arXiv subjects

Hiroki Arimura

Publications and source records attributed to Hiroki Arimura.

At least 19 recordsLinked to original sources

arXiv2026

Fully Persistent Dynamic LCE via AVL Trees and AVL Grammars

We study fully persistent dynamic strings with equality and longest common extension (LCE) queries. Straightforward full persistence is problematic for the splay-based FeST structure, since the same unbalanced past version can be reused indefinitely and the usual amortized analysis no longer applies. We give a fully persistent dynamic LCE structure, called FeAVL, based on path copying over AVL trees. For an operation involving string(s) of total length $n$, it supports split, concatenate, and single-character updates in worst-case $O(\log n)$ time, equality in worst-case $O(\log n)$ time w.h.p., and LCE in worst-case $O(\log n+\log^2\ell)$ time w.h.p., where $\ell$ is the answer; each update creates only $O(\log n)$ new permanent nodes. We also give a grammar-compressed instantiation via AVL grammars: starting from an initial grammar of size $g_0$, after $U$ updates, the total number of permanent grammar nodes is $O(g_0+I+U\log n_{\max})$, where $I$ is the number of inserted fresh characters and $n_{\max}$ is the maximum string length appearing during the update sequence.

cs.DS↗

arXiv2026

Output-Sensitive Construction of CDAWGs from BWT-Runs

The compact directed acyclic word graph (CDAWG) of a string can be viewed in two equivalent ways: as the edge-compacted DAWG of the string, and as the DAG obtained from the suffix tree by merging the nodes whose subtrees are isomorphic. By exploiting these two views in opposite directions, we show how to build, for the (reversed) input string of length $n$, the CDAWG with $e_L$ edges in $O(e_L\log n\log(n/r))$ time with $O(r\log(n/r)+e_L)$ words of working space, provided that the fully functional compressed suffix tree of Gagie, Navarro, and Prezza of size $O(r\log(n/r))$ is available. Here, $r$ denotes the number of BWT-runs of the input string.

cs.DS↗

arXiv2026

Algebraic Model Counting for Global Analysis of Optimal Decision Trees

Ensuring model reliability in Explainable AI requires a global assessment of the hypothesis space. We propose a formal framework for the exhaustive analysis of optimal and near-optimal decision trees, called Algebraic Decision Tree Counting (ADTC). Inspired by Algebraic Model Counting (AMC) in knowledge representation, ADTC reformulates diverse analytical tasks, such as optimization, counting, and sampling, into a unified sum-of-products computation over a semiring $R$. While the hypothesis space of decision trees is doubly exponential with respect to the maximum depth $Δ$, our dynamic programming algorithm achieves $O^*(n^{O(Δ)})$ time complexity in the number of features $n$, where $O^*$ suppresses polynomial factors. To handle complex constraints consisting of multiple tree metrics, we introduce model behavior tensors that aggregate semiring values via convolution products over a tensor semiring. This algebraic approach efficiently constructs a model profile that captures the global landscape and trade-offs between criteria such as accuracy, size, and fairness. We demonstrate the utility of our software, emtrees, on real-world datasets, illustrating how ADTC facilitates evidence-based model selection in sensitive domains.

cs.AI↗

arXiv2024

Finding Diverse Strings and Longest Common Subsequences in a Graph

In this paper, we study for the first time the Diverse Longest Common Subsequences (LCSs) problem under Hamming distance. Given a set of a constant number of input strings, the problem asks to decide if there exists some subset $\mathcal X$ of $K$ longest common subsequences whose diversity is no less than a specified threshold $Δ$, where we consider two types of diversities of a set $\mathcal X$ of strings of equal length: the Sum diversity and the Min diversity defined as the sum and the minimum of the pairwise Hamming distance between any two strings in $\mathcal X$, respectively. We analyze the computational complexity of the respective problems with Sum- and Min-diversity measures, called the Max-Sum and Max-Min Diverse LCSs, respectively, considering both approximation algorithms and parameterized complexity. Our results are summarized as follows. When $K$ is bounded, both problems are polynomial time solvable. In contrast, when $K$ is unbounded, both problems become NP-hard, while Max-Sum Diverse LCSs problem admits a PTAS. Furthermore, we analyze the parameterized complexity of both problems with combinations of parameters $K$ and $r$, where $r$ is the length of the candidate strings to be selected. Importantly, all positive results above are proven in a more general setting, where an input is an edge-labeled directed acyclic graph (DAG) that succinctly represents a set of strings of the same length. Negative results are proven in the setting where an input is explicitly given as a set of strings. The latter results are equipped with an encoding such a set as the longest common subsequences of a specific input string set.

