Jul 2026· Embedded Systems and Applications· pp. 105:1-105:19· 0 citations· 41 references
Computer Science
TL;DR
A near-linear-time algorithm is given and there is no combinatorial algorithm improving over the state-of-the-art $\mathcal{O}(m|E| + N)$ bound for any $b\ge 4$.
Abstract
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 $\mathcal{O}(m|E| + N)$ 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 $\Theta(|V|)$. 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. We provide a more fine-grained classification that essentially settles the complexity of $b$-SMBG parameterized by $b$: (1) We give a near-linear-time algorithm for $b=3$. (2) We show that there is no combinatorial algorithm improving over the state-of-the-art $\mathcal{O}(m|E| + N)$ bound for any $b\ge 4$. (3) We also present a fast matrix multiplication-based algorithm yielding an improvement for $b \in \mathcal{O}(1)$, which is conditionally optimal. (4) Finally, we show that under SETH, for any $b \in \omega(\log |V|)$, no algorithm can improve over the state of the art.
This paper investigates an information-limited version of the planted subgraph detection problem, in which the planted structure is an arbitrary sequence of graphs, where $\Gamma_n$ is embedded in an ambient graph on $n$ vertices, but the observer does not have access to the full adjacency matrix.
A novel exponential time algorithm is presented to compute the exact GED and a corresponding edit sequence in $O^*(4 + \varepsilon)^n$ time and polynomial space, provided one of the two graphs admits strictly sublinear balanced separators.
L. Bülte, Philip Mayer, Lars Müller et al.· 0 citations
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 $\mathcal{O}(|P|+|\textsf{Occ}_T(P)|)$ to support counting queries in time $\mathcal{O}(|P|)$, where $P$ is the queried pattern and $\textsf{Occ}_T(P)$ 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.
B. Bals, P. Charalampopoulos, Oded Lachish et al.· Embedded Systems and Applica...· 0 citations
It is proved that, as long as $p=o(\log n /n)$, for any $\varepsilon>0$, one can construct a confidence set of vertices of size $K(\varepsilon)$ that depends only on $\varepsilon$ and not on $n$, such that it contains the root with probability at least $1-\varepsilon$.
Luc Devroye, Gábor Lugosi, Neeladri Maitra· 1 citation· ⚡1
This paper provides the first truly linear-time approximation scheme for the Densest Subgraph Problem, and uses assignments arising from a flow-based formulation together with a structural carving lemma to progressively carve "sparse" parts of the graph while nearly preserving the densest subgraph.
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $\gamma^{+}$ (additions only) and the Cayley edit distance $\gamma_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $\gamma^{+}(G)\ge n\Delta^{*}/(2m)-1$, where $\Delta^{*}$ is the least $d\ge\Delta$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $\gamma^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $\gamma^{+}$, while $\gamma_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $\gamma^{+}(P_n)=\gamma^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $\gamma_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.
