Random generation of words with fixed occurrences of symbols in regular languages

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Random generation of words with fixed occurrences of symbols in regular languages

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Random generation of words with fixed occurrences of symbols in regular languages

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Random generation of words

with fixed occurrences of symbols

in regular languages

2

1

1

A. Bertoni P. Massazza R. Radicioni

1 Università degli Studi di Milano, Dipartimento di Scienze dell'Informazione, via Comelico 39, 20135 Milano, Italy.

2 Università dell’Insubria, Dipartimento di Informatica e Comunicazione, via Mazzini 5, 21100 Varese, Italy

MIUR-COFIN: Ravello, september 19-21

INPUT: Integers

OUTPUT: A random word

with occurrences of

THE RGFOL PROBLEM

Fix a language

- APPLICATIONS:
- Approximate counting
- Testing and complexity analysis
- Statistical analysis of biological sequences

Is there an algorithm ?

OUR CONTRIBUTION

Answer is YESforL regular, M = 2

RANDOM GENERATION AND COUNTING

Deep correlation with counting

[Flajolet, Van Cutsem, Zimmermann 1994]

Best algorithm for regular languages working in time [Denise, Roques, Termier 2000]

arithmetical operations (due to the

precomputation of all )

STANDARD TECHNIQUE

Minimal automaton for L

Language recognized by

ALTERNATIVE SOLUTION:

Arithmetical ops

(1’)

21

6

0

1

1

1

1

1

A VERY SIMPLE EXAMPLE

, minimal automaton has one state

STANDARD TECHNIQUE:

Arithmetical ops

1

1

(1)

1

4

1

3

6

1

2

3

4

0

1

1

1

1

1

(2)

Alternative solution uses equations of type

Theorem:If Lis regular, then there exist polynomials

s.t. verifies recurrences of type (2).

RECURRENCES WITH 1-dim SHIFTS

Standard technique uses equations of type

THE FUNCTION “MOVE”

Def.: Move( , s, sense) computes a matrix of coefficients

from M by means of recurrences of type (2),depending on direction s and on sense sense.

An Forward (Backward) move in the direction s uses

Example: M=2, Move( , 1 , forward)

A FIRST ATTEMPT

Given , we first compute (GB Bases)

Then, an algorithm holds if the coefficients

do not vanish in for

Phase 1:

Computation

of

Phase 2:

Random

generation

W

W

W

SW

SW

SW

S

S

S

What if the leading and the least coefficients vanish?

SOLUTION for M=2:

Consider the recurrence equation (with constant coefficients)

directly associated with a rational function

and define a procedureSmartMove(M,dir)that smartly uses recurrence (3) whenever it is not possible to compute from by means of recurrences of type (2).

Fact:A matrix of coefficients M can be computed by (3)

if are known.

Examples:

Theorem

RandomGen(n1,n2) runs in time (and space) O(n1+n2)

Fact 1

In the gridthere are O(n1+n2) points where the coefficients of recurrences of type (2) vanish.

Fact 2

RandomGen(n1,n2) calls SmartMove() O(n1+n2) times

Fact 3

The cost (time and space) of a call SmartMove (M(x,y),dir) that occurs inside RandomGen(n1,n2) is O(n1+n2)

Fact 4

The cost (time and space) of h calls SmartMove(M(xh,yh),dirh) that

occur inside RandomGen(n1,n2) is O(max(h, n1+n2))

CONCLUSIONS

There exists an O(n1+n2) algorithm for the RGFOL problem(under uniform cost criterion) when is regular.

Future Works

- Extension of the general case to arbitrary alphabets (M>2).

- Extension to unambiguous context-free languages.

- Deep investigation on the nature of recurrences

- Complexity analysis under log. cost criterion.