cs.DS↗

arXiv2024

Computing Minimal Absent Words and Extended Bispecial Factors with CDAWG Space

A string $w$ is said to be a minimal absent word (MAW) for a string $S$ if $w$ does not occur in $S$ and any proper substring of $w$ occurs in $S$. We focus on non-trivial MAWs which are of length at least 2. Finding such non-trivial MAWs for a given string is motivated for applications in bioinformatics and data compression. Fujishige et al. [TCS 2023] proposed a data structure of size $Θ(n)$ that can output the set $\mathsf{MAW}(S)$ of all MAWs for a given string $S$ of length $n$ in $O(n + |\mathsf{MAW}(S)|)$ time, based on the directed acyclic word graph (DAWG). In this paper, we present a more space efficient data structure based on the compact DAWG (CDAWG), which can output $\mathsf{MAW}(S)$ in $O(|\mathsf{MAW}(S)|)$ time with $O(\mathsf{e}_\min)$ space, where $\mathsf{e}_\min$ denotes the minimum of the sizes of the CDAWGs for $S$ and for its reversal $S^R$. For any strings of length $n$, it holds that $\mathsf{e}_\min < 2n$, and for highly repetitive strings $\mathsf{e}_\min$ can be sublinear (up to logarithmic) in $n$. We also show that MAWs and their generalization minimal rare words have close relationships with extended bispecial factors, via the CDAWG.

cs.DS↗

arXiv2023

Optimally Computing Compressed Indexing Arrays Based on the Compact Directed Acyclic Word Graph

In this paper, we present the first study of the computational complexity of converting an automata-based text index structure, called the Compact Directed Acyclic Word Graph (CDAWG), of size $e$ for a text $T$ of length $n$ into other text indexing structures for the same text, suitable for highly repetitive texts: the run-length BWT of size $r$, the irreducible PLCP array of size $r$, and the quasi-irreducible LPF array of size $e$, as well as the lex-parse of size $O(r)$ and the LZ77-parse of size $z$, where $r, z \le e$. As main results, we showed that the above structures can be optimally computed from either the CDAWG for $T$ stored in read-only memory or its self-index version of size $e$ without a text in $O(e)$ worst-case time and words of working space. To obtain the above results, we devised techniques for enumerating a particular subset of suffixes in the lexicographic and text orders using the forward and backward search on the CDAWG by extending the results by Belazzougui et al. in 2015.

cs.DS↗

arXiv2023

Minimum Consistent Subset for Trees Revisited

In a vertex-colored graph $G = (V, E)$, a subset $S \subseteq V$ is said to be consistent if every vertex has a nearest neighbor in $S$ with the same color. The problem of computing a minimum cardinality consistent subset of a graph is known to be NP-hard. On the positive side, Dey et al. (FCT 2021) show that this problem is solvable in polynomial time when input graphs are restricted to bi-colored trees. In this paper, we give a polynomial-time algorithm for this problem on $k$-colored trees with fixed $k$.

cs.DS↗

arXiv2022

Computing the Collection of Good Models for Rule Lists

Since the seminal paper by Breiman in 2001, who pointed out a potential harm of prediction multiplicities from the view of explainable AI, global analysis of a collection of all good models, also known as a `Rashomon set,' has been attracted much attention for the last years. Since finding such a set of good models is a hard computational problem, there have been only a few algorithms for the problem so far, most of which are either approximate or incomplete. To overcome this difficulty, we study efficient enumeration of all good models for a subclass of interpretable models, called rule lists. Based on a state-of-the-art optimal rule list learner, CORELS, proposed by Angelino et al. in 2017, we present an efficient enumeration algorithm CorelsEnum for exactly computing a set of all good models using polynomial space in input size, given a dataset and a error tolerance from an optimal model. By experiments with the COMPAS dataset on recidivism prediction, our algorithm CorelsEnum successfully enumerated all of several tens of thousands of good rule lists of length at most $\ell = 3$ in around 1,000 seconds, while a state-of-the-art top-$K$ rule list learner based on Lawler's method combined with CORELS, proposed by Hara and Ishihata in 2018, found only 40 models until the timeout of 6,000 seconds. For global analysis, we conducted experiments for characterizing the Rashomon set, and observed large diversity of models in predictive multiplicity and fairness of models.

