Utility functions on locally connected spaces

We investigate the role of local connectedness in utility theory and prove that any continuous total preorder on a locally connected separable space is continuously representable. This is a new simple criterion for the representability of continuous preferences, and is not a consequence of the standard theorems in utility theory that use conditions such as connectedness and separability, second countability, or path-connectedness. Finally we give applications to problems involving the existence of value functions in population ethics and to the problem of proving the existence of continuous utility functions in general equilibrium models with land as one of the commodities. (C) 2003 Elsevier B.V. All rights reserved.

Multiscale histograms: Summarizing topological relations in large spatial datasets

Summarizing topological relations is fundamental to many spatial applications including spatial query optimization. In this paper, we present several novel techniques to eectively construct cell density based spatial histograms for range (window) summarizations restricted to the four most important topological relations: contains, contained, overlap, and disjoint. We rst present a novel framework to construct a multiscale histogram composed of multiple Euler histograms with the guarantee of the exact summarization results for aligned windows in constant time. Then we present an approximate algorithm, with the approximate ratio 19/12, to minimize the storage spaces of such multiscale Euler histograms, although the problem is generally NP-hard. To conform to a limited storage space where only k Euler histograms are allowed, an effective algorithm is presented to construct multiscale histograms to achieve high accuracy. Finally, we present a new approximate algorithm to query an Euler histogram that cannot guarantee the exact answers; it runs in constant time. Our extensive experiments against both synthetic and real world datasets demonstrated that the approximate mul- tiscale histogram techniques may improve the accuracy of the existing techniques by several orders of magnitude while retaining the cost effciency, and the exact multiscale histogram technique requires only a storage space linearly proportional to the number of cells for the real datasets.