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We study online routing algorithms on the $\Theta$6-graph and the half-$\Theta$6-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex s and a target vertex t in the $\Theta$6-graph (resp. half-$\Theta$6-graph), there exists a deterministic online routing algorithm that finds a path from s to t whose length is at most 2 st (resp. 2.89 st) which is optimal in the worst case [Bose et al., siam J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the $\Theta$6-graph and half-$\Theta$6-graph defined on a Poisson point process. For the $\Theta$6-graph, our online routing algorithm has an expected routing ratio of 1.161 (when s and t random) and a maximum expected routing ratio of 1.22 (maximum for fixed s and t where all other points are random), much better than the worst-case routing ratio of 2. For the half-$\Theta$6-graph, our memoryless online routing algorithm has an expected routing ratio of 1.43 and a maximum expected routing ratio of 1.58. Our online routing algorithm that uses a constant amount of additional memory has an expected routing ratio of 1.34 and a maximum expected routing ratio of 1.40. The additional memory is only used to remember the coordinates of the starting point of the route. Both of these algorithms have an expected routing ratio that is much better than their worst-case routing ratio of 2.89.

A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probability distributions on order types defined in this way are heavily concentrated and therefore sample inefficiently the space of order types. We present two results on this question. First, we study experimentally the bias in the order types of $n$ random points chosen uniformly and independently in a square, for $n$ up to $16$. Second, we study algorithms for determining the order type of a point set in terms of the number of coordinate bits they require to know. We give an algorithm that requires on average $4n \log\_2 n+O(n)$ bits to determine the order type of $P$, and show that any algorithm requires at least $4n \log\_2 n - O(n \log\log n)$ bits. This implies that the concentration conjecture cannot be proven by an "efficient encoding" argument.

Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we propose to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasi-linear time by a very simple algorithm., Comment: 12 pages, 6 figures

We present a new oblivious walking strategy for convex subdivisions. Our walk is faster than the straight walk and more generally applicable than the visibility walk. To prove termination of our walk we use a novel monotonically decreasing distance measure.

We study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d \geq 2$, there is a set of $d+1$ paths that does not admit a monotone simultaneous geometric embedding.

Let $X:=X_n\cup\{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson point process of intensity $n$. We provide a first non-trivial lower bound for the distance between the expected length of the shortest path between $(0,0)$ and $(1,0)$ in the Delaunay triangulation associated with $X$ when the intensity of $X_n$ goes to infinity. Experimental values indicate that the correct value is about 1.04. We also prove that the expected number of Delaunay edges crossed by the line segment $[(0,0),(1,0)]$ is equivalent to $2.16\sqrt{n}$ and that the expected length of a particular path converges to 1.18 giving an upper bound on the stretch factor., Comment: 30 pages

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 little analysis is available for the properties of the paths generated and the computational resources required to generate them under a random distribution hypothesis for the input. In this paper we analyse a new deterministic planar navigation algorithm with constant competitiveness which follows vertex adjacencies in the Delaunay triangulation. We call this strategy cone walk. We prove that given $n$ uniform points in a smooth convex domain of unit area, and for any start point $z$ and query point $q$; cone walk applied to $z$ and $q$ will access at most $O(|zq|\sqrt{n} +\log^7 n)$ sites with complexity $O(|zq|\sqrt{n} \log \log n + \log^7 n)$ with probability tending to 1 as $n$ goes to infinity. We additionally show that in this model, cone walk is $(\log ^{3+\xi} n)$-memoryless with high probability for any pair of start and query point in the domain, for any positive $\xi$. We take special care throughout to ensure our bounds are valid even when the query points are arbitrarily close to the border.

Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is increasing in n. In dimension d>=3 we prove that if lim(E((f[d -1](Kn))/(An^c)->1 when n->infinity for some constants A and c > 0 then the function E(f[d-1](Kn)) is increasing for n large enough. In particular, the number of facets of the convex hull of n random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.

