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This paper presents the formulation of a discrete equation whose solutions have a strong combinatory nature. More formally, given two subsets Y and C, we are interested in finding all subsets X that satisfy the equation (called Minkowski Addition Equation) X ⊕ C = Y. One direct application of the solutions of this equation is that they can be used to find best representations for fast computation of erosions and dilations. The main (and original) result presented in this paper (which is a theoretical result) is an analytical solution formula for this equation. One important characteristic of this analytical formula is that all solutions (which can be in worst case exponential) are expressed in a compact representation.

In the store-and-forward routing problem, packets have to be routed along given paths such that the arrival time of the latest packet is minimized. A groundbreaking result of Leighton, Maggs and Rao says that this can always be done in time $O(\textrm{congestion} + \textrm{dilation})$, where the congestion is the maximum number of paths using an edge and the dilation is the maximum length of a path. However, the analysis is quite arcane and complicated and works by iteratively improving an infeasible schedule. Here, we provide a more accessible analysis which is based on conditional expectations. Like [LMR94], our easier analysis also guarantees that constant size edge buffers suffice. Moreover, it was an open problem stated e.g. by Wiese [Wie11], whether there is any instance where all schedules need at least $(1+\varepsilon)\cdot(\textrm{congestion}+\textrm{dilation})$ steps, for a constant e>0. We answer this question affirmatively by making use of a probabilistic construction.

In the paper an approach to pattern recognition based on a notion of similarity in linear semi-ordered (Kantorovitsch) space is presented. It is compared with other approaches based on the metric distance and on angular dilation measures in observation spaces. Basic assumptions of the Kantorovitsch space are shortly presented. It is shown that finite reference sets for pattern recognition take on in Kantorovitsch space formal structures presented by connectivity graphs which facilitate finding the reference vectors for pattern recognition.

The convalution operators on the euclidean spaces are only a particular case of convolution operators on arbitrary Lie groups. Pseudo-differential operators are roughly speaking convolution operators with variable coefficients. In classical theory of such operators it is important to have standard dilations on a euclidean space. So we pay attention only to Lie groups with dilation i.e. with 1-dimensional group of automorphisms converging to infinity when real parameter increases to infinity. It is well known that all Lie groups with dilations are nilpotent.

The spanning ratio and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and metric space, respectively. In this paper we show that computing the spanning ratio of a rectilinear path P in L 1 space has a lower bound of Ω(n logn) in the algebraic computation tree model and describe a deterministic O(n log2 n) time algorithm. On the other hand, we give a deterministic O(n log2 n) time algorithm for computing the maximum detour of a rectilinear path P in L 1 space and obtain an O(n) time algorithm when P is a monotone rectilinear path.

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find approximate farthest neighbors that are farther than a factor (1-e) of optimal in time O(logn) per query in D-dimensional Euclidean space for any constants D and e. As an application, we find an O(nlog n) expected time algorithm for choosing the center of a star topology network connecting a given set of points, so as to approximately minimize the maximum dilation between any pair of points.

Once a designed network is embedded into a newly designed network, a developed algorithm in the designed network is reusable in a newly designed network. Petersen-Torus has been designed recently, and Hyper Petersen has already been designed as a well-known interconnection network. In this study, it was proven that Hyper Petersen network whose degree increases with the increased number of nodes can be embedded into a Petersen-Torus network having fixed degree. Hyper Petersen log2 n 2 + 3 was embedded into Petersen-Torus PT(n,n) with expansion 1, dilation 1.5n+2, and congestion 5n. The embedding algorithm was designed for expansion 1, and congestion and dilation are in proportion to O(n) owing to the aspect of Hyper Petersen that the degree increases.

The area of knowledge merging is concerned with merging conflicting information while preserving as much as possible. Most proposals in the literature work with knowledge bases expressed in propositional logic. We propose a new framework for merging knowledge bases expressed in (subsets of) first-order logic. Dilation operators (a concept originally introduced by Bloch and Lang) are employed and developed, and by combining them with the concept of comparison orderings we obtain a framework that is driven by model-based intuitions but that can be implemented in a syntax-based manner. We demonstrate specific dilation operators and comparison orderings for use in applications. We also show how postulates from the literature on knowledge merging translate into our framework and provide the conditions that dilation operators and comparison orderings must satisfy in order for the respective merging operators to satisfy the new postulates.

Most peer-to-peer (P2P) networks proposed until now have either logarithmic degree and logarithmic dilation or constant degree and logarithmic dilation. In the latter case (which is optimal up to constant factors), the constant degree is achieved either in expectation or with high probability. We propose the first overlay network, called SkewCCC , with a maximum degree of 3 (minimum possible) and logarithmic dilation. Our approach can be viewed as a decentralized and distorted version of a Cube Connected Cycles network. Additionally, basic network operations such as join and leave take logarithmic time and are very simple to implement, which makes our construction viable in fields other than P2P networks. A very good example is scatternet construction for Bluetooth devices, in which case it is crucial to keep the degree at most 7.

