Testing (subclasses of) halfspaces

We address the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w . x − θ). We consider halfspaces over the continuous domain R n (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube { − 1,1} n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are ε-far from any halfspace using only poly(1) queries, independent of the dimension n. In contrast to the case of general halfspaces, we show that testing natural subclasses of halfspaces can be markedly harder; for the class of { − 1,1}-weight halfspaces, we show that a tester must make at least Ω(logn) queries. We complement this lower bound with an upper bound showing that O(√n) queries suffice.National Basic Research Program of China (grant 2007CB807900)National Basic Research Program of China (grant 2007CB807901)National Natural Science Foundation (China) (grant 60553001

Testing +/- 1-Weight Halfspaces

We consider the problem of testing whether a Boolean function f:{ − 1,1} [superscript n] →{ − 1,1} is a ±1-weight halfspace, i.e. a function of the form f(x) = sgn(w [subscript 1] x [subscript 1] + w [subscript 2] x [subscript 2 ]+ ⋯ + w [subscript n] x [subscript n] ) where the weights w i take values in { − 1,1}. We show that the complexity of this problem is markedly different from the problem of testing whether f is a general halfspace with arbitrary weights. While the latter can be done with a number of queries that is independent of n [7], to distinguish whether f is a ±-weight halfspace versus ε-far from all such halfspaces we prove that nonadaptive algorithms must make Ω(logn) queries. We complement this lower bound with a sublinear upper bound showing that $O(\sqrt{n}\cdot$poly$(\frac{1}{\epsilon}))$ queries suffice

Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For inputs in general position the number of bounded faces is O(n^d). For any fixed 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 polynomial number of linear programs

Searchable Sky Coverage of Astronomical Observations: Footprints and Exposures

Sky coverage is one of the most important pieces of information about astronomical observations. We discuss possible representations, and present algorithms to create and manipulate shapes consisting of generalized spherical polygons with arbitrary complexity and size on the celestial sphere. This shape specification integrates well with our Hierarchical Triangular Mesh indexing toolbox, whose performance and capabilities are enhanced by the advanced features presented here. Our portable implementation of the relevant spherical geometry routines comes with wrapper functions for database queries, which are currently being used within several scientific catalog archives including the Sloan Digital Sky Survey, the Galaxy Evolution Explorer and the Hubble Legacy Archive projects as well as the Footprint Service of the Virtual Observatory.Comment: 11 pages, 7 figures, submitted to PAS