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 Worboys-Bofakos tree model is perhaps the most related one to ours. Suppose @math is a bounded region that has a finite representation. In @cite_13 , each region @math is represented as a tree, where the root node represents, in our terms, the generalized region of @math , and each non-root node represents a simple region. If @math is a child node of @math , then @math is contained in @math . All child nodes of @math form a composite region (called a in @cite_13 ), which may separate @math into pieces.
@cite_13
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Consider a uniform random rooted tree on vertices labelled by @math , with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number @math of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider @math and let @math denote the event that all @math cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if @math , we have @math , whereas if @math we have @math . We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we are led to consider the following variant of the problem: take the tree to be the family tree of a Galton-Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson( @math ) number of cars arrive at each vertex. Let @math be the number of cars which visit the root of the tree. Then for @math , we have @math , whereas for @math , we have @math . This discontinuous phase transition turns out to be a generic phenomenon in settings with an arbitrary offspring distribution of mean at least 1 for the tree and arbitrary arrival distribution. Given a graph @math and a subset @math of terminals, a of @math is a tree that spans @math . In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a minimum weight Steiner tree of @math . We study a natural generalization of the VST problem motivated by multi-level graph construction, the (V-GSST), which can be stated as follows: given a graph @math and terminals @math , where each terminal @math requires a facility of a minimum grade of service @math , compute a Steiner tree @math by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in @math with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edge-weighted case, they have not been studied as well in the more general vertex-weighted case. We first describe a simple heuristic for the V-GSST problem whose approximation ratio depends on @math , the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the V-GSST problem admits a @math -approximation, where @math is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multi-grade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service. For @math -dimensional tensors with possibly large @math , an hierarchical data structure, called the Tree-Tucker format, is presented as an alternative to the canonical decomposition. It has asymptotically the same (and often even smaller) number of representation parameters and viable stability properties. The approach involves a recursive construction described by a tree with the leafs corresponding to the Tucker decompositions of three-dimensional tensors, and is based on a sequence of SVDs for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial “spatial” dimensions. It is shown how this format can be applied to the problem of multidimensional convolution. Convincing numerical examples are given. We consider the following problem. Given a rooted tree @math , label the nodes of @math in the most compact way such that, given the labels of two nodes @math and @math , one can determine in constant time, by looking only at the labels, whether @math is ancestor of @math . The best known labeling scheme is rather straightforward and uses labels of length at most @math bits each, where @math is the number of nodes in the tree. Our main result in this paper is a labeling scheme with maximum label length @math . Our motivation for studying this problem is enhancing the performance of web search engines. In the context of this application each indexed document is a tree, and the labels of all trees are maintained in main memory. Therefore even small improvements in the maximum label length are important.