Obstacle Numbers of Graphs

An obstacle representation of a graph G is a drawing of G in the plane with straight-line edges, together with a set of polygons (respectively, convex polygons) called obstacles, such that an edge exists in G if and only if it does not intersect an obstacle. The obstacle number (convex obstacle number) of G is the smallest number of obstacles (convex obstacles) in any obstacle representation of G. In this paper, we identify families of graphs with obstacle number 1 and construct graphs with arbitrarily large obstacle number (convex obstacle number). We prove that a graph has an obstacle representation with a single convex k-gon if and only if it is a circular arc graph with clique covering number at most k in which no two arcs cover the host circle. We also prove independently that a graph has an obstacle representation with a single segment obstacle if and only if it is the complement of an interval bigraph.     

Testing Homotopy for Paths in the Plane

In this paper we present an efficient algorithm to test if two given paths are homotopic; that is, whether they wind around obstacles in the plane in the same way. For paths specified by n line segments with obstacles described by n points, several standard ways achieve quadratic running time. For simple paths, our algorithm runs in O(n log n) time, which we show is tight. For self-intersecting paths the problem is related to Hopcrofts problem; our algorithm runs in O(n ^3/2log n) time.     

Optimality of the Delaunay triangulation in ℝ d

In this paper we present new optimality results for the Delaunay triangulation of a set of points in ℝ ^d . These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ℝ ^d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.     

Improved heuristics for the minimum weight triangulation problem

Given a planar point setS, a triangulation ofS is a maximal set of non-intersecting line segments connecting the points. The minimum weight triangulation problem is to find a triangulation ofS such that the sum of the lengths of the line segments in it is the smallest. No polynomial time algorithm is known to produce the optimal or even a constant approximation of the optimal solution, and it is also unknown whether the problem is NP-hard. In this paper, we propose two improved heuristics, which triangulate a set ofn points in a plane inO(n ³) time and never do worse than the minimum spanning tree triangulation algorithm given by Lingas and the greedy spanning tree triangulation algorithm given by Heath and Pemmaraju. These two algorithms both produce an optimal triangulation if the points are the vertices of a convex polygon, and also do the same in some special cases.     

Computational Complexity of the Vertex Cover Problem in the Class of Planar Triangulations

We study the computational complexity of the vertex cover problem in the class of planar graphs (planar triangulations) admitting a plane representation whose faces are triangles. It is shown that the problem is strongly NP-hard in the class of 4-connected planar triangulations in which the degrees of vertices are of order O(log n), where n is the number of vertices, and in the class of plane 4-connected Delaunay triangulations based on the Minkowski triangular distance. A pair of vertices in such a triangulation is adjacent if and only if there is an equilateral triangle ?(p, λ) with p ∈ R² and λ > 0 whose interior does not contain triangulation vertices and whose boundary contains this pair of vertices and only it, where ?(p, λ) = p + λ? = {x ∈ R²: x = p + λa, a ∈ ?}; here ? is the equilateral triangle with unit sides such that its barycenter is the origin and one of the vertices belongs to the negative y-axis. Keywords: computational complexity, Delaunay triangulation, Delaunay TD-triangulation.     

Covering and guarding polygons using L k -sets

The Art Gallery Problem is the problem of finding a minimum number of points (called guards) in a given polygon such that every point in the polygon is visible to at least one of the guards. Chvátal [5] was the first to show that, in the worst case, [n/3] such points will suffice for any polygon of n sides. O'Rourke [15] later showed that only [n/4] guards were needed if line segments, rather than points, were allowed as guards. In this paper, we unify these results, and extend them to many other classes of guards, while using a generalization of visibility known as link-visibility. In particular, we present the following theorems:

