Related papers: Fault-Equivalent Lowest Common Ancestors

We investigate the nearest common ancestor (NCA) function in rooted trees. As the main conceptual contribution, the paper introduces universal trees for the NCA function: For a given family of rooted trees, an NCA-universal tree $S$ is a…

A level-ancestor or LA query about a rooted tree $T$ takes as arguments a node $v$ in $T$, of depth $d_v$, say, and an integer $d$ with $0\le d\le d_v$ and returns the ancestor of $v$ in $T$ of depth $d$. The static LA problem is to process…

Phylogenetic networks provide a framework for representing evolutionary histories involving reticulate events such as hybridization or horizontal gene transfer. A central problem is to infer such networks from local structural information.…

Let $T$ be a rooted tree, and $V(T)$ its set of vertices. A subset $X$ of $V(T)$ is called an infima closed set of $T$ if for any two vertices $u,v\in X$, the first common ancestor of $u$ and $v$ is also in $X$. This paper determines the…

Consider a forest that evolves via $link$ operations that make the root of one tree the child of a node in another tree. Intermixed with $link$ operations are $nca$ operations, which return the nearest common ancestor of two given nodes…

We study the problem of designing a \emph{resilient} data structure maintaining a tree under the Faulty-RAM model [Finocchi and Italiano, STOC'04] in which up to $\delta$ memory words can be corrupted by an adversary. Our data structure…

Let $G=(V,E)$ be an $n$-vertex directed acyclic graph (DAG). A lowest common ancestor (LCA) of two vertices $u$ and $v$ is a common ancestor $w$ of $u$ and $v$ such that no descendant of $w$ has the same property. In this paper, we consider…

Motivated by the behavior of the trace pairing over tame cyclic number fields, we introduce the notion of tame lattices. Given an arbitrary non-trivial lattice $\mathcal{L}$ we construct a parametric family of full-rank sub-lattices…

A lattice is a set of all the integer linear combinations of certain linearly independent vectors. One of the most important concepts on lattice is the successive minima which is of vital importance from both theoretical and practical…

The success of gradient descent in ML and especially for learning neural networks is remarkable and robust. In the context of how the brain learns, one aspect of gradient descent that appears biologically difficult to realize (if not…

A rooted tree $T$ with vertex labels $t(v)$ and set-valued edge labels $\lambda(e)$ defines maps $\delta$ and $\varepsilon$ on the pairs of leaves of $T$ by setting $\delta(x,y)=q$ if the last common ancestor $\text{lca}(x,y)$ of $x$ and…

The CONNECTED VERTEX COVER problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity…

We give a coset realization of the vertex operator algebra $M(1)^+$ with central charge $\ell$. We realize $M(1)^+$ as a commutant of certain affine vertex algebras of level -1 in the vertex algebra $L_{C_{\ell}…

According to the classic Chv{\'{a}}tal's Lemma from 1977, a graph of minimum degree $\delta(G)$ contains every tree on $\delta(G)+1$ vertices. Our main result is the following algorithmic "extension" of Chv\'{a}tal's Lemma: For any…

A subset $F$ of edges in a connected graph $G$ is a $h$-extra edge-cut if $G-F$ is disconnected and every component has more than $h$ vertices. The $h$-extra edge-connectivity $\la^{(h)}(G)$ of $G$ is defined as the minimum cardinality over…

Federated Learning (FL) is enabling collaborative model training across institutions without sharing sensitive patient data. This approach is particularly valuable in low- and middle-income countries (LMICs), where access to trained medical…

The $\ell$-component connectivity (or $\ell$-connectivity for short) of a graph $G$, denoted by $\kappa_\ell(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a…

Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$…

A fundamental requirement for intelligent systems is the ability to learn continuously under changing environments. However, models trained in this regime often suffer from catastrophic forgetting. Leveraging pre-trained models has recently…

For a fixed finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem asks, given an $n$-vertex input graph $G,$ for the minimum number of vertices that intersect all minor models in $G$ of the graphs in ${\cal F}$. by…