An orthogonal network for a given set of n points in the plane is an axis-aligned planar straight line graph that connects all input points. We show that for any set of n points in the plane, there is an orthogonal network that (i) is short having a total edge length of O(|T|), where |T| denotes the length of a minimum Euclidean spanning tree for the point set; (ii) is small having O(n) vertices and edges; and (iii) has constant geometric dilation, which means that for any two points u and v in the network, the shortest path in the network between u and v is at most constant times longer than the (Euclidean) distance between u and v.

We establish in this paper the link between the two main approaches for fuzzy mathematical morphology, based on duality with respect to complementation and on the adjunction property, respectively. We also prove that the corresponding definitions of fuzzy dilation and erosion are the most general ones if a set of classical properties is required.

Given a geometric graph G=(S,E) in ${\mathbb R}^{d}$ with constant dilation t, and a positive constant e, we show how to construct a (1+e)-spanner of G with ${\mathcal O}(|S|)$ edges using ${\mathcal O}(sort(|E|))I/O$ I/O operations.

In this text we show how points, point pairs, lines, planes, circles, spheres, and rotation, translation and dilation operators and their uncertainty can be evaluated from uncertain data in a unified manner using the Geometric Algebra of conformal space. This extends previous work by Forstner et al. [3] from points, lines and planes to non-linear entities and operators, while keeping the linearity of the estimation method. We give a theoretical description of our approach and show the results of some synthetic experiments.

Let G be an embedded planar graph whose edges may be curves. The detour between two points, p and q (on edges or vertices) of G, is the ratio between the shortest path in G between p and q and their Euclidean distance. The supremum over all pairs of points of all these ratios is called the geometric dilation of G. Our research is motivated by the problem of designing graphs of low dilation. We provide a characterization of closed curves of constant halving distance (i.e., curves for which all chords dividing the curve length in half are of constant length) which are useful in this context. We then relate the halving distance of curves to other geometric quantities such as area and width. Among others, this enables us to derive a new upper bound on the geometric dilation of closed curves, as a function of D/w, where D and w are the diameter and width, respectively. We further give lower bounds on the geometric dilation of polygons with n sides as a function of n. Our bounds are tight for centrally symmetric convex polygons.

A.1 Stochastic Maps A.1.1 Terminology A.1.2 Kadison-Schwarz Inequality A.1.3 A Useful Setting A.2 Representation Theorems A.2.1 W.F. Stinespring and K. Kraus A.2.2 Stinespring Representation A.2.3 Kraus Decomposition A.2.4 Connection between Stinespring and Kraus A.2.5 Equivalence A.2.6 Non-minimal Decompositions A.3 The Isometry v A.3.1 A Shifted Point of View A.3.2 Subspaces Which Are Minimal for an Inclusion A.3.3 The Map \(a: {\cal P}\rightarrow {\cal B}({\cal G},{\cal H})\) A.3.4 Metric Operator Spaces A.3.5 Non-minimality Revisited A.4 The Preadjoints C and D A.4.1 Preadjoints A.4.2 The Associated Pair (C,D) A.4.3 Coordinates A.4.4 Formulas for Partial Traces A.4.5 A Useful Formula for Dilation Theory A.5 Absorbing Vector States A.5.1 Stochastic Maps and Vector States A.5.2 Criteria for Absorbing Vector States A.5.3 Absorbing Sequences

According to the principles of group theory, every group \(\mathcal G\) of transformations \(G_{\rho }\) can be represented by an isomorphic group \(\mathcal{G}_L\) of linear transformations \(\mathbf{G}_{\rho }\) of a vector space. The transformations of event and parameter, introduced in Chap. 6, are nonlinear. The probability distributions p and w are not elements of a vector space. Is it possible to define form invariance in terms of linear transformations of a vector space? This would give the possibility of distinguishing classes of transformations, such as orthogonal or unitary ones, that can occur in form invariance, from other ones that cannot occur. Section 6.4 has given a hint of the way of constructing linear representations: one must express probabilities by probability amplitudes. The present chapter pursues this route and defines linear representations of models with the symmetries of translation and dilation.

Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The maximum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph that contains the given points. In this paper we prove that a dilation of 1.678 is always sufficient, and that π/2 = 1.570... is sometimes necessary in order to accommodate a finite set of points.

We consider one-to-one embeddings of the n-dimensional hypercube into grids with 2n vertices and present lower and upper bounds and asymptotic estimates for minimal dilation, edge-congestion, and their mean values. We also introduce and study two new cost-measures for these embeddings, namely the sum over i=1, ..., n of dilations and the sum of edge-congestions caused by the hypercube edges of the ith dimension. It is shown that, in the simulation via the embedding approach, such measures are much more suitable for evaluating the slowdown of uniaxial hypercube algorithms then the traditional cost measures.

The use of selective therapeutic ERCP, indicated by ultrasonographically detected dilation of the common bile duct, is a safe procedure with low morbidity, no mortality and a significant reduction in the need for conversion to open cholecystectomy.