A class of simplicial restart fixed point algorithms without an extra dimension

In an earlier paper we introduced an algorithm for approximating a fixed point of a mapping on the product space of unit simplices. Ideas of that paper are used to construct a class of triangulations ofR ⁿ. More precisely, for somek, 1 k n, and positive integersm ₁ , m_k with sumn, a triangulation ofR ⁿ is obtained by triangulating the cells which are formed by taking the product of given triangulations ofR ^mj, j = 1, ,k. The triangulation of each cell will be defined in relation to an arbitrarily chosen pointv inR ⁿ, being the starting point of the algorithm. Fork = n we obtain theK triangulation originally due to Todd. Each element of the class can be used to find a simplex which approximates a fixed point of a mapping onR ⁿ by generating a unique path of adjacent simplices of variable dimension starting with the pointv. We also give convergence conditions. It is indicated how in casek = n a connected set of fixed points can be generated. Moreover, we give some computational experience.     

On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space

We show that the number of critical positions of a convex polygonal objectB moving amidst polygonal barriers in two-dimensional space, at which it makes three simultaneous contacts with the obstacles but does not penetrate into any obstacle isO(kn _s (kn)) for somes6, wherek is the number of boundary segments ofB,n is the number of wall segments, and _s (q) is an almost linear function ofq yielding the maximal number of breakpoints along the lower envelope (i.e., pointwise minimum) of a set ofq continuous functions each pair of which intersect in at mosts points (here a breakpoint is a point at which two of the functions simultaneously attain the minimum). We also present an example where the number of such critical contacts is (k ² n ²), showing that in the worst case our upper bound is almost optimal.Work on this paper by the second author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation.     

ON STABLE LINE SEGMENTS IN ALL TRIANGULATIONS OF A PLANAR POINT SET

Abstract. Let S be a set of finite plauar points. A llne segment L(p, q) with p, q E Sis called a stable line segment of S, if there is no Line segment with two endpoints in S intersecting L(p, q). In this paper, some geometric properties of the set of all stable line segments     

Nice Point Sets Can Have Nasty Delaunay Triangulations

   Abstract. We consider the complexity of Delaunay triangulations of sets of points in R ³ under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of n points in R ³ with spread Δ has complexity Ω(min{ Δ³, nΔ, n² }) and O(min{ Δ⁴, n ² }). For the case

Dynamic partition trees

In this paper we study dynamic variants of conjugation trees and related structures that have recently been introduced for performing various types of queries on sets of points and line segments, like half-planar range searching, shooting, intersection queries, etc. For most of these types of queries dynamic structures are obtained with an amortized update time ofO(log² n) (or less) with only minor increases in query times. As an application of the method we obtain an output-sensitive method for hidden surface removal in a set ofn triangles that runs in timeO(nlogn+n ^· k ) where=log₂((1+5)/2) 0.695 andk is the size of the visibility map obtained.Research of the second author was partially supported by the ESPRIT II Basic Research Actions Program of the EC, under contract No. 3075 (project ALCOM).     

Algorithms for graphs of bounded treewidth via orthogonal range searching

We show that, for any fixed constant k3, the sum of the distances between all pairs of vertices of an abstract graph with n vertices and treewidth at most k can be computed in O(nlog^k−1n) time.We also show that, for any fixed constant k2, the dilation of a geometric graph (i.e., a graph drawn in the plane with straight-line segments) with n vertices and treewidth at most k can be computed in O(nlog^k+1n) expected time. The dilation (or stretch-factor) of a geometric graph is defined as the largest ratio, taken over all pairs of vertices, between the distance measured along the graph and the Euclidean distance.The algorithms for both problems are based on the same principle: data structures for orthogonal range searching in bounded dimension provide a compact representation of distances in abstract graphs of bounded treewidth.     

Finding thek smallest spanning trees

We give improved solutions for the problem of generating thek smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes timeO(m log(m, n)=k ²); for planar graphs this bound can be improved toO(n+k ²). We also show that thek best spanning trees for a set of points in the plane can be computed in timeO(min(k ² n+n logn,k ²+kn log(n/k))). Thek best orthogonal spanning trees in the plane can be found in timeO(n logn+kn log log(n/k)+k ²).     

Every Set of Disjoint Line Segments Admits a Binary Tree

Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist configurations of line segments where any such tree requires degree 3. We provide an O(nlog n) time algorithm for constructing such a tree, and show that this is optimal. Received September 14, 1999, and in revised form January 17, 2001. Online publication August 29, 2001.     

Alternating Paths along Axis-Parallel Segments

It is shown that for a set S of n pairwise disjoint axis-parallel line segments in the plane there is a simple alternating path of length . This bound is best possible in the worst case. In the special case that the n pairwise disjoint axis-parallel line segments are protruded (that is, if the intersection point of the lines through every two nonparallel segments is not visible from both segments), there is a simple alternating path of length n. Work on this paper was partially supported by National Science Foundation grants CCR-0049093 and IIS-0121562. A preliminary version of this paper has appeared in the Proceedings of the 8th International Workshop on Algorithms and Data Structures (Ottawa, ON, 2003), vol. 2748 of Lecture Notes on Computer Science, Springer, Berlin, 2003, pp. 389–400.     

Tight Bounds for Connecting Sites Across Barriers

Given m points (sites) and n obstacles (barriers) in the plane, we address the problem of finding a straight line minimum cost spanning tree on the sites, where the cost is proportional to the number of intersections (crossings) between tree edges and barriers. If the barriers are infinite lines, it is known that there is a spanning tree such that every barrier is crossed by tree edges, and this bound is asymptotically optimal. Asano et al. showed that if the barriers are pairwise disjoint line segments, then there is a spanning tree such that every barrier crosses at most 4 tree edges and so the total cost is at most 4n. Lower bound constructions are known with 3 crossings per barrier and 2n total cost. We obtain tight bounds on the minimum cost of spanning trees in the special case where the barriers are interior disjoint line segments that form a convex subdivision of the plane and there is a point in every cell of the subdivision. In particular, we show that there is a spanning tree such that every barrier crosses at most 2 tree edges, and there is a spanning tree of total cost 5n/3. Both bounds are the best possible. Work by Eynat Rafalin and Diane Souvaine was supported by the National Science Foundation under Grant #CCF-0431027. E. Rafalin’s research conducted while at Tufts University.     

Some structural properties of minimally contraction-critically 5-connected graphs

An edge of a k-connected graph is said to be k-removable (resp. k-contractible) if the removal (resp. the contraction ) of the edge results in a k-connected graph. A k-connected graph with neither k-removable edge nor k-contractible edge is said to be minimally contraction-critically k-connected. We show that around an edge whose both end vertices have degree greater than 5 of a minimally contraction-critically 5-connected graph, there exists one of two specified configurations. Using this fact, we prove that each minimally contraction-critically 5-connected graph on n vertices has at least vertices of degree 5.     

An efficient motion-planning algorithm for a convex polygonal object in two-dimensional polygonal space

We present an efficient algorithm for planning the motion of a convex polygonal bodyB in two-dimensional space bounded by a collection of polygonal obstacles. Our algorithm extends and combines the techniques of Leven and Sharir and of Sifrony and Sharir used for the case in whichB is a line segment (a ladder). It also makes use of the results of Kedem and Sharir on the planning of translational motion ofB amidst polygonal obstacles, and of a recent result of Leven and Sharir on the number of free critical contacts ofB with such polygonal obstacles. The algorithm runs in timeO(kn ₆(kn) logkn), wherek is the number of sides ofB, n is the number of obstacle edges, and ,(q) is an almost linear function ofq yielding the maximal number of connected portions ofq continuous functions which compose the graph of their lower envelope, where it is assumed that each pair of these functions intersect in at mosts points.Work on this paper by the second author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation.     

A Construction of Almost Steiner Systems

Let n, k, and t be integers satisfying . A Steiner system with parameters t, k, and n is a k‐uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for . In this note we prove that for every and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k‐uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.