arXiv subjects

Stephane Durocher

Publications and source records attributed to Stephane Durocher.

25 records · Page 2Linked to original sources

arXiv2012

Robust Non-Parametric Data Approximation of Pointsets via Data Reduction

In this paper we present a novel non-parametric method of simplifying piecewise linear curves and we apply this method as a statistical approximation of structure within sequential data in the plane. We consider the problem of minimizing the average length of sequences of consecutive input points that lie on any one side of the simplified curve. Specifically, given a sequence $P$ of $n$ points in the plane that determine a simple polygonal chain consisting of $n-1$ segments, we describe algorithms for selecting an ordered subset $Q \subset P$ (including the first and last points of $P$) that determines a second polygonal chain to approximate $P$, such that the number of crossings between the two polygonal chains is maximized, and the cardinality of $Q$ is minimized among all such maximizing subsets of $P$. Our algorithms have respective running times $O(n^2\log n)$ when $P$ is monotonic and $O(n^2\log^2 n)$ when $P$ is an arbitrary simple polyline. Finally, we examine the application of our algorithms iteratively in a bootstrapping technique to define a smooth robust non-parametric approximation of the original sequence.

cs.CG↗

arXiv2011

Bounding Interference in Wireless Ad Hoc Networks with Nodes in Random Position

The interference at a wireless node s can be modelled by the number of wireless nodes whose transmission ranges cover s. Given a set of positions for wireless nodes, the interference minimization problem is to assign a transmission radius (equivalently, a power level) to each node such that the resulting communication graph is connected, while minimizing the maximum interference. We consider the model introduced by von Rickenback et al. (2005), in which each transmission range is represented by a ball and edges in the communication graph are symmetric. The problem is NP-complete in two dimensions (Buchin 2008) and no polynomial-time approximation algorithm is known. Furthermore, even in one dimension (the highway model), the problem's complexity is unknown and the maximum interference of a set of n wireless nodes can be as high as Theta(sqrt(n)) (von Rickenback et al. 2005). In this paper we show how to solve the problem efficiently in settings typical for wireless ad hoc networks. In particular, we show that if node positions are represented by a set P of n points selected uniformly and independently at random over a d-dimensional rectangular region, for any fixed d, then the topology given by the closure of the Euclidean minimum spanning tree of P has maximum interference O(log n) with high probability. We extend this bound to a general class of communication graphs over a broad set of probability distributions. Next we present a local algorithm that constructs a graph from this class; this is the first local algorithm to provide an upper bound on the expected maximum interference. Finally, we discuss an empirical evaluation of our algorithm with a suite of simulation results.

cs.CG↗

arXiv2011

A Simple Linear-Space Data Structure for Constant-Time Range Minimum Query

We revisit the range minimum query problem and present a new O(n)-space data structure that supports queries in O(1) time. Although previous data structures exist whose asymptotic bounds match ours, our goal is to introduce a new solution that is simple, intuitive, and practical without increasing costs for query time or space.

cs.DS↗

arXiv2011

Linear-Space Data Structures for Range Mode Query in Arrays

A mode of a multiset $S$ is an element $a \in S$ of maximum multiplicity; that is, $a$ occurs at least as frequently as any other element in $S$. Given a list $A[1:n]$ of $n$ items, we consider the problem of constructing a data structure that efficiently answers range mode queries on $A$. Each query consists of an input pair of indices $(i, j)$ for which a mode of $A[i:j]$ must be returned. We present an $O(n^{2-2ε})$-space static data structure that supports range mode queries in $O(n^ε)$ time in the worst case, for any fixed $ε\in [0,1/2]$. When $ε= 1/2$, this corresponds to the first linear-space data structure to guarantee $O(\sqrt{n})$ query time. We then describe three additional linear-space data structures that provide $O(k)$, $O(m)$, and $O(|j-i|)$ query time, respectively, where $k$ denotes the number of distinct elements in $A$ and $m$ denotes the frequency of the mode of $A$. Finally, we examine generalizing our data structures to higher dimensions.

cs.DS↗

arXiv2009

Improved Approximation Algorithms for Segment Minimization in Intensity Modulated Radiation Therapy

he segment minimization problem consists of finding the smallest set of integer matrices that sum to a given intensity matrix, such that each summand has only one non-zero value, and the non-zeroes in each row are consecutive. This has direct applications in intensity-modulated radiation therapy, an effective form of cancer treatment. We develop three approximation algorithms for matrices with arbitrarily many rows. Our first two algorithms improve the approximation factor from the previous best of $1+\log_2 h $ to (roughly) $3/2 \cdot (1+\log_3 h)$ and $11/6\cdot(1+\log_4{h})$, respectively, where $h$ is the largest entry in the intensity matrix. We illustrate the limitations of the specific approach used to obtain these two algorithms by proving a lower bound of $\frac{(2b-2)}{b}\cdot\log_b{h} + \frac{1}{b}$ on the approximation guarantee. Our third algorithm improves the approximation factor from $2 \cdot (\log D+1)$ to $24/13 \cdot (\log D+1)$, where $D$ is (roughly) the largest difference between consecutive elements of a row of the intensity matrix. Finally, experimentation with these algorithms shows that they perform well with respect to the optimum and outperform other approximation algorithms on 77% of the 122 test cases we consider, which include both real world and synthetic data.

cs.DS↗

arXiv2000

Toward the Rectilinear Crossing Number of $K_n$: New Drawings, Upper Bounds, and Asymptotics

Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear, and that no three edges intersect in a point unless that point is an endpoint of all three. The rectilinear crossing number of K_n is the fewest number of edge crossings attainable over all rectilinear drawings of K_n. For each n we construct a rectilinear drawing of K_n that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of drawings of K_n with good asymptotics. Finally, we mention some old and new open problems.

cs.DM↗

arXiv2000

The Rectilinear Crossing Number of K_10 is 62

A drawing of a graph G in the plane is said to be a rectilinear drawing of G if the edges are required to be line segments (as opposed to Jordan curves). We assume no three vertices are collinear. The rectilinear crossing number of G is the fewest number of edge crossings attainable over all rectilinear drawings of G. Thanks to Richard Guy, exact values of the rectilinear crossing number of K_n, the complete graph on n vertices, for n = 3,...,9, are known (Guy 1972, White and Beinke 1978, Finch 2000, Sloanes A014540). Since 1971, thanks to the work of David Singer (1971, Gardiner 1986), the rectilinear crossing number of K_10 has been known to be either 61 or 62, a deceptively innocent and tantalizing statement. The difficulty of determining the correct value is evidenced by the fact that Singer's result has withstood the test of time. In this paper we use a purely combinatorial argument to show that the rectilinear crossing number of K_10 is 62. Moreover, using this result, we improve an asymptotic lower bound for a related problem. Finally, we close with some new and old open questions that were provoked, in part, by the results of this paper, and by the tangled history of the problem itself.

cs.DM↗