Pseudorandom Walks: Looking Random in

The Long Run or All The Way?

Omer ReingoldOmer ReingoldWeizmannWeizmann InstituteInstitute

Graph WalksGraph 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 v0, v1,…vi,… s.t v0=s, and (vi,vi+1) is an edge in G.

……GG ss

v0

v4

v3

v2

v1

Alternatively a walk is a sequence of edge-labels a1, a2,…ai (each ai[D]).

For a given G and s, labels ā v0=s,v1,…,vi,

… with (vi-1,vi) beings the edge labeled ai out

of vi-1.

How to Walk a Graph?How to Walk a Graph?

• Random walk - when in doubt, flip a coin:

• At step i, follow edge labeled ai uniformly and independently of all previous labels.

• “If G is expanding enough, walk converges quickly to unique stationary distribution.”

What is a Pseudorandom-Walk What is a Pseudorandom-Walk Generator?Generator?

Short random Short random seedseed

Efficient Efficient Deterministic Deterministic GeneratorGenerator

a1, a2,…ai,…

N,D, pseudorandom walk

• But what is a pseudorandom walk?• First option: v0, v1,…vi 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 vi

• A distribution W on labels a1, a2,…,ai implies a distribution on vertices v0, v1[W],…vi[W].

• W is pseudorandom if for every G (size ≤N and degree D), s and i, the distributions vi[W] and vi[U] are statistically close.

--

--

Walks & Space-Bounded Walks & Space-Bounded ComputationsComputations

• Walking a graph requires log N memory – i’th state: (vi,i).

• Already know pseudorandom generators that fool space-bounded algorithms [AKS87, BNS89, Nisan90, INW94]. .

• Shortest seed – log2 N bits.• In fact, pseudorandom walks PRGs that

fool space-bounded algorithms …

R/WR/W

Space Bounded AlgorithmsSpace Bounded Algorithms

.. .. .. ... read only read only

MM

..

.. .. write onlywrite only

..

..

Input

Work

Output:

One Way

RandomRandom

coinscoins

0/1

Space

LL:: deterministic deterministic loglogspacespace

LLkk: deterministic space : deterministic space O(logO(logk k n)n)

RLRL: randomized : randomized loglogspace, space, poly-time one-sided error poly-time one-sided error

((BPLBPL: randomized : randomized loglogspace, space, poly-time two-sided error)poly-time two-sided error)

RL (BPL) vs. LDoes randomness help

space bounded computation?

RL (BPL) vs. LDoes randomness help

space bounded computation?

Given an RL algorithm for language P & input x configuration graph:

0

01

1

11

1

0

0

0poly(|x|)configs

transitions oncurrent random bit

T (running time) ≤ poly(|x|) times

s = start config

t = accept config

PR Walks Derandomize RLPR 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:• xP algorithm accepts w.p. ≥ ½ - • xP algorithm never accepts

A PR walk generator that runs in logspace with seed length O(log N) RL == BPL == L

By defn of RL:• xP random walk from s ends at t w.p. ≥ ½• xP t unreachable from s

Where are we at?Where are we at?

• PR walks on directed graphs PRGs that fool space-bounded algorithms.

• Sufficiently efficient such PRGs imply RL = L via “Oblivious Derandomization”

• Nisan’s generator [Nisan 90] - O(log2 N)-long seed RL L2

• ([Saks,Zhou 95] RL L3/2 Oblivious derandomiztion is not the most powerful)

• Pseudorandom generators for space bounded computations – powerful derandomization tool

UndirectedUndirected Connectivity 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 1960’s (context of AI, mazes, wiring of circuits, …).

Work also for the directed case. Require linear space

……GG ss tt

Undirected Connectivity in Undirected Connectivity in RLRL• [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 LUndirected Connectivity in L

Assume G regular and non-bipartite

The Algorithm [R]The Algorithm [R]

• ĜĜ has constant degree. • Each connected component of ĜĜ an

expander. v in GG define the set Cv={<v,*>} in ĜĜ.

• u and v are connected Cu and Cv are in the same connected component.

……GGss

tt

Assume G regular and non-bipartite

……ĜĜ

logspace transformation

highly connected; logarithmic diameter; random walk converges to uniform in logarithmic number of steps

What about PR Walks? What about PR Walks?

• An edge between Cu and Cv in ĜĜ “projects” to a polynomial path between u and v in GG

• GG is connected ĜĜ an expander log path in ĜĜ converges to uniform projects to a poly path in GG that converges to uniform

• The projection is logspace • “Oblivious of G”G”, if GG is consistently labelled

……GGss

tt ……ĜĜ

Cu

Cvvu

Labellings of Regular Labellings of Regular DigraphsDigraphs

• 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)

32 1 1

243

4u v

Pseudo-Converging WalksPseudo-Converging Walks

Goal: walk Gen(U) converges to stationary distribution - vt[Gen(U)] -close to stationary.

Thm1[Thm1[R04,RTV05R04,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 Short random seedseed

Efficient Efficient Deterministic Deterministic Generator Generator Gen

a1, a2,…ai,…

at

N,D,,…,…

polypoly((1/1/))..log(ND/)

Do Pseudo-Converging Walks Do Pseudo-Converging Walks Suffice for Derandomize RL?Suffice for Derandomize RL?

• [RTV05] A new complete promise problem for RL: st connectivity on rapidly-mixing digraphs…

s

t

• xP random walk from s ends at t w.p. ≥ ½• xP t unreachable from s

Graph G that is poly mixing with stationary distribution:• Layer uniform• Within layer i = distribution of alg’s config at time i

Graph G that is poly mixing• xP polynomial weight of both s and t• xP t unreachable from s ( t has zero weight)

For Oblivious Derandomization For Oblivious Derandomization – Regular is Good Enough– 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 What is it About Consistent Labelling?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 toprove RL=L

Summary on RL vs. LSummary 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 Fixed Walks with Pseudorandom PropertiesProperties

• 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-Nlog N UTS.• Explicit polynomial-length UTS only for very

few and limited cases (e.g., cycles [Istrail88]).

Fixed Walks with Pseudorandom Fixed Walks with Pseudorandom PropertiesProperties Cont. 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 RemarksConcluding 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.