G regular and non-bipartite

The Algorithm [R] ĜĜ has constant degree. ĜEach connected component of Ĝ an expander. GĜ v in G define the set Cv={ } in Ĝ. u and v are connected Cu and Cv are in the same connected component. … G s t Assume G regular and non-bipartite … Ĝ logspace transformation highly connected; logarithmic diameter; random walk converges to uniform in logarithmic number of steps

What about PR Walks? Ĝ GAn edge between Cu and Cv in Ĝ projects to a polynomial path between u and v in G GĜ Ĝ GG is connected Ĝ an expander log path in Ĝ converges to uniform projects to a poly path in G that converges to uniform The projection is logspace GGOblivious of G, if G is consistently labelled … G s t … Ĝ Cu Cv vu

Labellings of Regular Digraphs Denote by i (v) the ith neighbor of v Inconsistently labelled: u,v,i s.t. i (u)= i (v) Consistently labelled: i i is a permutation (Every regular digraph has a consistent labelling) u v

Pseudo-Converging Walks Goal: walk Gen( U ) converges to stationary distribution - v t [ Gen( U ) ] -close to stationary. Thm1[ R04,RTV05 ] Thm1[ R04,RTV05 ] Gen that is Pseudo-Converging: D-regular, N-vertex, connected,consistently- labelled digraph G, start vertex s, Space and seed length - log(ND/ ) Walk length t = poly(mixing time) … Short random seed Efficient Deterministic Generator Efficient Deterministic Generator Gen a 1, a 2,…a i,… a t,… N,D,,… poly(1/ ). poly(1/ ). log(ND/ )

Do Pseudo-Converging Walks Suffice for Derandomize RL? [RTV05] A new complete promise problem for RL: st connectivity on rapidly-mixing digraphs… s t x P random walk from s ends at t w.p. ½ x P t unreachable from s Graph G that is poly mixing with stationary distribution: Layer uniform Within layer i = distribution of algs config at time i Graph G that is poly mixing x P polynomial weight of both s and t x P t unreachable from s ( t has zero weight)

For Oblivious Derandomization – Regular is Good Enough Thm1 [ R04,RTV05 ] Pseudo-Converging generator for walks on regular consistently-labelled digraphs. Thm2 [ RTV05 ] Pseudo-Converging generator for any regular digraph RL = L Graph labelling is at the heart of oblivious derandomization

What is it About Consistent Labelling? Consider a walk from u and a walk from v following a fixed sequence of labels (e.g., 0000…) If the graph is consistently labelled the two walks will never merge More generally, a walk on a consistently labelled graph never loses entropy. In fact, for RL = L, enough to give a long-enough walk for regular digraphs that (when starting at a uniform vertex) stays uniform.

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 Summary on RL vs. L It is not about reversibility but about regularity In fact it is about having estimates on stationary probabilities [CRV07]

Fixed Walks with Pseudorandom Properties Universal-traversal sequence (UTS) introduced by Cook in the late 70's with the motivation of proving USTCON in logspace. (N,D)-UTS: a sequence of edge labels in [D]. Guides a walk through all of the vertices of any D-regular graph on N vertices. [AKLLR79] poly-long UTS exist (probabilistic). [Nisan 90] Explicit, length-N log N UTS. Explicit polynomial-length UTS only for very few and limited cases (e.g., cycles [Istrail88]).

Fixed Walks with Pseudorandom Properties Cont. Universal-traversal sequences Universal-exploration sequences [Koucky 01]. Like traversal sequences but directions are relative. Thm [ R04 ] log-space constructible universal exploration sequences Thm [ R04,RTV05 ] log-space constructible universal traversal sequences for consistently- labelled digraphs. Open: full fledged universal traversal sequences

Concluding Remarks Considered two flavors of pseudorandom walks Pseudorandom all the way – equivalent to PRG that fool space bounded computations Pseudorandom in the limit – sufficient to derandomize RL Surprising applications of PRG for space bounded computations [ Ind00,Siv02,KNR05,HHR06,… ] 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.