Related papers: Near-Linear-Time Deterministic Plane Steiner Spann…
We use exponential start time clustering to design faster and more work-efficient parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of…
We introduce a new structure for a set of points in the plane and an angle $\alpha$, which is similar in flavor to a bounded-degree MST. We name this structure $\alpha$-MST. Let $P$ be a set of points in the plane and let $0 < \alpha \le…
We provide a spectrum of new theoretical insights and practical results for finding a Minimum Dilation Triangulation (MDT), a natural geometric optimization problem of considerable previous attention: Given a set $P$ of $n$ points in the…
The 2-Opt heuristic is one of the simplest algorithms for finding good solutions to the metric Traveling Salesman Problem. It is the key ingredient to the well-known Lin-Kernighan algorithm and often used in practice. So far, only upper and…
Solving linear programs is often a challenging task in distributed settings. While there are good algorithms for solving packing and covering linear programs in a distributed manner (Kuhn et al.~2006), this is essentially the only class of…
Traveling salesman problem is a NP-hard problem. Until now, researchers have not found a polynomial time algorithm for traveling salesman problem. Among the existing algorithms, dynamic programming algorithm can solve the problem in time…
We describe an algorithm to construct an intrinsic Delaunay triangulation of a smooth closed submanifold of Euclidean space. Using results established in a companion paper on the stability of Delaunay triangulations on $\delta$-generic…
This paper gives simple distributed algorithms for the fundamental problem of computing graph distances in the Congested Clique model. One of the main components of our algorithms is fast matrix multiplication, for which we show an…
An important variant of the classic Traveling Salesman Problem (TSP) is the Dynamic TSP, in which a system with dynamic constraints is tasked with visiting a set of n target locations (in any order) in the shortest amount of time. Such…
We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning…
We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a $(1+\epsilon)$…
The (unweighted) point-separation problem asks, given a pair of points $s$ and $t$ in the plane, and a set of candidate geometric objects, for the minimum-size subset of objects whose union blocks all paths from $s$ to $t$. Recent work has…
Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. However, whilst a number of algorithms and existence proofs have been proposed, very…
In this paper, we develop deterministic fully dynamic algorithms for computing approximate distances in a graph with worst-case update time guarantees. In particular, we obtain improved dynamic algorithms that, given an unweighted and…
We propose a new refinement algorithm to generate size-optimal quality-guaranteed Delaunay triangulations in the plane. The algorithm takes $O(n \log n + m)$ time, where $n$ is the input size and $m$ is the output size. This is the first…
We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second…
We study the structure of solutions to linear programming formulations for the traveling salesperson problem (TSP). We perform a detailed analysis of the support of the subtour elimination linear programming relaxation, which leads to…
We present a new $4$-approximation algorithm for the Combinatorial Motion Planning problem which runs in $\mathcal{O}(n^2\alpha(n^2,n))$ time, where $\alpha$ is the functional inverse of the Ackermann function, and a fully distributed…
The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution…
We consider the k-outconnected directed Steiner tree problem (k-DST). Given a directed edge-weighted graph $G=(V,E,w)$, where $V=\{r\}\cup S \cup T$, and an integer $k$, the goal is to find a minimum cost subgraph of $G$ in which there are…