arXiv subjects

Christos Levcopoulos

Publications and source records attributed to Christos Levcopoulos.

9 recordsLinked to original sources

arXiv2024

The Voronoi Diagram of Weakly Smooth Planar Point Sets in $O(\log n)$ Deterministic Rounds on the Congested Clique

We study the problem of computing the Voronoi diagram of a set of $n^2$ points with $O(\log n)$-bit coordinates in the Euclidean plane in a substantially sublinear in $n$ number of rounds in the congested clique model with $n$ nodes. Recently, Jansson et al. have shown that if the points are uniformly at random distributed in a unit square then their Voronoi diagram within the square can be computed in $O(1)$ rounds with high probability (w.h.p.). We show that if a very weak smoothness condition is satisfied by an input set of $n^2$ points with $O(\log n)$-bit coordinates in the unit square then the Voronoi diagram of the point set within the unit square can be computed in $O(\log n)$ rounds in this model.

cs.CG↗

arXiv2023

Convex Hulls, Triangulations, and Voronoi Diagrams of Planar Point Sets on the Congested Clique

We consider geometric problems on planar $n^2$-point sets in the congested clique model. Initially, each node in the $n$-clique network holds a batch of $n$ distinct points in the Euclidean plane given by $O(\log n)$-bit coordinates. In each round, each node can send a distinct $O(\log n)$-bit message to each other node in the clique and perform unlimited local computations. We show that the convex hull of the input $n^2$-point set can be constructed in $O(\min\{ h,\log n\})$ rounds, where $h$ is the size of the hull, on the congested clique. We also show that a triangulation of the input $n^2$-point set can be constructed in $O(\log^2n)$ rounds on the congested clique. Finally, we demonstrate that the Voronoi diagram of $n^2$ points with $O(\log n)$-bit coordinates drawn uniformly at random from a unit square can be computed within the square with high probability in $O(1)$ rounds on the congested clique.

cs.DC↗

arXiv2022

Perpetual maintenance of machines with different urgency requirements

A garden $G$ is populated by $n\ge 1$ bamboos $b_1, b_2, ..., b_n$ with the respective daily growth rates $h_1 \ge h_2 \ge \dots \ge h_n$. It is assumed that the initial heights of bamboos are zero. The robotic gardener maintaining the garden regularly attends bamboos and trims them to height zero according to some schedule. The Bamboo Garden Trimming Problem (BGT) is to design a perpetual schedule of cuts to maintain the elevation of the bamboo garden as low as possible. The bamboo garden is a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time. The objective is to design a perpetual schedule of servicing which minimizes the maximum (weighted) waiting time for servicing. We consider two variants of BGT. In discrete BGT the robot trims only one bamboo at the end of each day. In continuous BGT the bamboos can be cut at any time, however, the robot needs time to move from one bamboo to the next. For discrete BGT, we show tighter approximation algorithms for the case when the growth rates are balanced and for the general case. The former algorithm settles one of the conjectures about the Pinwheel problem. The general approximation algorithm improves on the previous best approximation ratio. For continuous BGT, we propose approximation algorithms which achieve approximation ratios $O(\log \lceil h_1/h_n\rceil)$ and $O(\log n)$.

cs.DS↗

arXiv2022

Local Routing in Sparse and Lightweight Geometric Graphs

Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no $O(1)$-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin [Online routing in triangulations. SIAM Journal on Computing, 33(4):937-951, 2004] showed that there exists an online routing algorithm that is $O(1)$-competitive. However, a Delaunay triangulation can have $Ω(n)$ vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tree. We show a simple construction, given a set $V$ of $n$ points in the Euclidean plane, of a planar geometric graph on $V$ that has small weight (within a constant factor of the weight of a minimum spanning tree on $V$), constant degree, and that admits a local routing strategy that is $O(1)$-competitive. Moreover, the technique used to bound the weight works generally for any planar geometric graph whilst preserving the admission of an $O(1)$-competitive routing strategy.

