  In this paper, we explore worst-case solutions for the problems of single and
multiple matching on strings in the word RAM model with word length w. In the
first problem, we have to build a data structure based on a pattern p of length
m over an alphabet of size sigma such that we can answer to the following
query: given a text T of length n, where each character is encoded using
log(sigma) bits return the positions of all the occurrences of p in T (in the
following we refer by occ to the number of reported occurrences). For the
multi-pattern matching problem we have a set S of d patterns of total length m
and a query on a text T consists in finding all positions of all occurrences in
T of the patterns in S. As each character of the text is encoded using log
sigma bits and we can read w bits in constant time in the RAM model, we assume
that we can read up to (w/log sigma) consecutive characters of the text in one
time step. This implies that the fastest possible query time for both problems
is O((n(log sigma/w)+occ). In this paper we present several different results
for both problems which come close to that best possible query time. We first
present two different linear space data structures for the first and second
problem: the first one answers to single pattern matching queries in time
O(n(1/m+log sigma/w)+occ) while the second one answers to multiple pattern
matching queries to O(n((log d+log y+log log d)/y+log sigma/w)+occ) where y is
the length of the shortest pattern in the case of multiple pattern-matching. We
then show how a simple application of the four russian technique permits to get
data structures with query times independent of the length of the shortest
pattern (the length of the only pattern in case of single string matching) at
the expense of using more space.
