Sébastien Collette

Integrating job parallelism in real-time scheduling theory

We investigate the global scheduling of sporadic, implicit deadline, real-time task systems on multiprocessor platforms. We provide a task model which integrates job parallelism. We prove that the time-complexity of the feasibility problem of these systems is linear relatively to the number of (sporadic) tasks for a fixed number of processors. We propose a scheduling algorithm theoretically optimal (i.e., preemptions and migrations neglected). Moreover, we provide an exact feasibility utilization bound. Lastly, we propose a technique to limit the number of migrations and preemptions.

Decomposition of Multiple Coverings into More Parts

We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any locally finite αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth. The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery life.

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 .

Empty region graphs

A family of proximity graphs, called Empty Region Graphs (ERG) is presented. The vertices of an ERG are points in the plane, and two points are connected if their neighborhood, defined by a region, does not contain any other point. The region defining the neighborhood of two points is a parameter of the graph. This way of defining graphs is not new, and ERGs include several known proximity graphs such as Nearest Neighbor Graphs, @b-Skeletons or @Q-Graphs. The main contribution is to provide insight and connections between the definition of ERG and the properties of the corresponding graphs. We give conditions on the region defining an ERG to ensure a number of properties that might be desirable in applications, such as planarity, connectivity, triangle-freeness, cycle-freeness, bipartiteness and bounded degree. These conditions take the form of what we call tight regions: maximal or minimal regions that a region must contain or be contained in to make the graph satisfy a given property. We show that every monotone property has at least one corresponding tight region; we discuss possibilities and limitations of this general model for constructing a graph from a point set.

Reconfiguration of Cube-Style Modular Robots Using O(logn) Parallel Moves

We consider a model of reconfigurable robot, introduced and prototyped by the robotics community. The robot consists of independently manipulable unit-square atoms that can extend/contract arms on each side and attach/detach from neighbors. The optimal worst-case number of sequential moves required to transform one connected configuration to another was shown to be θ(n) at ISAAC 2007. However, in principle, atoms can all move simultaneously. We develop a parallel algorithm for reconfiguration that runs in only O(logn) parallel steps, although the total number of operations increases slightly to θ(n logn). The result is the first (theoretically) almost-instantaneous universally reconfigurable robot built from simple units.

Non-crossing matchings of points with geometric objects

Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point-object pairs. In this paper, we address the algorithmic problem of determining whether a non-crossing matching exists between a given point-object pair. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their size is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete.

Optimal location of transportation devices

We consider algorithms for finding the optimal location of a simple transportation device, that we call a moving walkway, consisting of a pair of points in the plane between which the travel speed is high. More specifically, one can travel from one endpoint of the walkway to the other at speed v>1, but can only travel at unit speed between any other pair of points. The travel time between any two points in the plane is the minimum between the actual geometric distance, and the time needed to go from one point to the other using the walkway. A location for a walkway is said to be optimal with respect to a given finite set of points if it minimizes the maximum travel time between any two points of the set. We give a simple linear-time algorithm for finding an optimal location in the case where the points are on a line. We also give an @W(nlogn) lower bound for the problem of computing the travel time diameter of a set of n points on a line with respect to a given walkway. Then we describe an O(nlogn) algorithm for locating a walkway with the additional restriction that the walkway must be horizontal. This algorithm is based on a recent generic method for solving quasiconvex programs with implicitly defined constraints. It is used in a (1+@e)-approximation algorithm for optimal location of a walkway of arbitrary orientation.

Coloring hypergraphs induced by dynamic point sets and bottomless rectangles

We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color. We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k)=3k−2. This can be interpreted as coloring point sets in ℝ2 with k colors such that any bottomless rectangle containing at least 3k−2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set. For this problem, we also prove a lower bound p(k)>ck, where c>1.67. Hence, for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function p(k) exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).

Entropy, triangulation, and point location in planar subdivisions

A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D, is &Htilde; + O(&Htilde;1/2+1) where &Htilde;=&Htilde;(G,D) is a lower bound on the expected number of point-line comparisons performed by any linear decision tree for point location in G under the query distribution D. The preprocessing algorithm runs in O(n log n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has near-minimum entropy.

Some properties of k-Delaunay and k-Gabriel graphs

We consider two classes of higher order proximity graphs defined on a set of points in the plane, namely, the k-Delaunay graph and the k-Gabriel graph. We give bounds on the following combinatorial and geometric properties of these graphs: spanning ratio, diameter, connectivity, chromatic number, and minimum number of layers necessary to partition the edges of the graphs so that no two edges of the same layer cross.