For integers m,d,D with m ge 3, d ge 2, and D ge 2, let T (m) be a 2--dimensional quadratic toroidal grid with side length m and let B (d,D) be the base d, dimension D de Bruijn graph; assume that |T ( m ) | = | B (d,D ) |. The starting point for our inves-ti-ga-tions is the observation that, for m, D even, embeddings f : T (m) rightarrow B (d,D) with load 1, expansion 1, and dilation D/2 can easily be found (and have previously been described in the literature). In the present paper, we pose the question whether or not there exist embeddings f:T (m) rightarrow B (d,D) with these properties em and with clocked congestion 1. We prove results implying a positive answer to this question when d is greater than two. For d=2, we do not have a complete answer, but present partial results.

For Cartesian products G=G_1 times dots times G_m (m geq 2) of nontrivial connected graphs G_i and the n-dimensional base B de Bruijn graph D=D_B(n), we investigate whether or not there exists a spanning subgraph of D which is isomorphic to G. We show that G is never a spanning subgraph of D when n is greater than three or when n equals three and m is greater than two. For n=3 and m=2, we can show for wide classes of graphs that G cannot be a spanning subgraph of D. In particular, these non-existence results imply that D_B(n) never contains a torus (i.e., the Cartesian product of mge2 cycles) as a spanning subgraph when n is greater than two. For n=2 the situation is quite different: we present a sufficient condition for a Cartesian product G to be a spanning subgraph of D=D_B(2). As one of the corollaries we obtain that a torus G=G_1 timesdotstimes G_m is a spanning subgraph of D=D_B(2) provided that |G|=|D| and that the G_i are even cycles of length ge 4. In addition we apply our results to obtain embeddings of relatively small dilation of popular processor networks (as tori, meshes and hypercubes) into de Bruijn graphs of fixed small base.

The hypercube is a particularly versatile network for parallel computing. It is well-known that 2-dimensional grid machines can be simulated on a hypercube with a small constant communication overhead. We introduce new easily computable functions which embed many 3-dimensional grids into their optimal hypercubes with dilation 2. Moreover, we show that one can reduce the open problem to recognize whether it is possible to embed every 3-dimensional grid into its optimal hypercube with dilation at most 2 by constructing embeddings for a particular class of grids. We embed some of these grids, and thus for the first time one can guarantee that every 3-dimensional grid with at most 29–18 nodes is embeddable into its optimal hypercube with dilation 2.

It is shown that, for every k, and for every k-dimensional mesh M, there is a one-to-one embedding of M into its optimum size hypercube with dilation at most 2k-1.

The notion of negative shape is that it is an artifice suggested in algebra on various occasions when geometric problems are translated into the language of algebra. Though the notion appears to be very useful, its significance and potential have hardly been explored till this day. In this paper two such cases are considered, namely, an algebraic formulation of Minkowski addition of two geometric objects and an analytic formulation of the area/volume of an object, and it is shown how such formulations indicate the negative shape notion. By means of a mixed area/volume concept, it is shown that negative shape notions derived from the two formulations are exactly identical. The usefulness of the negative shape concept is also briefly indicated.

Star networks have been proposed as a possible interconnection network for massively parallel computers. In this paper we investigate embeddings of complete binary trees into star networks. Let G and H be two networks represented by simple undirected graphs. An embedding of G into H is an injective mapping f from the vertices of G into the vertices of H. The dilation of the embedding is the maximum distance between f(u), f(v) taken over all edges (u,v) of G.

The nearest neighbor graph (NNG), defined for a set of points in Euclidean space, has found many uses in computational geometry and clustering analysis. Yet there seems to be surprisingly little knowledge about some basic properties of this graph. In this talk, we ask some natural questions that are motivated by geometric embedding problems. For example, in the simulation of many-body systems, it is desirable to map a set of particles to a regular data array so as to preserve neighborhood relations, i.e., to minimize the dilation of NNG. We will derive bounds on the dilation by studying the diameter of NNG. Other properties and applications of NNG will also be discussed. (Joint work with Mike Paterson)

We consider optimum simulations of large mesh networks by hypercubes. For any arbitrary mesh M, let "M's optimum hypercube" be the smallest hypercube that has at least as many processors as M and, for any k>0, let Q(M/2k) be "M's 1/2k-size hypercube", which has 1/2k as many processors as M's optimum hypercube. The ratio MI/Q(M/2k) is called M's 1/2 k - density. We show that (a) for every 2-D mesh M, if M's 1/2-density≤1.828, then M can be embedded into its 1/2-size hypercube with dilation 1 and load factor 2, (b) for every 2-D mesh M, if M's 1/4-density≤3.809, then M can be embedded into its 1/4-size hypercube with dilation 1 and load factor 4, and if M's 1/4-density≤2.8125, then M can be embedded into its 1/4-size hypercube with dilation 1 and load factor 3, (c) If every 2-D mesh M with 1/2k1-density≤a can be embedded into its 1/2k1-size hypercube with dilation 1 and load factor l1, and every 2-D mesh M with 1/2k2-density ≤b can be embedded into its 1/2k2-size hypercube with dilation 1 and load factor l2, then we can obtain the densities for load factor l1+l2 and load factor l1×l2 based on a, b.

We construct and discuss some explicit examples of Markov dilations for semigroups of completely positive operators on the W*-algebra of 2×2 matrices. In particular, we obtain a continuous Markov dilation for a semigroup of non-quasifree operators.