Topological information are the most important kind of qualitative spatial information. Current formalisms for the topological aspect of space focus on relations between regions or properties of regions. This work provides a qualitative model for representing the topological internal structure of complex regions, which could be of multiple pieces and or have holes and islands to any finite level. We propose a layered graph model for representing the internal structure of complex plane regions, where each node represents the closure of a connected component of the interior or the exterior of a complex region. The model provides a complete representation in the sense that the (global) nine-intersections between the interiors, the boundaries, and the exteriors of two complex regions can be determined by the (local) RCC8 topological relations between the associated simple regions. Moreover, this graph model has an inherent hierarchy which could be exploited for map generalization.
The notion of simple region with holes and its generalized region are first introduced in @cite_3 . Our definition of simple region with holes is slightly different from that given in @cite_3 , but is consistent with the one in @cite_15 .
@cite_15 @cite_3
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The 4-intersection, a model for the representation of topological relations between 2-dimensional objects with connected boundaries and connected interiors, is extended to cover topological relations between 2-dimensional objects with arbitrary holes, called regions with holes. Each region with holes is represented by its generalized region—the union of the object and its holes — and the closure of each hole. The topological relation between two regions with holes, A and B, is described by the set of all individual topological relations between (1) A ’s generalized region and B’s generalized region, (2) A ’s generalized region and each of B’s holes, (3) B’s generalized region with each of A ’s holes, and (4) each of A ’s holes with each of B’s holes. As a side product, the same formalism applies to the description of topological relations between 1-spheres. An algorithm is developed that minimizes the number of individual topological relations necessary to describe a configuration completely. This model of representing complex topological relations is suitable for a multi-level treatment of topological relations, at the least detailed level of which the relation between the generalized regions prevails. It is shown how this model applies to the assessment of consistency in multiple representations when, at a coarser level of less detail, regions are generalized by dropping holes. We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R_1,...,R_n in the plane and finite point sets P_i subset of R_i for i=1,...,n, are there non-crossing connectors y_i for (R_i,P_i), i.e., arc-connected sets y_i with P_i subset of y_i subset of R_i for every i=1,...,n, such that y_i and y_j are disjoint for all i different from j? We prove that non-crossing connectors do always exist if the regions form a collection of pseudo-disks, i.e., the boundaries of every pair of regions intersect at most twice. We provide a simple polynomial-time algorithm if the regions are axis-aligned rectangles. Finally we prove that the general problem is NP-complete, even if the regions are convex, the boundaries of every pair of regions intersect at most four times and P_i consists of only two points on the boundary of R_i for i=1,...,n. For generalized Reed-Solomon codes, it has been proved [7] that the problem of determining if a received word is a deep hole is co-NP-complete. The reduction relies on the fact that the evaluation set of the code can be exponential in the length of the code - a property that practical codes do not usually possess. In this paper, we first present a much simpler proof of the same result. We then consider the problem for standard Reed-Solomon codes, i.e. the evaluation set consists of all the nonzero elements in the field. We reduce the problem of identifying deep holes to deciding whether an absolutely irreducible hypersurface over a finite field contains a rational point whose coordinates are pairwise distinct and nonzero. By applying Cafure-Matera estimation of rational points on algebraic varieties, we prove that the received vector (f(α))α∈Fpfor the Reed-Solomon [p - 1, k]p, k < p1 4-Ɛ, cannot be a deep hole, whenever f(x) is a polynomial of degree k + d for 1 ≤ d ≤ p3 13-Ɛ. We present a framework for extending coverage algorithms from convex to nonconvex domains. We are particularly interested in coverage in the presence of obstacles, that is domains that contain holes. We identify a class of connected regions in Ropf2 than can be mapped through a diffeomorphism to an almost convex region in Ropf2 - a convex region from which a finite (possibly empty) set of points has been subtracted. The transformation allows us to solve the coverage problem using the approach of , and obtain the solution of the original problem through inverse transformation. We provide a formal analysis of the approach and demonstrate its effectiveness through simulations. We conclude the paper with a discussion on possible extensions of the work that we are currently undertaking.