cs.LG↗

arXiv2022

Cartesian Tree Subsequence Matching

Park et al. [TCS 2020] observed that the similarity between two (numerical) strings can be captured by the Cartesian trees: The Cartesian tree of a string is a binary tree recursively constructed by picking up the smallest value of the string as the root of the tree. Two strings of equal length are said to Cartesian-tree match if their Cartesian trees are isomorphic. Park et al. [TCS 2020] introduced the following Cartesian tree substring matching (CTMStr) problem: Given a text string $T$ of length $n$ and a pattern string of length $m$, find every consecutive substring $S = T[i..j]$ of a text string $T$ such that $S$ and $P$ Cartesian-tree match. They showed how to solve this problem in $\tilde{O}(n+m)$ time. In this paper, we introduce the Cartesian tree subsequence matching (CTMSeq) problem, that asks to find every minimal substring $S = T[i..j]$ of $T$ such that $S$ contains a subsequence $S'$ which Cartesian-tree matches $P$. We prove that the CTMSeq problem can be solved efficiently, in $O(m n p(n))$ time, where $p(n)$ denotes the update/query time for dynamic predecessor queries. By using a suitable dynamic predecessor data structure, we obtain $O(mn \log \log n)$-time and $O(n \log m)$-space solution for CTMSeq. This contrasts CTMSeq with closely related order-preserving subsequence matching (OPMSeq) which was shown to be NP-hard by Bose et al. [IPL 1998].

cs.DS↗

arXiv2021

Ordered Counterfactual Explanation by Mixed-Integer Linear Optimization

Post-hoc explanation methods for machine learning models have been widely used to support decision-making. One of the popular methods is Counterfactual Explanation (CE), also known as Actionable Recourse, which provides a user with a perturbation vector of features that alters the prediction result. Given a perturbation vector, a user can interpret it as an "action" for obtaining one's desired decision result. In practice, however, showing only a perturbation vector is often insufficient for users to execute the action. The reason is that if there is an asymmetric interaction among features, such as causality, the total cost of the action is expected to depend on the order of changing features. Therefore, practical CE methods are required to provide an appropriate order of changing features in addition to a perturbation vector. For this purpose, we propose a new framework called Ordered Counterfactual Explanation (OrdCE). We introduce a new objective function that evaluates a pair of an action and an order based on feature interaction. To extract an optimal pair, we propose a mixed-integer linear optimization approach with our objective function. Numerical experiments on real datasets demonstrated the effectiveness of our OrdCE in comparison with unordered CE methods.

cs.LG↗

arXiv2020

Efficient Constrained Pattern Mining Using Dynamic Item Ordering for Explainable Classification

Learning of interpretable classification models has been attracting much attention for the last few years. Discovery of succinct and contrasting patterns that can highlight the differences between the two classes is very important. Such patterns are useful for human experts, and can be used to construct powerful classifiers. In this paper, we consider mining of minimal emerging patterns from high-dimensional data sets under a variety of constraints in a supervised setting. We focus on an extension in which patterns can contain negative items that designate the absence of an item. In such a case, a database becomes highly dense, and it makes mining more challenging since popular pattern mining techniques such as fp-tree and occurrence deliver do not efficiently work. To cope with this difficulty, we present an efficient algorithm for mining minimal emerging patterns by combining two techniques: dynamic variable-ordering during pattern search for enhancing pruning effect, and the use of a pointer-based dynamic data structure, called dancing links, for efficiently maintaining occurrence lists. Experiments on benchmark data sets showed that our algorithm achieves significant speed-ups over emerging pattern mining approach based on LCM, a very fast depth-first frequent itemset miner using static variable-ordering.

cs.DB↗

arXiv2019

Constant Amortized Time Enumeration of Independent Sets for Graphs with Bounded Clique Number

In this study, we address the independent set enumeration problem. Although several efficient enumeration algorithms and careful analyses have been proposed for maximal independent sets, no fine-grained analysis has been given for the non-maximal variant. From the main result, we propose an algorithm $\texttt{EIS}$ for the non-maximal variant that runs in $O(q)$ amortized time and linear space, where $q$ is the clique number, i.e., the maximum size of a clique in an input graph. Note that $\texttt{EIS}$ works correctly even if the exact value of $q$ is unknown. Despite its simplicity, $\texttt{EIS}$ is optimal for graphs with a bounded clique number, such as, triangle-free graphs, planar graphs, bounded degenerate graphs, locally bounded expansion graphs, and $F$-free graphs for any fixed graph $F$, where a $F$-free graph is a graph that has no copy of $F$ as a subgraph.

cs.DS↗

arXiv2019

Enumeration of Distinct Support Vectors for Interactive Decision Making

In conventional prediction tasks, a machine learning algorithm outputs a single best model that globally optimizes its objective function, which typically is accuracy. Therefore, users cannot access the other models explicitly. In contrast to this, multiple model enumeration attracts increasing interests in non-standard machine learning applications where other criteria, e.g., interpretability or fairness, than accuracy are main concern and a user may want to access more than one non-optimal, but suitable models. In this paper, we propose a K-best model enumeration algorithm for Support Vector Machines (SVM) that given a dataset S and an integer K>0, enumerates the K-best models on S with distinct support vectors in the descending order of the objective function values in the dual SVM problem. Based on analysis of the lattice structure of support vectors, our algorithm efficiently finds the next best model with small latency. This is useful in supporting users's interactive examination of their requirements on enumerated models. By experiments on real datasets, we evaluated the efficiency and usefulness of our algorithm.

