arXiv subjects

Michiel Smid

Publications and source records attributed to Michiel Smid.

67 records · Page 4Linked to original sources

arXiv2013

Network Farthest-Point Diagrams

Consider the continuum of points along the edges of a network, i.e., an undirected graph with positive edge weights. We measure distance between these points in terms of the shortest path distance along the network, known as the network distance. Within this metric space, we study farthest points. We introduce network farthest-point diagrams, which capture how the farthest points---and the distance to them---change as we traverse the network. We preprocess a network G such that, when given a query point q on G, we can quickly determine the farthest point(s) from q in G as well as the farthest distance from q in G. Furthermore, we introduce a data structure supporting queries for the parts of the network that are farther away from q than some threshold R > 0, where R is part of the query. We also introduce the minimum eccentricity feed-link problem defined as follows. Given a network G with geometric edge weights and a point p that is not on G, connect p to a point q on G with a straight line segment pq, called a feed-link, such that the largest network distance from p to any point in the resulting network is minimized. We solve the minimum eccentricity feed-link problem using eccentricity diagrams. In addition, we provide a data structure for the query version, where the network G is fixed and a query consists of the point p.

cs.CG↗

arXiv2012

Fixed-Orientation Equilateral Triangle Matching of Point Sets

Given a point set $P$ and a class $\mathcal{C}$ of geometric objects, $G_\mathcal{C}(P)$ is a geometric graph with vertex set $P$ such that any two vertices $p$ and $q$ are adjacent if and only if there is some $C \in \mathcal{C}$ containing both $p$ and $q$ but no other points from $P$. We study $G_{\bigtriangledown}(P)$ graphs where $\bigtriangledown$ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-$Θ_6$ graphs and TD-Delaunay graphs. The main result in our paper is that for point sets $P$ in general position, $G_{\bigtriangledown}(P)$ always contains a matching of size at least $\lceil\frac{n-2}{3}\rceil$ and this bound cannot be improved above $\lceil\frac{n-1}{3}\rceil$. We also give some structural properties of $G_{\davidsstar}(P)$ graphs, where $\davidsstar$ is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of $G_{\davidsstar}(P)$ is simply a path. Through the equivalence of $G_{\davidsstar}(P)$ graphs with $Θ_6$ graphs, we also derive that any $Θ_6$ graph can have at most $5n-11$ edges, for point sets in general position.

cs.CG↗

arXiv2010

Improved Methods For Generating Quasi-Gray Codes

Consider a sequence of bit strings of length d, such that each string differs from the next in a constant number of bits. We call this sequence a quasi-Gray code. We examine the problem of efficiently generating such codes, by considering the number of bits read and written at each generating step, the average number of bits read while generating the entire code, and the number of strings generated in the code. Our results give a trade-off between these constraints, and present algorithms that do less work on average than previous results, and that increase the number of bit strings generated.

cs.DM↗

arXiv2010

Pi/2-Angle Yao Graphs are Spanners

We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8(29+23sqrt(2)). Enroute to this, we also show that the Yao graph Y4 in the Linf metric is a planar spanner with stretch factor 8.

cs.CG↗

arXiv2009

An Ω(n log n) lower bound for computing the sum of even-ranked elements

Given a sequence A of 2n real numbers, the Even-Rank-Sum problem asks for the sum of the n values that are at the even positions in the sorted order of the elements in A. We prove that, in the algebraic computation-tree model, this problem has time complexity Θ(n log n). This solves an open problem posed by Michael Shamos at the Canadian Conference on Computational Geometry in 2008.

cs.DS↗

arXiv2009

Algorithms for Marketing-Mix Optimization

Algorithms for determining quality/cost/price tradeoffs in saturated markets are considered. A product is modeled by $d$ real-valued qualities whose sum determines the unit cost of producing the product. This leads to the following optimization problem: given a set of $n$ customers, each of whom has certain minimum quality requirements and a maximum price they are willing to pay, design a new product and select a price for that product in order to maximize the resulting profit. An $O(n\log n)$ time algorithm is given for the case, $d=1$, of linear products, and $O(n(\log n)^{d+1})$ time approximation algorithms are given for products with any constant number, $d$, of qualities. To achieve the latter result, an $O(nk^{d-1})$ bound on the complexity of an arrangement of homothetic simplices in $\R^d$ is given, where $k$ is the maximum number of simplices that all contain a single points.

