Sequence binary decision diagram: Minimization, relationship to acyclic automata, and complexities of Boolean set operations

Abstract

The manipulation of large sequence data is one of the most important problems in string processing. In this paper, we discuss a new data structure for storing and manipulating sets of strings, called Sequence Binary Decision Diagrams (sequence BDDs), which has recently been introduced by Loekito et al. (Knowl. Inf. Syst., 24(2), 235-268, 2009) as a descendant of both acyclic DFAs (ADFAs) and binary decision diagrams (BDDs). Sequence BDDs can compactly represent sets of sequences similarly to minimal ADFAs, and allow efficient set operations inherited from BDDs. We study fundamental properties of sequence BDDs, such as the characterization of minimal sequence BDDs by reduced sequence BDDs, non-trivial relationships between sizes of minimal sequence BDDs and minimal ADFAs, the complexities of minimization, Boolean set operations, and sequence BDD construction. We also show experimental results for real and artificial data sets.

Extraction of templates from phrases using Sequence Binary Decision Diagrams

“…This contraction requires O (|δ|), computational time complexity where δ is the number of edges. In contrast, as noted earlier, the application of Loekito et al (2010) and Denzumi et al (2016) effects a direct, incremental construction of the minimal representation.…”

“…Let |X| denote the number of elements of a given set X. It has been recently proven (Denzumi, Yoshinaka, Arimura, and Minato, 2016), that minDFA requires O(|δ|) whereas SeqBDD O(|V |) computational space complexity, and furthermore, |V | ≤ |δ| where minDFA can be |Σ| times larger than the former (Denzumi, Yoshinaka, Arimura, and Minato, 2016). In other words, a SeqBDD is theoretically never larger than the equivalent minDFA.…”

“…We provide the former in Section 3.1, and the latter in Section 3.2. The basis of the SeqBDD can be found in Loekito, Bailey and Pei (2010), Denzumi et al (2016), and the definitions and algorithms necessary to replicate our method are all provided in this article.…”

“…In order to cope with this strictness, we relax the algorithm given by (Denzumi et al 2011): We call the proposed graph structure Relaxed SeqBDD and explain it below. Since we cannot expect all examples to occur in reality, we instead look at False phrases represented by the graph.…”

DenseZDD: A Compact and Fast Index for Families of Sets †

“…ZDD nodes correspond to labeled edges in the automaton. However, the size of such an automaton via straightforward translation can be Θ(n) times larger than the original ZDD [11] in the worst case. Therefore, we want to perform member(v, S) operations in O(|S|) time on ZDDs.…”