Related papers: Probabilistic Analysis of RRT Trees
A rectilinear Steiner tree for a set $P$ of points in $\mathbb{R}^2$ is a tree that connects the points in $P$ using horizontal and vertical line segments. The goal of Minimal Rectilinear Steiner Tree is to find a rectilinear Steiner tree…
We present a new approximation algorithm for the treewidth problem which finds an upper bound on the treewidth and constructs a corresponding tree decomposition as well. Our algorithm is a faster variation of Reed's classical algorithm. For…
The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an…
We consider a search problem on trees in which the goal is to find an adversarially placed treasure, while relying on local, partial information. Specifically, each node in the tree holds a pointer to one of its neighbors, termed…
We destroy a finite tree of size $n$ by cutting its edges one after the other and in uniform random order. Informally, the associated cut-tree describes the genealogy of the connected components created by this destruction process. We…
We compute the second order correction for the cover time of the binary tree of depth $n$ by (continuous-time) random walk, and show that with probability approaching 1 as $n$ increases, $\sqrt{\tau_{\mathrm{cov}}}=\sqrt{|E|}[\sqrt{2\log…
The $\lambda$-biased random walk on a binary tree of depth $n$ is the continuous-time Markov chain that has unit mean holding times and, when at a vertex other than the root or a leaf of the tree in question, has a probability of jumping to…
Rapidly-exploring random tree (RRT) has been applied for autonomous parking due to quickly solving high-dimensional motion planning and easily reflecting constraints. However, planning time increases by the low probability of extending…
Let $b$ be an integer greater than 1 and let $W^{\ee}=(W^{\ee}_n; n\geq 0)$ be a random walk on the $b$-ary rooted tree $\U_b$, starting at the root, going up (resp. down) with probability $1/2+\epsilon$ (resp. $1/2 -\epsilon$), $\epsilon…
We propose an algorithm whose input are parameters $k$ and $r$ and a hypergraph $H$ of rank at most $r$. The algorithm either returns a tree decomposition of $H$ of generalized hypertree width at most $4k$ or 'NO'. In the latter case, it is…
The asymptotically optimal version of Rapidly-exploring Random Tree (RRT*) is often used to find optimal paths in a high-dimensional configuration space. The well-known issue of RRT* is its slow convergence towards the optimal solution. A…
We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an…
We present improved learning-augmented algorithms for finding an approximate minimum spanning tree (MST) for points in an arbitrary metric space. Our work follows a recent framework called metric forest completion (MFC), where the learned…
Motivated by questions in social networks, distributed computing and probabilistic combinatorics, the last few years have seen increasing interest in network evolution models where new vertices entering the system need to make decisions…
The Radial Spanning Tree (RST) in dimension $d\geq2$ is a random geometric graph constructed on a homogeneous Poisson point process $\mathcal N$ in $\mathbb R^d$ augmented by the origin, with edges connecting each $x\in\mathcal N$ to the…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total…
Due to their efficiency and small size, decision trees and random forests are popular machine learning models used for classification on resource-constrained systems. In such systems, the available execution time for inference in a random…
We study the problem of collective tree exploration (CTE) where a team of $k$ agents is tasked to traverse all the edges of an unknown tree as fast as possible, assuming complete communication between the agents. In this paper, we present…
We show an $\widetilde{O}(m^{1.5} \epsilon^{-1})$ time algorithm that on a graph with $m$ edges and $n$ vertices outputs its spanning tree count up to a multiplicative $(1+\epsilon)$ factor with high probability, improving on the previous…
Given $n \in \mathbb{N}$ and $\mu \in \mathbb{R}$, a $\textit{$\mu$-height-biased tree of size $n$}$ is a random plane tree $\mathbf{\mathbf{T}}_n$ with $n$ vertices with law given by $\mathbb{P}(\mathbf{T}=t) \propto e^{-\mu h(t)}$, where…