arXiv subjects

Matei Tinca

Publications and source records attributed to Matei Tinca.

3 recordsLinked to original sources

arXiv2026

Cut Query Reachability for DAGs with Subquadratic Queries

In the cut-query model, we have access to a (directed) graph via an oracle and we can query the size of the (directed) cut of a given subset of the vertices. One of the most elementary tasks in this model is to decide if there is a path two fixed vertices $s$ and $t$. While many results are known for undirected graphs, much less in understood for directed graphs in the cut query model. Even for the basic task of $s$-$t$ reachability, the best known randomized algorithm, is to reconstruct the entire graph with a technique by Grebinski and Kucherov using $O(n^2 / \log n)$ queries [Grebinski and Kucherov, 2000]. We restrict our attention to directed acyclic graphs (DAGs) and obtain a deterministic single-source reachability algorithm using $O(n \sqrt{n \log n})$ queries. The result is based on a topological sort algorithm, and can also be adapted to compute single-source shortest paths in DAGs.

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arXiv2026

Optimal Enumeration of Eulerian Trails in Directed Graphs

The BEST theorem, due to de Bruijn, van Aardenne-Ehrenfest, Smith, and Tutte, is a classical tool from graph theory that links the Eulerian trails in a directed graph $G=(V,E)$ with the arborescences in $G$. In particular, one can use the BEST theorem to count the Eulerian trails in $G$ in polynomial time. For enumerating the Eulerian trails in $G$, one could naturally resort to first enumerating the arborescences in $G$ and then exploiting the insight of the BEST theorem to enumerate the Eulerian trails in $G$: every arborescence in $G$ corresponds to at least one Eulerian trail in $G$. For over two decades, the fastest algorithm for enumerating arborescences in $G$ took $O(m\log n + n + z_A \log^2 n)$ time, where $n=|V|$, $m=|E|$, and $z_A$ is the number of arborescences in $G$ [Uno, ISAAC 1998]. Since Uno's algorithm does not lead to an optimal enumeration of Eulerian trails in directed graphs, we were motivated to develop a direct algorithm for this problem. Our central contribution is a remarkably simple algorithm to directly enumerate the $z_T$ Eulerian trails in $G$ in the optimal $O(m + z_T)$ time. As a consequence, our result improves on an implementation of the BEST theorem for counting Eulerian trails in $G$ when $z_T=o(n^2)$, and also unconditionally improves the combinatorial $O(m\cdot z_T)$-time algorithm of Conte et al. [FCT 2021] for the same task. Moreover, we show that, with some care, our algorithm can be extended to enumerate Eulerian trails in directed multigraphs in optimal time, enabling applications in bioinformatics and data privacy.

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arXiv2025

Subtree Mode and Applications

The mode of a collection of values (i.e., the most frequent value in the collection) is a key summary statistic. Finding the mode in a given range of an array of values is thus of great importance, and constructing a data structure to solve this problem is in fact the well-known Range Mode problem. In this work, we introduce the Subtree Mode (SM) problem, the analogous problem in a leaf-colored tree, where the task is to compute the most frequent color in the leaves of the subtree of a given node. SM is motivated by several applications in domains such as text analytics and biology, where the data are hierarchical and can thus be represented as a (leaf-colored) tree. Our central contribution is a time-optimal algorithm for SM that computes the answer for every node of an input $N$-node tree in $O(N)$ time. We further show how our solution can be adapted for node-colored trees, or for computing the $k$ most frequent colors, for any given $k=O(1)$, in the optimal $O(N)$ time. Moreover, we prove that a similarly fast solution for when the input is a sink-colored directed acyclic graph instead of a leaf-colored tree is highly unlikely. Our experiments on real datasets with trees of up to $7.3$ billion nodes demonstrate that our algorithm is faster than baselines by at least one order of magnitude and much more space efficient. They also show that it is effective in pattern mining, sequence-to-database search, and biology applications.

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