Online Algorithms for Finding Distinct Substrings with Length and Multiple Prefix and Suffix Conditions

Let two static sequences of strings P and S, representing prefix and suffix conditions respectively, be given as input for preprocessing. For the query, let two positive integers k₁ and k₂ be given, as well as a string T given in an online manner, such that T_i represents the length-i prefix of T for 1 ≤ i≤ | T|. In this paper we are interested in computing the set ans_i of distinct substrings w of T_i such that k₁≤ | w| ≤ k₂, and w contains some p∈ P as a prefix and some s∈ S as a suffix. More specifically, the counting problem is to output | ans_i|, whereas the reporting problem is to output all elements of ans_i, for each iteration i. Let σ denote the alphabet size, and for a sequence of strings A, ‖ A‖ = ∑ _u∈A| u|. Then, we show that after O((‖ P‖ + ‖ S‖ ) log σ) -time preprocessing, the solutions for the counting and reporting problems for each iteration up to i can be output in O(| T_i| log σ) and O(| T_i| log σ+ | ans_i| ) total time. The preprocessing time can be reduced to O(‖ P‖ + ‖ S‖ ) for integer alphabets of size polynomial with regard to ‖ P‖ + ‖ S‖. Our algorithms have possible applications to network traffic classification.