Online algorithms for finding distinct substrings with length and multiple prefix and suffix conditions

TL;DR

This paper presents efficient online algorithms for finding distinct substrings within a string that meet specific prefix and suffix conditions, with applications in network traffic classification.

Contribution

The authors introduce new algorithms with optimal preprocessing and query times for counting and reporting substrings satisfying prefix and suffix constraints in an online setting.

Findings

Preprocessing time is $O(( ext{length}(P)+ ext{length}(S)) imes ext{log} ext{sigma})$.

Query time for counting is $O(|T_i| imes ext{log} ext{sigma})$.

Algorithms are applicable to network traffic classification.

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 $1 \leq i \leq |T|$. In this paper we are interested in computing the set $\mathit{ans_i}$ of distinct substrings $w$ of $T_i$ such that $k_1 \leq |w| \leq k_2$, 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 $|\mathit{ans_i}|$, whereas the reporting problem is to output all elements of $\mathit{ans_i}$, 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 $O(|T_i| \log\sigma + |\mathit{ans_i}|)$ 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.