In this article, we provide the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson [1979, 1983] is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (i.e., where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger [1999] already provided a Maple package treating the nonreduced case, without giving an expression of the generating function or a detailed proof. We provide a complete proof validating the use of the inclusion-exclusion principle. Some formulæ for expected values, variance, and covariance for number of occurrences when considering two arbitrary sets of finite words are given as an application of our methodology.

Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to the (internal) structure of stored words and several splitting procedures used in diverse contexts. The profile of a trie is a parameter that represents the number of nodes (either internal or external) with the same distance from the root. It is a function of the number of strings stored in a trie and the distance from the root. Several, if not all, trie parameters such as height, size, depth, shortest path, and fill-up level can be uniformly analyzed through the (external and internal) profiles. Although profiles represent one of the most fundamental parameters of tries, they have hardly been studied in the past. The analysis of profiles is surprisingly arduous, but once it is carried out it reveals unusually intriguing and interesting behavior. We present a detailed study of the distribution of the profiles in a trie built over random strings generated by a memoryless source. We first derive recurrences satisfied by the expected profiles and solve them asymptotically for all possible ranges of the distance from the root. It appears that profiles of tries exhibit several fascinating phenomena. When moving from the root to the leaves of a trie, the growth of the expected profiles varies. Near the root, the external profiles tend to zero at an exponential rate, and then the rate gradually rises to being logarithmic; the external profiles then abruptly tend to infinity, first logarithmically and then polynomially; they then tend polynomially to zero again. Furthermore, the expected profiles of asymmetric tries are oscillating in a range where profiles grow polynomially, while symmetric tries are nonoscillating, in contrast to most shape parameters of random tries studied previously. Such a periodic behavior for asymmetric tries implies that the depth satisfies a central limit theorem but not a local limit theorem of the usual form. Also the widest levels in symmetric tries contain a linear number of nodes, differing from the order $n/\sqrt{\log n}$ for asymmetric tries, n being the size of the trees. Finally, it is observed that profiles satisfy central limit theorems when the variance goes unbounded, while near the height they are distributed according to Poisson laws. As a consequence of these results we find typical behaviors of the height, shortest path, fill-up level, and depth. These results are derived here by methods of analytic algorithmics such as generating functions, Mellin transform, Poissonization and de-Poissonization, the saddle-point method, singularity analysis, and uniform asymptotic analysis.

SUMMARY: We present a Web server where the SYSTERS cluster set of the non-redundant protein database consisting of sequences from SWISS-PROT and PIR is being made available for querying and browsing. The cluster set can be searched with a new sequence using the SSMAL search tool. Additionally, a multiple alignment is generated for each cluster and annotated with domain information from the Pfam protein family database. AVAILABILITY: The server address is http://www.dkfz-heidelberg.de/tbi/services/cluster/ systersform

MOTIVATION: We want to provide biologists with a fast and sensitive scanning tool for searching local alignments of a protein query sequence against databases of protein multiple alignments, such as ProDom. Conversely, we want to provide a tool for locally aligning a protein multiple alignment query against a protein database such as SWISSPROT. RESULTS: We developed the program SSMAL (Shuffling Similarities with Multiple Alignments) which utilizes features of the Blast (Altschul et al., J. Mol. Biol., 215, 403-410, 1990) algorithm and part of the Blast code. Our software allows both scanning of multiple alignments and searching with a multiple alignment. Deletions in the multiple alignment only are handled and a SSMAL search may miss some similarities found by a profile search. However, an SSMAL scan of a database such as ProDom would be 20-30 times faster that a profile scan. In the worst case, a SSMAL search is approximately 9 times faster than a profile search. AVAILABILITY: http://www.dkfz-heidelberg.de/tbi/ people/nicodeme and follow the hyperlink SSMAL. CONTACT: [email protected]