2,746 research outputs found
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
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 x 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).Comment: 39 pages, 15 figure
Robust Procedures for Obtaining Assembly Contact State Extremal Configurations
Two important components in the selection of an admittance that facilitates force-guided assembly are the identification of: 1) the set of feasible contact states, and 2) the set of configurations that span each contact state, i.e., the extremal configurations. We present a procedure to automatically generate both sets from CAD models of the assembly parts. In the procedure, all possible combinations of principle contacts are considered when generating hypothesized contact states. The feasibility of each is then evaluated in a genetic algorithm based optimization procedure. The maximum and minimum value of each of the 6 configuration variables spanning each contact state are obtained by again using genetic algorithms. Together, the genetic algorithm approach, the hierarchical data structure containing the states, the relationships among the states, and the extremals within each state are used to provide a reliable means of identifying all feasible contact states and their associated extremal configurations
Almost-Fisher families
A classic theorem in combinatorial design theory is Fisher's inequality,
which states that a family of subsets of with all pairwise
intersections of size can have at most non-empty sets. One may
weaken the condition by requiring that for every set in , all but
at most of its pairwise intersections have size . We call such
families -almost -Fisher. Vu was the first to study the maximum
size of such families, proving that for the largest family has
sets, and characterising when equality is attained. We substantially refine his
result, showing how the size of the maximum family depends on . In
particular we prove that for small one essentially recovers Fisher's
bound. We also solve the next open case of and obtain the first
non-trivial upper bound for general .Comment: 27 pages (incluiding one appendix
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