Linear-size suffix tries

Abstract

Please cite this article in press as: M. Crochemore et al., Linear-size suffix tries, Theoret. Comput. Sci. (2016), http://dx.doi.org/10.1016/j.tcs. 2016.04.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Linear-Size Suffix TriesSuffix trees are highly regarded data structures for text indexing and string algorithms [MCreight 76, Weiner 73]. For any given string w of length n = |w|, a suffix tree for w takes O(n) nodes and links. It is often presented as a compacted version of a suffix trie for w, where the latter is the trie (or digital search tree) built on the suffixes of w. Here the compaction process replaces each maximal chain of unary nodes with a single arc. For this, the suffix tree requires that the labels of its arcs are substrings encoded as pointers to w (or equivalent information). On the contrary, the arcs of the suffix trie are labeled by single symbols but there can be Θ(n 2 ) nodes and links for suffix tries in the worst case because of their unary nodes. It is an interesting question if the suffix trie can be stored using O(n) nodes. We present the linear-size suffix trie, which guarantees O(n) nodes. We use a new technique for reducing the number of unary nodes to O(n), that stems from some results on antidictionaries. For instance, by using the linear-size suffix trie, we are able to check whether a pattern p of length m = |p| occurs in w in O(m log |Σ|) time and we can find the longest common substring of two strings w 1 and w 2 in O((|w 1 | + |w 2 |) log |Σ|) time for an alphabet Σ.

Fast Label Extraction in the CDAWG

“…to edges in the suffix tree of T . This parallels the grammar implicit in [6] and explicit in [21], whose nonterminals correspond to unary paths in the suffix trie of T , i.e. to edges in the suffix tree of T .…”

“…We achieve this by dropping the run-length-encoded representation of the Burrows-Wheeler transform of T , used in [2], and by exploiting the fact that the reversed CDAWG induces a context-free grammar that produces T and only T , as described in [1]. A related grammar, already implicit in [6], has been concurrently exploited in [21] to achieve similar bounds to ours. Note that in some strings, for example in the family T i for i ≥ 0, where T 0 = 0 and T i = T i−1 iT i−1 , the length of the string grows exponentially in the size of the CDAWG, thus shaving an O(log log n) term is identical to shaving an O(log e T ) term.…”

Linear-Size CDAWG: New Repetition-Aware Indexing and Grammar Compression

“…Proposition 1 (Crochemore et al [5]). For a text T of length n, the linear-size suffix trie LSTrie(T ) for T can be stored in O(n log n) bits of space supporting reconstruction of the label of a given edge in O(ℓ) time, where ℓ is the length of the edge label.…”

“…Recently, Crochemore et al [5] proposed a compact variant of a suffix trie, called linear-size suffix trie (or LSTrie, for short), denoted LSTrie(T ). It is a compacted tree with the topology and the size similar to STree(T ), but has no indirect references to a text T (See Fig.…”

“…Crochemore et al's method [5] does not regard the order of decoding characters on an edge label. This implies that LSTrie(T ) needs O(ℓ) worst case time to read any prefix of an edge label of length ℓ.…”