Geographic information retrieval (GIR) systems allow users to specify a geographic context, in addition to a more traditional query, enabling the system to pinpoint interesting search results whose relevancy is location-dependent. In particular local search services have become a widely used mechanism to find businesses, such as hotels, restaurants, and shops, which satisfy a geographical restriction. Unfortunately, many useful types of geographic restrictions are currently not supported in these systems, including restrictions that specify the neighborhood in which the business should be located. As the boundaries of city neighborhoods are not readily available, automated techniques to construct representations of the spatial extent of neighborhoods are required to support this kind of restrictions. In this paper, we propose such a technique, using fuzzy footprints to cope with the inherent vagueness of most neighborhood boundaries, and we provide experimental results that demonstrate the potential of our technique in a local search setting.

We examine a recolouring scheme ostensibly used to assist in classifying geographic data. Given a drawing of a graph with bi-chromatic vertices, a vertex can be recoloured if it is surrounded by neighbours of the opposite colour. The notion of surrounded is defined as a contiguous subset of neighbours that span an angle greater than 180 degrees. The recolouring of surrounded vertices continues in sequence, in no particular order, until no vertex remains surrounded. We show that for some classes of graphs the process terminates in a polynomial number of steps. On the other hand, there are classes of graphs where the process never terminates.

Given a collection of points representing geographic data we consider the task of delineating boundaries based on the features of the points. Assuming that the features are binary, for example, red or blue, this can be viewed as determining red and blue regions, or states. Due to regional anomalies or sampling error, we may find that reclassifying, or recolouring, some points may lead to a more rational delineation of boundaries. In this note we study the maximal length of recolouring sequences where recolouring rules are based on neighbour relations and neighbours are defined by a geometric graph. We show that the difference in the maximal length of recolouring sequences is striking, as it can range from a linear bound for all trees, to an infinite sequence for some planar graphs. 1

We study variants of the potato peeling problem on meshed (triangulated) polygons. Given a polygon with holes, and a triangular mesh that covers its interior (possibly using additional vertices), we want to find a largest-area connected set of triangles of the mesh that is convex, or has some other shape-related property. In particular, we consider (i) convexity, (ii) monotonicity, (iii) bounded backturn, and (iv) bounded total turning angle. The first three problems are solved in polynomial time, whereas the fourth problem is shown to be NP-hard. 1

© The Author(s) 2009. This article is published with open access at Springerlink.com Abstract We study variants of the potato peeling problem on meshed (triangulated) polygons. Given a polygon with holes, and a triangular mesh that covers its interior (possibly using additional vertices), we want to find a largest-area connected set of triangles of the mesh that is convex, or has some other shape-related property. In particular, we consider (i) convexity, (ii) monotonicity, (iii) bounded backturn, and (iv) bounded total turning angle. The first three problems are solved in polynomial time, whereas the fourth problem is shown to be NP-hard.