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.
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Random generation of words
with fixed occurrences of symbols
in regular languages
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
OUTPUT: A random word
with occurrences of
THE RGFOL PROBLEM
Fix a language
Is there an algorithm ?
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 )
Minimal automaton for L
Language recognized by
A VERY SIMPLE EXAMPLE
, minimal automaton has one state
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
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.
RandomGen(n1,n2) runs in time (and space) O(n1+n2)
In the gridthere are O(n1+n2) points where the coefficients of recurrences of type (2) vanish.
RandomGen(n1,n2) calls SmartMove() O(n1+n2) times
The cost (time and space) of a call SmartMove (M(x,y),dir) that occurs inside RandomGen(n1,n2) is O(n1+n2)
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))
There exists an O(n1+n2) algorithm for the RGFOL problem(under uniform cost criterion) when is regular.