cs.CG↗

arXiv2008

Communication-Efficient Construction of the Plane Localized Delaunay Graph

Let $V$ be a finite set of points in the plane. We present a 2-local algorithm that constructs a plane $\frac{4 π\sqrt{3}}{9}$-spanner of the unit-disk graph $\UDG(V)$. This algorithm makes only one round of communication and each point of $V$ broadcasts at most 5 messages. This improves the previously best message-bound of 11 by Araújo and Rodrigues (Fast localized Delaunay triangulation, Lecture Notes in Computer Science, volume 3544, 2004).

cs.CG↗

arXiv2008

On the Stretch Factor of Convex Delaunay Graphs

Let C be a compact and convex set in the plane that contains the origin in its interior, and let S be a finite set of points in the plane. The Delaunay graph DG_C(S) of S is defined to be the dual of the Voronoi diagram of S with respect to the convex distance function defined by C. We prove that DG_C(S) is a t-spanner for S, for some constant t that depends only on the shape of the set C. Thus, for any two points p and q in S, the graph DG_C(S) contains a path between p and q whose Euclidean length is at most t times the Euclidean distance between p and q.

cs.CG↗

arXiv2007

Spanners of Complete $k$-Partite Geometric Graphs

We address the following problem: Given a complete $k$-partite geometric graph $K$ whose vertex set is a set of $n$ points in $\mathbb{R}^d$, compute a spanner of $K$ that has a ``small'' stretch factor and ``few'' edges. We present two algorithms for this problem. The first algorithm computes a $(5+ε)$-spanner of $K$ with O(n) edges in $O(n \log n)$ time. The second algorithm computes a $(3+ε)$-spanner of $K$ with $O(n \log n)$ edges in $O(n \log n)$ time. The latter result is optimal: We show that for any $2 \leq k \leq n - Θ(\sqrt{n \log n})$, spanners with $O(n \log n)$ edges and stretch factor less than 3 do not exist for all complete $k$-partite geometric graphs.

cs.CG↗

arXiv2007

Geometric Spanners With Small Chromatic Number

Given an integer $k \geq 2$, we consider the problem of computing the smallest real number $t(k)$ such that for each set $P$ of points in the plane, there exists a $t(k)$-spanner for $P$ that has chromatic number at most $k$. We prove that $t(2) = 3$, $t(3) = 2$, $t(4) = \sqrt{2}$, and give upper and lower bounds on $t(k)$ for $k>4$. We also show that for any $ε>0$, there exists a $(1+ε)t(k)$-spanner for $P$ that has $O(|P|)$ edges and chromatic number at most $k$. Finally, we consider an on-line variant of the problem where the points of $P$ are given one after another, and the color of a point must be assigned at the moment the point is given. In this setting, we prove that $t(2) = 3$, $t(3) = 1+ \sqrt{3}$, $t(4) = 1+ \sqrt{2}$, and give upper and lower bounds on $t(k)$ for $k>4$.

cs.CG↗

arXiv2007

Sparse geometric graphs with small dilation

Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.

cs.CG↗

arXiv2007

On a family of strong geometric spanners that admit local routing strategies

We introduce a family of directed geometric graphs, denoted $\paz$, that depend on two parameters $λ$ and $θ$. For $0\leq θ<\fracπ{2}$ and ${1/2} < λ< 1$, the $\paz$ graph is a strong $t$-spanner, with $t=\frac{1}{(1-λ)\cosθ}$. The out-degree of a node in the $\paz$ graph is at most $\lfloor2π/\min(θ, \arccos\frac{1}{2λ})\rfloor$. Moreover, we show that routing can be achieved locally on $\paz$. Next, we show that all strong $t$-spanners are also $t$-spanners of the unit disk graph. Simulations for various values of the parameters $λ$ and $θ$ indicate that for random point sets, the spanning ratio of $\paz$ is better than the proven theoretical bounds.

cs.CG↗

arXiv2003

Range Mode and Range Median Queries on Lists and Trees

We consider algorithms for preprocessing labelled lists and trees so that, for any two nodes u and v we can answer queries of the form: What is the mode or median label in the sequence of labels on the path from u to v.

cs.DS↗