We consider directed graphs in which the nodes are labeled with strings. A walk in such a graph naturally corresponds to the concatenation of the visited nodes' labels. These graphs are widely used in bioinformatics to compactly describe large collections of highly similar genomes. Given such a graph G = (V,E) and a pattern of length m, we seek a walk whose corresponding string has an occurrence of the pattern. We call this the SMLG problem. Amir et al. [J. Algorithms, 2000] showed that SMLG can be solved in 𝒪(m |E| + N) time, where N is the total length of all node labels. Equi et al. [ACM Trans. Algorithms, 2023] showed that this is essentially optimal (under SETH). The existing lower bound assumes that the sought walk is of length Θ(|V|). Thus, we might be able to bypass this lower bound by restricting the walk length to b-1, which naturally reduces to having as input a directed graph whose set of nodes is partitioned into b blocks. Then, we seek a walk in this graph that starts in the first block and ends in the last block. We call this the b-SMBG problem. Equi et al. [Algorithmica, 2023] showed that, if we impose no restriction on b, the existing algorithm of Amir et al. is essentially optimal for b-SMBG (again under SETH). We provide a more fine-grained classification that essentially settles the complexity of b-SMBG parameterized by b: 1) For b = 2, Pissis [SOSA 2025] already provided a simple 𝒪(m + |E|+N)-time algorithm. 2) We design a new 𝒪̃(m + |E| + N)-time algorithm for b = 3. As a direct implication of this result, the SMLG problem for b ≤ 3 (walks of length at most 2) also admits near-linear-time complexity. 3) There is no 𝒪((m |E|)^{1-ε} + N)-time combinatorial algorithm, for any b ≥ 4 and ε > 0. 4) There is an algorithm working in 𝒪(max(|V|, m)^ω+N) time, where ω is the matrix multiplication exponent, which is conditionally optimal for graphs with b ≥ 4 blocks. 5) Under SETH, no 𝒪((m |E|)^{1-ε} + N)-time algorithm exists, for any b = ω(log |V|) and ε > 0. Although our motivation is primarily of a theoretical nature, we stress that our algorithms are simple to implement. As such, they may contribute to practical advancements in applications where the SMLG problem is an important primitive, such as in the analysis of pangenome graphs.

The reachability diameter (ReachDiam) of a directed graph is the maximum distance over all pairs u,v where v is reachable from u. This notion is present in the definition of shortcut sets, and the name was recently coined in that context by Haeupler, Jiang, and Saranurak [SOSA 2026]. While this is a very natural notion of diameter in directed graphs, and especially DAGs, it is so far not computationally explored. Other definitions of diameter in directed graphs are either trivial (infinite) in graphs that are not strongly connected (e.g., the classical definition) or are non-trivial only in highly restrictive graph classes (e.g., Min-Diameter). We initiate the problem of computing the (approximate) reachability diameter from a fine-grained complexity point of view. Under certain fine-grained assumptions, we prove that there is no algorithm in time 𝒪(n^{ω - ε}) that gives any approximation of ReachDiam in weighted graphs. Similarly, there is no algorithm with better than 2-approximation for unweighted graphs in this time. To supplement this, we provide algorithmic upper bounds that lead to additive approximation of ReachDiam for unweighted graphs. Hence, we establish a strong separation between the weighted and unweighted cases, which makes this type of diameter different in nature than other known notions. Considering the hardness in general weighted graphs, we also study special graph classes and get small constant approximations for DAGs with bounded width or graphs with bounded treewidth. Interestingly, our techniques also lead to exact hopsets with hopbound 2 for bounded treewidth graphs. This and some of our upper bounds for general graphs show technical connections between approximating ReachDiam and computing shortcut sets and hopsets.

We prove an elementary yet powerful combinatorial lemma: in any rooted tree with L leaves, the number of nodes whose depth is smaller than the number of their leaf descendants is at most L. For any string T of length n, a direct application of this lemma to the suffix trie of T yields that the number of substrings of T whose length is smaller than their number of occurrences in T is at most n. This combinatorial insight leads to space-efficient data structures with optimal query times for string counting problems via the following algorithmic framework: store the counts for the at most n "frequent" substrings of T in a preprocessing step, and use a reporting query to count for the "infrequent" substrings. Our framework acts as a convenient black box, lifting indexes with reporting time 𝒪(|P|+|Occ_T(P)|) to support counting queries in time 𝒪(|P|), where P is the queried pattern and Occ_T(P) is the set of occurrences of P in T. As applications, we show efficient indexes for consecutive occurrences, weighted sequences, strings with utilities, and non-overlapping occurrences.

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 log²n) 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(n²), and also unconditionally improves the combinatorial 𝒪(m⋅z_T)-time algorithm of Conte et al. [TKDD 2026] 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.

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.

Researchers, policy makers, and engineers need to make sense of data from spreading processes as diverse as rumor spreading in social networks, viral infections, and water contamination. Classical questions include predicting infection behavior in a given network or deducing the network structure from infection data. Most of the research on network infections studies static graphs, that is, the connections in the network are assumed to not change. More recently, temporal graphs, in which connections change over time, have been used to more accurately represent real-world infections, which rarely occur in unchanging networks. We propose a model for temporal graph discovery that is consistent with previous work on static graphs and embraces the greater expressiveness of temporal graphs. For this model, we give algorithms and lower bounds which are often tight. We analyze different variations of the problem, which make our results widely applicable and it also clarifies which aspects of temporal infections make graph discovery easier or harder. We round off our analysis with an experimental evaluation of our algorithm on real-world interaction data from the Stanford Network Analysis Project and on temporal Erdős-Renyi graphs. On Erdős-Renyi graphs, we uncover a threshold behavior, which can be explained by a novel connectivity parameter that we introduce during our theoretical analysis.

Many real-world networks, such as transportation or trade networks, are dynamic in the sense that the edge set may change over time, but these changes are known in advance. This behavior is captured by the temporal graphs model, which has recently become a trending topic in theoretical computer science. A core open problem in the field is to prove the existence of linear-size temporal spanners in temporal cliques, i.e., sparse subgraphs of complete temporal graphs that ensure all-pairs reachability via temporal paths. So far, the best known result is the existence of temporal spanners with $\mathcal{O}(n\log n)$ many edges. We present significant progress towards proving that linear-size temporal spanners exist in all temporal cliques. We adapt techniques used in previous works and heavily expand and generalize them to provide a simpler and more intuitive proof of the $\mathcal{O}(n\log n)$ bound. Moreover, we use our novel approach to show that a large class of temporal cliques, called edge-pivot graphs, admit linear-size temporal spanners. To contrast this, we investigate other classes of temporal cliques that do not belong to the class of edge-pivot graphs. We introduce two such graph classes and we develop novel techniques for establishing the existence of linear temporal spanners in these graph classes as well.

Human lives are increasingly influenced by algorithms, which therefore need to meet higher standards not only in accuracy but also with respect to explainability. This is especially true for high-stakes areas such as real estate valuation. Unfortunately, the methods applied there often exhibit a trade-off between accuracy and explainability.