arXiv subjects

Carlo Tosoni

Publications and source records attributed to Carlo Tosoni.

5 recordsLinked to original sources

arXiv2026

Faster Cache-Efficient Pattern Matching for Deterministic Wheeler Pangenome Graphs

Pattern matching on strings is regarded as one of the core operations in computer science. Although researchers proposed several solutions to this problem, some of the most elegant and widely used approaches are based on the renowned Burrows-Wheeler transform (BWT). The success of the BWT lies in its pattern matching algorithm known as backward search, which is not only near-optimal in the RAM model, but also runs directly on a compressed representation of the input string. More recently, the backward search has been generalized to Wheeler deterministic finite automata (DFAs), a subclass of standard DFAs, without losing its near-optimal time efficiency. Similarly to the case of strings, this pattern matching algorithm for Wheeler DFAs has found applications in bioinformatics, where researchers have shown that specific pangenome graphs of human chromosomes can be transformed into Wheeler DFAs and consequently indexed using this strategy. However, this BWT-based index on Wheeler DFAs inherited a significant drawback from the original backward search, namely the high number of I/O operations triggered during the algorithm execution, which are in the worst-case lower-bounded by the length of the pattern. In this paper, we address this limitation by proposing the first cache-friendly algorithm specifically designed for Wheeler DFAs. Our new data structure reduces the number of I/O operations by employing a strategy analogous to the suffix array: it interleaves a binary search with fast sequential scans of the automaton. We empirically validate this new indexing strategy by running our algorithm on real-world Wheeler pangenome graphs. We show that while our data structure can use up to 15 times the space required by the backward search, it can also be 500 times faster and able to process a single character of the pattern in less than 3 ns.

cs.DS↗

arXiv2025

New Entropy Measures for Tries with Applications to the XBWT

Entropy quantifies the number of bits required to store objects under certain given assumptions. While this is a well established concept for strings, in the context of tries the state-of-the-art regarding entropies is less developed. The standard trie worst-case entropy considers the set of tries with a fixed number of nodes and alphabet size. However, this approach does not consider the frequencies of the symbols in the trie, thus failing to capture the compressibility of tries with skewed character distributions. On the other hand, the label entropy [FOCS '05], proposed for node-labeled trees, does not take into account the tree topology, which has to be stored separately. In this paper, we introduce two new entropy measures for tries - worst-case and empirical - which overcome the two aforementioned limitations. Notably, our entropies satisfy similar properties of their string counterparts, thereby becoming very natural generalizations of the (simpler) string case. Indeed, our empirical entropy is closely related to the worst-case entropy and is reachable through a natural extension of arithmetic coding from strings to tries. Moreover we show that, similarly to the FM-index for strings [JACM '05], the XBWT of a trie can be compressed and efficiently indexed within our k-th order empirical entropy plus o(n) bits, with n being the number of nodes. Interestingly, the space usage of this encoding includes the trie topology and the upper-bound holds for every k sufficiently small, simultaneously. This XBWT encoding is always strictly smaller than the original one [JACM '09] and we show that in certain cases it is asymptotically smaller.

cs.DS↗

arXiv2025

Indexing Tries within Entropy-Bounded Space

We study the problem of indexing and compressing tries using a BWT-based approach. Specifically, we consider a succinct and compressed representation of the XBWT of Ferragina et al.\ [FOCS '05, JACM '09] corresponding to the analogous of the FM-index [FOCS '00, JACM '05] for tries. This representation allows to efficiently count the number of nodes reached by a given string pattern. To analyze the space complexity of the above trie index, we propose a proof for the combinatorial problem of counting the number of tries with a given symbol distribution. We use this formula to define a worst-case entropy measure for tries, as well as a notion of k-th order empirical entropy. In particular, we show that the relationships between these two entropy measures are similar to those between the corresponding well-known measures for strings. We use these measures to prove that the XBWT of a trie can be encoded within a space bounded by our k-th order empirical entropy plus a o(n) term, with n being the number of nodes in the trie. Notably, as happens for strings, this space bound can be reached for every sufficiently small k simultaneously. Finally, we compare the space complexity of the above index with that of the r-index for tries proposed by Prezza [SODA '21] and we prove that in some cases the FM-index for tries is asymptotically smaller.

cs.DS↗

arXiv2025

Encoding Co-Lex Orders of Finite-State Automata in Linear Space

The Burrows-Wheeler transform (BWT) is a string transformation that enhances string indexing and compressibility. Cotumaccio and Prezza [SODA '21] extended this transformation to nondeterministic finite automata (NFAs) through co-lexicographic partial orders, i.e., by sorting the states of an NFA according to the co-lexicographic order of the strings reaching them. As the BWT of an NFA shares many properties with its original string variant, the transformation can be used to implement indices for locating specific patterns on the NFA itself. The efficiency of the resulting index is influenced by the width of the partial order on the states: the smaller the width, the faster the index. The most efficient index for arbitrary NFAs currently known in the literature is based on the coarsest forward-stable co-lex (CFS) order of Becker et al. [SPIRE '24]. In this paper, we prove that this CFS order can be encoded within linear space in the number of states in the automaton. The importance of this result stems from the fact that encoding such an order in linear space represents a big first step in the direction of building the index based on this order in near-linear time -- the biggest open research question in this context. The currently most efficient known algorithm for this task run in quadratic time in the number of transitions in the NFA and are thus infeasible to be run on very large graphs (e.g., pangenome graphs). At this point, a near-linear time algorithm is solely known for the simpler case of deterministic automata [Becker et al., ESA '23] and, in fact, this algorithmic result was enabled by a linear space encoding for deterministic automata [Kim et al., CPM '23].

cs.DS↗

arXiv2024

Indexing Finite-State Automata Using Forward-Stable Partitions

An index on a finite-state automaton is a data structure able to locate specific patterns on the automaton's paths and consequently on the regular language accepted by the automaton itself. Cotumaccio and Prezza [SODA '21], introduced a data structure able to solve pattern matching queries on automata, generalizing the famous FM-index for strings of Ferragina and Manzini [FOCS '00]. The efficiency of their index depends on the width of a particular partial order of the automaton's states, the smaller the width of the partial order, the faster is the index. However, computing the partial order of minimal width is NP-hard. This problem was mitigated by Cotumaccio [DCC '22], who relaxed the conditions on the partial order, allowing it to be a partial preorder. This relaxation yields the existence of a unique partial preorder of minimal width that can be computed in polynomial time. In the paper at hand, we present a new class of partial preorders and show that they have the following useful properties: (i) they can be computed in polynomial time, (ii) their width is never larger than the width of Cotumaccio's preorders, and (iii) there exist infinite classes of automata on which the width of Cotumaccio's pre-order is linearly larger than the width of our preorder.

cs.FL↗