Viterbi Algorithm Generalized for n-Tape Best-Path Search

We present a generalization of the Viterbi algorithm for identifying the path with minimal (resp. maximal) weight in a n-tape weighted finite-state machine (n-WFSM), that accepts a given n-tuple of input strings (s_1,... s_n). It also allows us to compile the best transduction of a given input n-tuple by a weighted (n+m)-WFSM (transducer) with n input and m output tapes. Our algorithm has a worst-case time complexity of O(|s|^n |E| log (|s|^n |Q|)), where n and |s| are the number and average length of the strings in the n-tuple, and |Q| and |E| the number of states and transitions in the n-WFSM, respectively. A straight forward alternative, consisting in intersection followed by classical shortest-distance search, operates in O(|s|^n (|E|+|Q|) log (|s|^n |Q|)) time.

[1]  André Kempe,et al.  WFSM Auto-intersection and Join Algorithms , 2005, FSMNLP.

[2]  Arto Salomaa,et al.  Semirings, Automata, Languages , 1985, EATCS Monographs on Theoretical Computer Science.

[3]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[4]  Stephen Warshall,et al.  A Theorem on Boolean Matrices , 1962, JACM.

[5]  Jorge E. Mezei,et al.  On Relations Defined by Generalized Finite Automata , 1965, IBM J. Res. Dev..

[6]  Michael J. Fischer,et al.  The String-to-String Correction Problem , 1974, JACM.

[7]  Kalervo Järvelin,et al.  Fuzzy translation of cross-lingual spelling variants , 2003, SIGIR.

[8]  Jason Eisner,et al.  A Note on Join and Auto-Intersection of n-ary Rational Relations , 2004 .

[9]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[10]  Andrew J. Viterbi,et al.  Error bounds for convolutional codes and an asymptotically optimum decoding algorithm , 1967, IEEE Trans. Inf. Theory.

[11]  Tero Harju,et al.  The Equivalence Problem of Multitape Finite Automata , 1991, Theor. Comput. Sci..

[12]  André Kempe,et al.  A Class of Rational n-WFSM Auto-intersections , 2005, CIAA.

[13]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[14]  Mehryar Mohri,et al.  A Rational Design for a Weighted Finite-State Transducer Library , 1997, Workshop on Implementing Automata.

[15]  Martin Kay,et al.  Nonconcatenative Finite-State Morphology , 1987, EACL.

[16]  Dana S. Scott,et al.  Finite Automata and Their Decision Problems , 1959, IBM J. Res. Dev..

[17]  André Kempe,et al.  Multi-tape Automata with Symbol Classes , 2006, CIAA.

[18]  Seth Pettie,et al.  A new approach to all-pairs shortest paths on real-weighted graphs , 2004, Theor. Comput. Sci..

[19]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[20]  Richard Bellman,et al.  ON A ROUTING PROBLEM , 1958 .

[21]  Martin Kay,et al.  Regular Models of Phonological Rule Systems , 1994, CL.

[22]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956 .

[23]  George Anton Kiraz,et al.  Multitiered nonlinear morphology using multitape finite automata: a case study on Syriac and Arabic , 2000, CL.

[24]  Mehryar Mohri,et al.  Semiring Frameworks and Algorithms for Shortest-Distance Problems , 2002, J. Autom. Lang. Comb..