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In this paper, we propose a physically inspired graph-theoretical clustering method, which first makes the data points organized into an attractive graph, called In-Tree, via a physically inspired rule, called Nearest Descent (ND). In…
Quantum algorithms for several problems in graph theory are considered. Classical algorithms for finding the lowest weight path between two points in a graph and for finding a minimal weight spanning tree involve searching over some space.…
The Aldous--Broder algorithm provides a way of sampling a uniformly random spanning tree for finite connected graphs using simple random walk. Namely, start a simple random walk on a connected graph and stop at the cover time. The tree…
We construct forests that span $\mathbb{Z}^d$, $d\geq2$, that are stationary and directed, and whose trees are infinite, but for which the subtrees attached to each vertex are as short as possible. For $d\geq3$, two independent copies of…
The basic idea of the kd-tree algorithm is to recursively partition a point set P by hyperplanes, and to store the obtained partitioning in a binary tree. Due to its immense popularity, many applications in astronomy have been implemented.…
Robots are still poor at traversing cluttered large obstacles required for important applications like search and rescue. By contrast, animals are excellent at doing so, often using direct physical interaction with obstacles rather than…
Since many NP-complete graph problems have been shown polynomial-time solvable when restricted to claw-free graphs, we study the problem of determining the distance of a given graph to a claw-free graph, considering vertex elimination as…
The nonlinear programming (NLP) problem to solve distribution-level optimal power flow (D-OPF) poses convergence issues and does not scale well for unbalanced distribution systems. The existing scalable D-OPF algorithms either use…
Weight-balanced trees are a popular form of self-balancing binary search trees. Their popularity is due to desirable guarantees, for example regarding the required work to balance annotated trees. While usual weight-balanced trees perform…
This paper presents a gentle tutorial and a structured reformulation of Bock's 1971 Algol procedure for constructing minimum directed spanning trees. Our aim is to make the original algorithm readable and reproducible for modern readers,…
{\em Reoptimization} is a setting in which we are given an (near) optimal solution of a problem instance and a local modification that slightly changes the instance. The main goal is that of finding an (near) optimal solution of the…
Markov chains based on spanning trees have been hugely influential in algorithms for assessing fairness in political redistricting. The input graph represents the geographic building blocks of a jurisdiction. The goal is to output a large…
It is required to find an optimal order of constructing the edges of a network so as to minimize the sum of the weighted connection times of relevant pairs of vertices. Construction can be performed anytime anywhere in the network, with a…
Depth first search (DFS) tree is a fundamental data structure for solving various graph problems. The classical DFS algorithm requires $O(m+n)$ time for a graph having $n$ vertices and $m$ edges. In the streaming model, an algorithm is…
We show that, for any graph optimization problem in which the feasible solutions can be expressed by a formula in monadic second-order logic describing sets of vertices or edges and in which the goal is to minimize the sum of the weights in…
In this paper we study the question of whether or not a static search tree should ever be unbalanced. We present several methods to restructure an unbalanced k-ary search tree $T$ into a new tree $R$ that preserves many of the properties of…
We present a tree structure algorithm for optimal control problems with state constraints. We prove a convergence result for a discrete time approximation of the value function based on a novel formulation of the constrained problem. Then…
Collision-free flight in cluttered environments is a critical capability for autonomous quadrotors. Traditional methods often rely on detailed 3D map construction, trajectory generation, and tracking. However, this cascade pipeline can…
We give a deterministic algorithm to construct a graph with no loops (a tree or a forest) whose vertices are the points of a d-dimensional stationary Poisson process S, subset of R^d. The algorithm is independent of the origin of…
We consider the problem of uniformly generating a spanning tree, of a connected undirected graph. This process is useful to compute statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle…