Raimund Seidel

How good are convex hull algorithms

Abstract A convex polytope P can be specified in two ways: as the convex hull of the vertex set V of P , or as the intersection of the set H of its facet-inducing halfspaces. The vertex enumeration problem is to compute V from H>. The facet enumeration problem is to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in | H | + | V | and the dimension. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability to deal well with “degeneracies”, or, inability to control the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fat-lattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly; dwarfed polytopes cause algorithms with bad intermediate size control to perform badly. We also present computational experience with trying to solve these problem on these hard polytopes, using various implementations of the main algorithms.

On the all-pairs-shortest-path problem in unweighted undirected graphs

We present an algorithm, APD, that solves the distance version of the all-pairs-shortest-path problem for undirected, unweighted n-vertex graphs in time O(M(n) log n), where M(n) denotes the time necessary to multiply two n x n matrices of small integers (which is currently known to be o(n 2.376 )). We also address the problem of actually finding a shortest path between each pair of vertices and present a randomized algorithm that matches APD in its simplicity and in its expected running time.

Small-dimensional linear programming and convex hulls made easy

We present two randomized algorithms. One solves linear programs involvingm constraints ind variables in expected timeO(m). The other constructs convex hulls ofn points in ℝd,d>3, in expected timeO(n[d/2]). In both boundsd is considered to be a constant. In the linear programming algorithm the dependence of the time bound ond is of the formd!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses.

Linear programming and convex hulls made easy

We present two randomized algorithms. One solves linear programs involving <italic>m</italic> constraints in <italic>d</italic> variables in expected time <italic>&Ogr;</italic>(<italic>m</italic>). The other constructs convex hulls of <italic>n</italic> points in R<italic><supscrpt>d</supscrpt>, d</italic> > 3, in expected time <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>⌈<italic>d</italic>/2⌉</supscrpt>). In both bounds <italic>d</italic> is considered to be a constant. In the linear programming algorithm the dependence of the time bound on <italic>d</italic> is of the form <italic>d</italic>!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses.

Four results on randomized incremental constructions

We prove four results on randomized incremental constructions (RICs): an analysis of the expected behavior under insertion and deletions, a fully dynamic data structure for convex hull maintenance in arbitrary dimensions, a tail estimate for the space complexity of RICs, a lower bound on the complexity of a game related to RICs.

Constructing higher-dimensional convex hulls at logarithmic cost per face

We exhibit a new approach for dealing with higher dimensional convex hull problems, such as enumerating all facets of the convex hull of a finite point set or constructing the facial lattice of such a convex hull. For fixed dimensions our new algorithms have worst case time complexity O(m 2 -tFlogm), where m is the size of the input point set and F is the size of the output produced. Such a dependence on the output size is desirable since F can range between ~(1) and O(mld/2|). Our time bound is an improvement over the best previously achieved bounds for a large range of values of F. The main tool in our new approach is the notion of a straight line shelling of a polytope.

Efficiently computing and representing aspect graphs of polyhedral objects

An efficient algorithm and a data structure for computing and representing the aspect graph of polyhedral objects under orthographic projection are presented. The aspect graph is an approach to representing 3-D objects by a set of 2-D views, for the purpose of object recognition. In this approach the viewpoint space is partitioned into regions such that in each region the qualitative structure of the line drawing does not change. The viewing data of an object is the partition of the viewpoint space together with a representative view in each region. The algorithm computes the viewing data for line drawings of polyhedral objects under orthographic projection. >

Randomized search trees

A randomized strategy for maintaining balance in dynamically changing search trees that has optimal expected behavior is presented. In particular, in the expected case an update takes logarithmic time and requires fewer than two rotations. Moreover, the update time remains logarithmic, even if the cost of a rotation is taken to be proportional to the size of the rotated subtree. The approach generalizes naturally to weighted trees, where the expected time bounds for accesses and updates again match the worst case time bounds of the best deterministic methods. The balancing strategy and algorithms are exceedingly simple and should be fast in practice.<<ETX>>

Voronoi diagrams and arrangements

We propose a uniform and general framework for defining and dealing with Voronoi Diagrams. In this framework a Voronoi Diagram is a partition of a domain D induced by a finite number of real valued functions on D. Valuable insight can be gained when one considers how these real valued functions partition DXR. With this view it turns out that the standard Euclidean Voronoi Diagram of point sets in R along with its order-&kgr; generalizations are intimately related to certain arrangements of hyperplanes. This fact can be used to obtain new Voronoi Diagram algorithms. We also discuss how the formalism of arrangements can be used to solve certain intersection and union problems.

On the difficulty of triangulating three-dimensional Nonconvex Polyhedra

A number of different polyhedraldecomposition problems have previously been studied, most notably the problem of triangulating a simple polygon. We are concerned with thepolyhedron triangulation problem: decomposing a three-dimensional polyhedron into a set of nonoverlapping tetrahedra whose vertices must be vertices of the polyhedron. It has previously been shown that some polyhedra cannot be triangulated in this fashion. We show that the problem of deciding whether a given polyhedron can be triangulated is NP-complete, and hence likely to be computationally intractable. The problem remains NP-complete when restricted to the case of star-shaped polyhedra. Various versions of the question of how many Steiner points are needed to triangulate a polyhedron also turn out to be NP-hard.