For a polyhedron @math P let @math B(P) denote the polytopal complex that is formed by all bounded faces of @math P. If @math P is the intersection of @math n halfspaces in @math RD, but the maximum dimension @math d of any face in @math B(P) is much smaller, we show that the combinatorial complexity of @math P cannot be too high; in particular, that it is independent of @math D. We show that the number of vertices of @math P is @math O(nd) and the total number of bounded faces of the polyhedron is @math O(nd2). For inputs in general position the number of bounded faces is @math O(nd). We show that for certain specific values of @math d and @math D, our bounds are tight. For any fixed @math d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a number of linear programs that is polynomial in @math n.
One motivation for studying bounded subcomplexes comes from the construction, a canonical method of embedding any metric space into a continuous space with properties similar to those of @math spaces @cite_23 @cite_11 @cite_9 . One way of defining the tight span, for a finite metric space with @math points @math , is to coordinatize @math -dimensional @math space by @math variables @math , and to define a polyhedral subset of the space by the @math linear inequalities [ x_i+x_j dist (p_i,p_j) ] for each possible pair @math . Then, the tight span is the bounded subcomplex of this polyhedron. For metric spaces satisfying an appropriate general position assumption, the dimension of the bounded subcomplex is between @math and @math @cite_10 , but certain combinatorially defined metrics, such as the metrics of distances on certain classes of planar graphs, can have tight spans of much lower dimension @cite_20 . Our results bound the complexity of these low-dimensional tight spans and allow them to be constructed efficiently, generalizing our previous algorithms for constructing tight spans when they are homeomorphic to subsets of the plane @cite_16 .
@cite_9 @cite_23 @cite_16 @cite_10 @cite_20 @cite_11
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Oxyalkylated condensation products prepared from aliphatic and aromatic amines, epihalohydrin and alkylene oxides are used in the preparation of polyurethane foams. We propose a new algorithm, called Equipoise, for the k-server problem, and we prove that it is two-competitive for two servers and 11-competitive for three servers. For k = 3, this is a substantial improvement over previously known constants. The algorithm uses several general techniques - convex hulls, work functions, and forgiveness. We describe a data structure, a rectangular complex, that can be used represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this data structure to construct the tight span of a metric space given as an n × n distance matrix, when the tight span is homeomorphic to a subset of the plane, in time O ( n 2 ), and to add a single point to a planar tight span in time O ( n ). As an application of this construction, we show how to test whether a given ﬁnite metric space embeds isometrically into the Manhattan plane in time O( n 2 ), and add a single point to the space and re-test whether it has such an embedding in time O ( n ). Given a finite metric, one can construct its tight span, a geometric object representing the metric. The dimension of a tight span encodes, among other things, the size of the space of explanatory trees for that metric; for instance, if the metric is a tree metric, the dimension of the tight span is one. We show that the dimension of the tight span of a generic metric is between @math and @math that both bounds are tight. Abstract The concept of tight extensions of a metric space is introduced, the existence of an essentially unique maximal tight extension T x —the “tight span,” being an abstract analogon of the convex hull—is established for any given metric space X and its properties are studied. Applications with respect to (1) the existence of embeddings of a metric space into trees, (2) optimal graphs realizing a metric space, and (3) the cohomological dimension of groups with specific length functions are discussed.