We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing of $G$ that is crossing-free and internally (weakly) convex, on a regular grid $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in the periodic representation) with $n$ vertices, we can compute in linear time a periodic straight-line drawing of $G$ that is crossing-free and (weakly) convex, on a periodic regular grid $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$., Comment: 37 pages

Robustness problems due to the substitution of the exact computation on real numbers by the rounded floating point arithmetic are often an obstacle to obtain practical implementation of geometric algorithms. If the adoption of the --exact computation paradigm--[Yap et Dube] gives a satisfactory solution to this kind of problems for purely combinatorial algorithms, this solution does not allow to solve in practice the case of algorithms that cascade the construction of new geometric objects. In this report, we consider the problem of rounding the intersection of two polygonal regions onto the integer lattice with inclusion properties. Namely, given two polygonal regions A and B having their vertices on the integer lattice, the inner and outer rounding modes construct two polygonal regions with integer vertices which respectively is included and contains the true intersection. We also prove interesting results on the Hausdorff distance, the size and the convexity of these polygonal regions.

We investigate the lines tangent to four triangles in R^3. By a construction, there can be as many as 62 tangents. We show that there are at most 162 connected components of tangents, and at most 156 if the triangles are disjoint. In addition, if the triangles are in (algebraic) general position, then the number of tangents is finite and it is always even., Comment: 13 pages, added missing case in Lemma 5

The compression of geometric structures is a relatively new field of data compression. Since about 1995, several articles have dealt with the coding of meshes, using for most of them the following approach: the vertices of the mesh are coded in an order such that it contains partially the topology of the mesh. In the same time, some simple rules attempt to predict the position of the current vertex from the positions of its neighbours that have been previously coded. In this article, we describe a compression algorithm whose principle is completely different: the order of the vertices is used to compress their coordinates, and then the topology of the mesh is reconstructed from the vertices. This algorithm, particularly suited for terrain models, achieves compression factors that are slightly greater than those of the currently available algorithms, and moreover, it allows progressive and interactive transmission of the meshes., Comment: 16 pages, 10 figures

Given a finite set of non-collinear points in the plane, there exists a line that passes through exactly two points. Such a line is called an ordinary line. An efficient algorithm for computing such a line was proposed by Mukhopadhyay et al. In this note we extend this result in two directions. We first show how to use this algorithm to compute an ordinary conic, that is, a conic passing through exactly five points, assuming that all the points do not lie on the same conic. Both our proofs of existence and the consequent algorithms are simpler than previous ones. We next show how to compute an ordinary hyperplane in three and higher dimensions., Comment: 7 pages, 2 figures

A circle $C$ separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An Theta(n log n) optimal algorithm is proposed to find all largest circles separating two given sets of line segments when line segments are allowed to meet only at their endpoints. In the general case, when line segments may intersect $\Omega(n^2)$ times, our algorithm can be adapted to work in O(n alpha(n) log n) time and O(n \alpha(n)) space, where alpha(n) represents the extremely slowly growing inverse of the Ackermann function., Comment: 14 pages, 3 figures, abstract presented at 8th Canadian Conference on Computational Geometry, 1996

We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with $n$ vertices. We present an O(m+n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest possible., Comment: 11 pages, 5 figures, abstract presented at European Symposium on Algorithms 1993

Two planar sets are circularly separable if there exists a circle enclosing one of the sets and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A linear-time algorithm is proposed to decide if two polygons are circularly separable. The algorithm outputs the smallest separating circle. The second problem asks for the largest circle included in a preprocessed, convex polygon, under some point and/or line constraints. The resulting circle must contain the query points and it must lie in the halfplanes delimited by the query lines., Comment: 23 pages, 7 figures, abstract presented at SODA95

