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

Abstract

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 $k_1$ and $k_2$ 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 $1 \le i \le |T|$. In this paper we are interested in computing the set $ans_i$ of distinct substrings w of $T_i$ such that $k_1 \le |w| \le k_2$, and w contains some $p \in P$ as a prefix and some $s \in S$ 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 $\sigma$ denote the alphabet size, and for a sequence of strings A, $\Vert A\Vert =\sum _{u\in A}|u|$. Then, we show that after $O((\Vert P\Vert +\Vert S\Vert )\log \sigma )$-time preprocessing, the solutions for the counting and reporting problems for each iteration up to i can be output in $O(|T_i| \log \sigma )$ and $O(|T_i| \log \sigma + | ans_i |)$ total time. The preprocessing time can be reduced to $O(\Vert P\Vert +\Vert S\Vert )$ for integer alphabets of size polynomial with regard to $\Vert P\Vert +\Vert S\Vert$. Our algorithms have possible applications to network traffic classification.

A Appendix

Below, we show the input sets that match the each of the application signatures described in [19] (Table 1).

Table 1. Input sets corresponding to application signatures