cs.LG↗

arXiv2019

An Efficient Algorithm for Enumerating Chordal Bipartite Induced Subgraphs in Sparse Graphs

In this paper, we propose a characterization of chordal bipartite graphs and an efficient enumeration algorithm for chordal bipartite induced subgraphs. A chordal bipartite graph is a bipartite graph without induced cycles with length six or more. It is known that the incident graph of a hypergraph is chordal bipartite graph if and only if the hypergraph is $β$-acyclic. As the main result of our paper, we show that a graph $G$ is chordal bipartite if and only if there is a special vertex elimination ordering for $G$, called CBEO. Moreover, we propose an algorithm ECB which enumerates all chordal bipartite induced subgraphs in $O(ktΔ^2)$ time per solution on average, where $k$ is the degeneracy, $t$ is the maximum size of $K_{t,t}$ as an induced subgraph, and $Δ$ is the degree. ECB achieves constant amortized time enumeration for bounded degree graphs.

cs.DS↗

arXiv2018

Efficient Enumeration of Dominating Sets for Sparse Graphs

A dominating set $D$ of a graph $G$ is a set of vertices such that any vertex in $G$ is in $D$ or its neighbor is in $D$. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a $k$-degenerate graph in $O(k)$ time per solution using $O(n + m)$ space, where $n$ and $m$ are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, $H$-minor free graphs with some fixed $H$. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.

cs.DS↗

arXiv2018

Fully-Online Suffix Tree and Directed Acyclic Word Graph Construction for Multiple Texts

We consider construction of the suffix tree and the directed acyclic word graph (DAWG) indexing data structures for a collection $\mathcal{T}$ of texts, where a new symbol may be appended to any text in $\mathcal{T} = \{T_1, \ldots, T_K\}$, at any time. This fully-online scenario, which arises in dynamically indexing multi-sensor data, is a natural generalization of the long solved semi-online text indexing problem, where texts $T_1, \ldots, T_{k}$ are permanently fixed before the next text $T_{k+1}$ is processed for each $1 \leq k < K$. We present fully-online algorithms that construct the suffix tree and the DAWG for $\mathcal{T}$ in $O(N \log σ)$ time and $O(N)$ space, where $N$ is the total lengths of the strings in $\mathcal{T}$ and $σ$ is their alphabet size. The standard explicit representation of the suffix tree leaf edges and some DAWG edges must be relaxed in our fully-online scenario, since too many updates on these edges are required in the worst case. Instead, we provide access to the updated suffix tree leaf edge labels and the DAWG edges to be redirected via auxiliary data structures, in $O(\log σ)$ time per added character.

cs.DS↗

arXiv2018

Efficient Enumeration of Subgraphs and Induced Subgraphs with Bounded Girth

The girth of a graph is the length of its shortest cycle. Due to its relevance in graph theory, network analysis and practical fields such as distributed computing, girth-related problems have been object of attention in both past and recent literature. In this paper, we consider the problem of listing connected subgraphs with bounded girth. As a large girth is index of sparsity, this allows to extract sparse structures from the input graph. We propose two algorithms, for enumerating respectively vertex induced subgraphs and edge induced subgraphs with bounded girth, both running in $O(n)$ amortized time per solution and using $O(n^3)$ space. Furthermore, the algorithms can be easily adapted to relax the connectivity requirement and to deal with weighted graphs. As a byproduct, the second algorithm can be used to answer the well known question of finding the densest $n$-vertex graph(s) of girth $k$.

cs.DS↗

arXiv2017

On the Model Shrinkage Effect of Gamma Process Edge Partition Models

The edge partition model (EPM) is a fundamental Bayesian nonparametric model for extracting an overlapping structure from binary matrix. The EPM adopts a gamma process ($Γ$P) prior to automatically shrink the number of active atoms. However, we empirically found that the model shrinkage of the EPM does not typically work appropriately and leads to an overfitted solution. An analysis of the expectation of the EPM's intensity function suggested that the gamma priors for the EPM hyperparameters disturb the model shrinkage effect of the internal $Γ$P. In order to ensure that the model shrinkage effect of the EPM works in an appropriate manner, we proposed two novel generative constructions of the EPM: CEPM incorporating constrained gamma priors, and DEPM incorporating Dirichlet priors instead of the gamma priors. Furthermore, all DEPM's model parameters including the infinite atoms of the $Γ$P prior could be marginalized out, and thus it was possible to derive a truly infinite DEPM (IDEPM) that can be efficiently inferred using a collapsed Gibbs sampler. We experimentally confirmed that the model shrinkage of the proposed models works well and that the IDEPM indicated state-of-the-art performance in generalization ability, link prediction accuracy, mixing efficiency, and convergence speed.

stat.ML↗