On the Comparison Complexity of the String Prefix-Matching Problem

In this paper we study the exact comparison complexity of the string prefix-matching problem in the deterministic sequential comparison model with equality tests. We derive almost tight lower and upper bounds on the number of symbol comparisons required in the worst case by on-line prefix-matching algorithms for any fixed pattern and variable text. Unlike previous results on the comparison complexity of string-matching and prefix-matching algorithms, our bounds are almost tight for any particular pattern. We also consider the special case where the pattern and the text are the same string. This problem, which we call the string self-prefix problem, is similar to the pattern preprocessing step of the Knuth-Morris-Pratt stringmatching algorithm that is used in several comparison efficient stringmatching and prefix-matching algorithms, including in our new algorithm. We obtain roughly tight lower and upper bounds on the number of symbol comparisons required in the worst case by on-line self-prefix algorithms. Our algorithms can be implemented in linear time and space in the standard uniform-cost random-access-machine model. ∗BRICS – Basic Research in Computer Science, Centre of the Danish National Research Foundation, Department of Computer Science, University of Aarhus, DK-8000 Aarhus C, Denmark. Partially supported by the ESPRIT Basic Research Action Program of the EC under contract #7141 (ALCOM II). Part of the research reported in the paper was carried out while this author was visiting at the Istituto di Elaborazione dell’Informazione, Consiglio Nazionale delle Ricerche, Pisa, Italy, with the support of the European Research Consortium for Informatics and Mathematics postdoctoral fellowship. †Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Belzoni 7, I-35131 Padova, Italy. Partially supported by “Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo” of the Italian National Research Councile under grant number 89.00026.69. ‡Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Belzoni 7, I-35131 Padova, Italy. Parts of the research reported in this paper were carried out while this author was visiting at the Institut Gaspard Monge, Université de Marne-la-Vallée, Noisy-leGrand, France, supported by “Borsa di studi per attività di perfeziomento all’estero” from the University of Padua, and at BRICS, Department of Computer Science, University of Aarhus, Aarhus, Denmark, supported by the Gini Foundation of Padua.