2. 1 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 1 1 2 MIUR-COFIN: Ravello, september 19-21

3. 2 Fix a language INPUT: Integers OUTPUT: A random word with occurrences of APPLICATIONS: Approximate counting Testing and complexity analysis Statistical analysis of biological sequences THE RGFO L PROBLEM

4. 3 Deep correlation with counting [Flajolet, Van Cutsem, Zimmermann 1994] Best algorithm for regular languages working in time [Denise, Roques, Termier 2000] Is there an algorithm ? OUR CONTRIBUTION Answer is YES for L regular, M = 2 RANDOM GENERATION AND COUNTING

5. 4 STANDARD TECHNIQUE Minimal automaton for L Language recognized by arithmetical operations (due to the precomputation of all )

6. 5 STANDARD TECHNIQUE: Arithmetical ops, minimal automaton has one state (1) (2) 1 1 A VERY SIMPLE EXAMPLE 1 1 1 0 4 3 2 11 3 6 4 111 ALTERNATIVE SOLUTION: Arithmetical ops 011111 6 21 (1’)

7. 6 Standard technique uses equations of type RECURRENCES WITH 1-dim SHIFTS Alternative solution uses equations of type Theorem: If L is regular, then there exist polynomials s.t. verifies recurrences of type (2).

8. 7 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. Example: M=2, Move(, 1, forward) An Forward (Backward) move in the direction s uses THE FUNCTION “MOVE”

9. 8 Given, we first compute (GB Bases) do not vanish in for Then, an algorithm holds if the coefficients A FIRST ATTEMPT Phase 1: Computation of Phase 2: Random generation

10. 9 SOLUTION for M=2: Consider the recurrence equation (with constant coefficients) directly associated with a rational function and define a procedure SmartMove(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: SSW W S W S W What if the leading and the least coefficients vanish?

11. 10 Fact 1 In the grid there are O(n 1 +n 2 ) points where the coefficients of recurrences of type (2) vanish. Fact 3 The cost (time and space) of a call SmartMove (M(x,y),dir) that occurs inside RandomGen(n 1,n 2 ) is O(n 1 +n 2 ) Fact 4 The cost (time and space) of h calls SmartMove(M(x h,y h ),dir h ) that occur inside RandomGen(n 1,n 2 ) is O(max(h, n 1 +n 2 )) Fact 2 RandomGen(n 1,n 2 ) calls SmartMove() O(n 1 +n 2 ) times Theorem RandomGen(n 1,n 2 ) runs in time (and space) O(n 1 +n 2 )

12. 11 There exists an O( n 1 +n 2 ) algorithm for the RGFO L 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. CONCLUSIONS