Computing Distributions using Random Walks on Graphs Guy Kindler Guy Kindler DIMACS Dan Romik Dan Romik Weizmann Institute of Science

Computing distributions [Knuth, Yao 76] Given a source of random bits, Given a source of random bits, output a sample with given distribution D. output a sample with given distribution D.

1 1 FormalizationFormalization Example: Compute D={(“0”,1/2), (“1”,1/4), (“2”,1/4)} “H”=right “T”=left “1” 0 0 Interpretation: Computing a distribution using a random walk on a binary tree.

Infinite trees are sometimes needed D={(“0”,2/3), (“1”,1/3)} D={(“0”,2/3), (“1”,1/3)} oRequirement: Output is reached with probability 1 o[Knuth, Yao 76] Output can be reached in expected time Ent(D)+O(1) Tight!Tight!

Some other models o[Romik ’99] Generate dist. B from dist. A in optimal time. o[von Neumann ’51] Generate unbiased coins from biased ones (when bias is unknown). o[Keane & O’Brien ’94] Generate f(p) -biased coins from p - biased ones. o[Peres & Nacu ’03] Generate f(p) -biased in “good time”. o[Mossel & Peres ’03] Generate f(p) -biased coins from p - biased, using a finite graph.

Finite state generators “0” “1” “0” “1” “0” Output: 110… [Knuth+Yao]:

Finite state generators “0” “1” “0” “1” “0” oInterpretation – binary representation: Generating a random variable on [0,1] using a random walk on a graph oDefinition: A distribution function is computable, if it is the output distribution of some f.s.g. oQuestion [Knuth+Yao]: which distributions are computable? smooth/analytic Output: 110…

History of the problem o[Knuth+Yao ’76] Computable analytic density functions must be polynomials with rational coefficients o[Yao ’84] The roots of such functions must be rational o[this work] 1.All functions with above properties can be computed 2.Allowing smooth functions does not add computable functions.

We’ll discuss… [Theorem] Let D be a distribution with density function f. If of is a non-negative polynomia l owith rational coefficients oand no irrational roots in [0,1], then D is computable. “0” “1” “0” “1” “0”

Generating some distributions uniform distribution: Generating max(X,Y): o run two f.s.g’s “in parallel” o output the maximum “0” “1” oAll distributions with density of the form: f(x)=c x m (1-x) n [Knuth+Yao ’76] oAll order statistics of independent uniform variables

More distributions [Knuth+Yao ’76]: uniform on [a,b], for a, b rational. All distributions with density of the form: f(x)=c (x-a) m (b-x) n 1 [a,b] (x) Generating max(X,Y) : orun two f.s.g’s “in parallel” ooutput the maximum

oeasy: if f 1,.., f k are computable, then so is a 1 f 1 +…a k f k (for a i rational) All distributions Answer: all polynomials with rational coefficients, and no irrational roots in [0,1] ! Answer: all polynomials with rational coefficients, and no irrational roots in [0,1] ! All distributions with density of the form: f(x)=c (x-a) m (b-x) n 1 [a,b] (x) Question: what is the set of rational mixtures of such functions ? Proof: 1. Geometric in nature 2. Non-constructive Proof: 1. Geometric in nature 2. Non-constructive Q.E.D. !

ConclusionsConclusions oWe solved the computability problem in the f.s.g. model, for smooth functions. oWe have no good bounds on complexity (size of graph) in this model. Open problems oSolve for other computational models (stack automaton? [Yao84] ) oSolve the general computablity question (no smoothness restriction) oSolve the complexity question

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