Light Spanners for High Dimensional Norms via Stochastic Decompositions

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with O~(n^{1+1/t^2}) edges, little is known. In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of l_p with 1<p <=2. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of l_p for 1<p <=2 has an O(t)-spanner with n^{1+O~(1/t^p)} edges and lightness n^{O~(1/t^p)}. In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)-spanner with lightness O(n^{1/t}). We exhibit the following tradeoff: metrics with decomposability parameter nu=nu(t) admit an O(t)-spanner with lightness O~(nu^{1/t}). For example, n-point Euclidean metrics have nu <=n^{1/t}, metrics with doubling constant lambda have nu <=lambda, and graphs of genus g have nu <=g. While these families do admit a (1+epsilon)-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch

Guidelines for the verification and validation of expert system software and conventional software: Bibliography. Volume 8

This volume contains all of the technical references found in Volumes 1-7 concerning the development of guidelines for the verification and validation of expert systems, knowledge-based systems, other AI systems, object-oriented systems, and conventional systems

Classifiers for modeling of mineral potential

[Extract] Classification and allocation of land-use is a major policy objective in most countries. Such an undertaking, however, in the face of competing demands from different stakeholders, requires reliable information on resources potential. This type of information enables policy decision-makers to estimate socio-economic benefits from different possible land-use types and then to allocate most suitable land-use. The potential for several types of resources occurring on the earth's surface (e.g., forest, soil, etc.) is generally easier to determine than those occurring in the subsurface (e.g., mineral deposits, etc.). In many situations, therefore, information on potential for subsurface occurring resources is not among the inputs to land-use decision-making [85]. Consequently, many potentially mineralized lands are alienated usually to, say, further exploration and exploitation of mineral deposits. Areas with mineral potential are characterized by geological features associated genetically and spatially with the type of mineral deposits sought. The term 'mineral deposits' means .accumulations or concentrations of one or more useful naturally occurring substances, which are otherwise usually distributed sparsely in the earth's crust. The term 'mineralization' refers to collective geological processes that result in formation of mineral deposits. The term 'mineral potential' describes the probability or favorability for occurrence of mineral deposits or mineralization. The geological features characteristic of mineralized land, which are called recognition criteria, are spatial objects indicative of or produced by individual geological processes that acted together to form mineral deposits. Recognition criteria are sometimes directly observable; more often, their presence is inferred from one or more geographically referenced (or spatial) datasets, which are processed and analyzed appropriately to enhance, extract, and represent the recognition criteria as spatial evidence or predictor maps. Mineral potential mapping then involves integration of predictor maps in order to classify areas of unique combinations of spatial predictor patterns, called unique conditions [51] as either barren or mineralized with respect to the mineral deposit-type sought

An Efficient Normalisation Procedure for Linear Temporal Logic and Very Weak Alternating Automata

In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form $\bigwedge_{i=1}^n \mathbf{G}\mathbf{F} \varphi_i \vee \mathbf{F}\mathbf{G} \psi_i$, where $\varphi_i$ and $\psi_i$ contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalises the formula, translates it into a special very weak alternating automaton, and applies a simple determinisation procedure, valid only for these special automata.Comment: This is the extended version of the referenced conference paper and contains an appendix with additional materia