This paper presents a method for direct building of minimal acyclic finite states automaton which recognizes a given finite list of words in lexicographical order. The size of the temporary automata which are necessary for the construction is less than the size of the resulting minimal automata plus the length of one of the longest words in the list. This property is the main advantage of our method.

This paper presents automaton construction algorithms based on the method for direct building of minimal acyclic finite states automaton for a given list [Mi98]. A detailed presentation of the base algorithm with correctness and complexity proofs is given. The memory complexity of the base algorithm is O(m) and the worst-case time complexity is O(n log(m)), where n is the total number of letters in the input list, m is the size of the resulting minimal automaton. Further we present algorithms for direct construction of minimal automaton presenting the union, intersection and difference of acyclic automata. In the cases of intersection and difference only the first input automaton has to be acyclic. The memory complexity of those construction algorithms is O(m), and the time complexity is O(n log(m)) for union and O(n1 + n log(m)) for intersection and difference, where n1 is the total number of letters in the first automaton language, n is the number of all letters in the resulting aut...

At STOC 2006 and CRYPTO 2007, Beimel et al. introduced a set of privacy requirements for algorithms that solve search problems. In this paper, we consider the longest common subsequence (LCS) problem as a private search problem, where the task is to find a string of (or embedding corresponding to) an LCS. We show that deterministic selection strategies do not meet the privacy guarantees considered for private search problems and, in fact, may “leak ” an amount of information proportional to the entire input. We then put forth and investigate several privacy structures for the LCS problem and design new and efficient output sampling and equivalence protecting algorithms that provably meet the corresponding privacy notions. Along the way, we also provide output sampling and equivalence protecting algorithms for finite regular languages, which may be of independent interest. 1