We study the problem of computing the free space F of a simple legged robot called the spider robot. The body of this robot is a single point and the legs are attached to the body. The robot is subject to two constraints: each leg has a maximal extension R (accessibility constraint) and the body of the robot must lie above the convex hull of its feet (stability constraint). Moreover, the robot can only put its feet on some regions, called the foothold regions. The free space F is the set of positions of the body of the robot such that there exists a set of accessible footholds for which the robot is stable. We present an efficient algorithm that computes F in O(n2 log n) time using O(n2 alpha(n)) space for n discrete point footholds where alpha(n) is an extremely slowly growing function (alpha(n) <= 3 for any practical value of n). We also present an algorithm for computing F when the foothold regions are pairwise disjoint polygons with n edges in total. This algorithm computes F in O(n2 alpha8(n) log n) time using O(n2 alpha8(n)) space (alpha8(n) is also an extremely slowly growing function). These results are close to optimal since Omega(n2) is a lower bound for the size of F., Comment: 29 pages, 22 figures, prelininar results presented at WAFR94 and IEEE Robotics & Automation 94

The assumption of real-number arithmetic, which is at the basis of conventional geometric algorithms, has been seriously challenged in recent years, since digital computers do not exhibit such capability. A geometric predicate usually consists of evaluating the sign of some algebraic expression. In most cases, rounded computations yield a reliable result, but sometimes rounded arithmetic introduces errors which may invalidate the algorithms. The rounded arithmetic may produce an incorrect result only if the exact absolute value of the algebraic expression is smaller than some (small) varepsilon, which represents the largest error that may arise in the evaluation of the expression. The threshold varepsilon depends on the structure of the expression and on the adopted computer arithmetic, assuming that the input operands are error-free. A pair (arithmetic engine,threshold) is an "arithmetic filter". In this paper we develop a general technique for assessing the efficacy of an arithmetic filter. The analysis consists of evaluating both the threshold and the probability of failure of the filter. To exemplify the approach, under the assumption that the input points be chosen randomly in a unit ball or unit cube with uniform density, we analyze the two important predicates "which-side" and "insphere". We show that the probability that the absolute values of the corresponding determinants be no larger than some positive value V, with emphasis on small V, is Theta(V) for the which-side predicate, while for the insphere predicate it is Theta(V^(2/3)) in dimension 1, O(sqrt(V)) in dimension 2, and O(sqrt(V) ln(1/V)) in higher dimensions. Constants are small, and are given in the paper., Comment: 22 pages 7 figures Results for in sphere test inproved in cs.CG/9907028

An efficient technique to solve precision problems consists in using exact computations. For geometric predicates, using systematically expensive exact computations can be avoided by the use of filters. The predicate is first evaluated using rounding computations, and an error estimation gives a certificate of the validity of the result. In this note, we studies the statistical efficiency of filters for cosphericity predicate with an assumption of regular distribution of the points. We prove that the expected value of the polynomial corresponding to the in sphere test is greater than epsilon with probability O(epsilon log 1/epsilon) improving the results of a previous paper by the same authors., Comment: 7 pages 2 figures presented at the 15th European Workshop Comput. Geom., 113--116, 1999 improve previous results (in other paper)

This paper presents how the space of spheres and shelling may be used to delete a point from a $d$-dimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we notice that this number only applies to low cost operations, while time consuming computations are only done a linear number of times. This algorithm may be viewed as a variation of Heller's algorithm, which is popular in the geographic information system community. Unfortunately, Heller algorithm is false, as explained in this paper., Comment: 15 pages 5 figures. in Proc. 15th Annu. ACM Sympos. Comput. Geom., 181--188, 1999

We provide a lower bound construction showing that the union of unit balls in three-dimensional space has quadratic complexity, even if they all contain the origin. This settles a conjecture of Sharir., Comment: 5 pages, 5 figures

We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the levels below. Point location is done by marching in a triangulation to determine the nearest neighbor of the query at that level, then the march restarts from that neighbor at the level below. Using a small sample (3%) allows a small memory occupation; the march and the use of the nearest neighbor to change levels quickly locate the query., Comment: 19 pages, 7 figures Proc. 14th Annu. ACM Sympos. Comput. Geom., 106--115, 1998

We use here the results on the influence graph by Boissonnat et al. to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from O(n log n) to O(n log star n). This technique applies in the following applications: triangulation of a simple polygon, skeleton of a simple polygon, Delaunay triangulation of points knowing the EMST (euclidean minimum spanning tree)., Comment: 16 pages, 6 figures, Proc. 3rd Canad. Conf. Comput. Geom., 1991