Related papers: The wired minimal spanning forest on the Poisson-w…
This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a…
We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result…
A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the…
We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is of exponential form $e^{-\mu h}$. Defining the total weight for such trees of fixed size to be $Z^{(\mu)}_N$,…
We define generic ensembles of infinite trees. These are limits as $N\to\infty$ of ensembles of finite trees of fixed size $N$, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices…
We introduce a new spanning tree model called the random spanning tree in random environment (RSTRE), which interpolates between the uniform spanning tree and the minimum spanning tree as the inverse temperature (disorder strength) $\beta$…
We study the properties of random walks on complex trees. We observe that the absence of loops reflects in physical observables showing large differences with respect to their looped counterparts. First, both the vertex discovery rate and…
We consider the number of common edges in two independent random spanning trees of a graph $G$. For complete graphs $K_n$, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson…
We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and…
In this paper, we propose a discrete circular distribution obtained by extending the wrapped Poisson distribution. This new distribution, the Invariant Wrapped Poisson (IWP), enjoys numerous advantages: simple tractable density,…
A particular continuous-time multitype branching process is considered, it is the continuous-time embedding of a discrete-time process which is very popular in theoretical computer science: the m-ary search tree (m is an integer). There is…
We study the expected value of the length $L_n$ of the minimum spanning tree of the complete graph $K_n$ when each edge $e$ is given an independent uniform $[0,1]$ edge weight. We sharpen the result of Frieze \cite{F1} that…
We obtain the complete physical spectrum of the $W_N$ string, for arbitrary $N$. The $W_N$ constraints freeze $N-2$ coordinates, while the remaining coordinates appear in the currents only {\it via} their energy-momentum tensor. The…
Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space, $r\in\mathcal{T}$ is…
We study iterated weighted residual (WR) splittings generated by a positive operator $R_{0}\in B\left(H\right)_{+}$ and a finite family of contractions $C_{1},\dots,C_{m}$ in $B\left(H\right)$. The associated residual update $R\mapsto…
We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that…
In this paper, we show that a Galton-Watson tree conditioned to have a fixed number of particles in generation $n$ converges in distribution as $n\rightarrow\infty$, and with this tool we study the span and gap statistics of a branching…
In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…
We study random two-component spanning forests ($2$SFs) of finite graphs, giving formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an…
We investigate the effective resistance $R_n$ and conductance $C_n$ between the root and leaves of a binary tree of height $n$. In this electrical network, the resistance of each edge $e$ at distance $d$ from the root is defined by…