cs.CG↗

arXiv2021

Efficient Assignment of Identities in Anonymous Populations

We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size $n$ of the population is embedded in the transition function. Our labeling protocols are silent w.h.p., i.e., eventually each agent reaches its final state and remains in it forever w.h.p., as well as safe, i.e., never update the label assigned to any single agent. We first present a fast, silent w.h.p.and safe labeling protocol for which the required number of interactions is asymptotically optimal, i.e., $O(n \log n/ε)$ w.h.p. It uses $(2+ε)n+O(n^c)$ states, for any $c<1,$ and the label range $1,\dots,(1+ε)n.$ Furthermore, we consider the so-called pool labeling protocols that include our fast protocol. We show that the expected number of interactions required by any pool protocol is $\ge \frac{n^2}{r+1}$, when the labels range is $1,\dots, n+r<2n.$ Next, we provide a protocol which is silent and safe once a unique leader is provided, and uses only $n+5\sqrt n +O(n^c)$ states, for any $c<1,$ and draws labels from the range $1,\dots,n.$ The expected number of interactions required by the protocol is $O(n^3).$ On the other hand, we show that (even if a unique leader is given in advance) any silent protocol that produces a valid labeling and is safe with probability $>1-\frac 1n$, uses $\ge n+\sqrt {n-1} -1$ states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. Furthermore, we show that for any silent and safe labeling protocol utilizing $n+t<2n$ states the expected number of interactions required to achieve a valid labeling is $\ge \frac{n^2}{t+1}$.

cs.DC↗

arXiv2017

Shortcuts for the Circle

Let $C$ be the unit circle in $\mathbb{R}^2$. We can view $C$ as a plane graph whose vertices are all the points on $C$, and the distance between any two points on $C$ is the length of the smaller arc between them. We consider a graph augmentation problem on $C$, where we want to place $k\geq 1$ \emph{shortcuts} on $C$ such that the diameter of the resulting graph is minimized. We analyze for each $k$ with $1\leq k\leq 7$ what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of~$k$. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is $2 + Θ(1/k^{\frac{2}{3}})$ for any~$k$.

math.MG↗

arXiv2016

Efficiently Correcting Matrix Products

We study the problem of efficiently correcting an erroneous product of two $n\times n$ matrices over a ring. Among other things, we provide a randomized algorithm for correcting a matrix product with at most $k$ erroneous entries running in $\tilde{O}(n^2+kn)$ time and a deterministic $\tilde{O}(kn^2)$-time algorithm for this problem (where the notation $\tilde{O}$ suppresses polylogarithmic terms in $n$ and $k$).

cs.DS↗

arXiv2016

A Fire Fighter's Problem

Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed $v>1$. How large must $v$ be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results. First, we analyze the natural curve $\mbox{FF}_v$ that develops when the fighter keeps building, at speed $v$, a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function $(e^{w Z} - s \, Z)^{-1}$, where $w$ and $s$ are real functions of $v$. For $v>v_c=2.6144 \ldots$ all zeroes are complex conjugate pairs. If $ϕ$ denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs $Θ( 1/ϕ)$ rounds before the fire is contained. As $v$ decreases towards $v_c$ these two zeroes merge into a real one, so that argument $ϕ$ goes to~0. Thus, curve $\mbox{FF}_v$ does not contain the fire if the fighter moves at speed $v=v_c$. (That speed $v>v_c$ is sufficient for containing the fire has been proposed before by Bressan et al. [7], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.) Second, we show that any curve that visits the four coordinate half-axes in cyclic order, and in inreasing distances from the origin, needs speed $v>1.618\ldots$, the golden ratio, in order to contain the fire. Keywords: Motion Planning, Dynamic Environments, Spiralling strategies, Lower and upper bounds

cs.CG↗

arXiv2011

Quickest Path Queries on Transportation Network

This paper considers the problem of finding a quickest path between two points in the Euclidean plane in the presence of a transportation network. A transportation network consists of a planar network where each road (edge) has an individual speed. A traveller may enter and exit the network at any point on the roads. Along any road the traveller moves with a fixed speed depending on the road, and outside the network the traveller moves at unit speed in any direction. We give an exact algorithm for the basic version of the problem: given a transportation network of total complexity n in the Euclidean plane, a source point s and a destination point t, and the quickest path between s and t. We also show how the transportation network can be preprocessed in time O(n^2 log n) into a data structure of size O(n^2) such that (1 + ε)-approximate cheapest path cost queries between any two points in the plane can be answered in time O(1ε^4 log n).

cs.CG↗