Sphere of influence graphs and the L∞-metric

Michael, T. S.; Quint, Thomas

doi:10.1016/s0166-218x(02)00246-9

Cited by 11 publications

(13 citation statements)

References 10 publications

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“…Note that if P is a d-dimensional SIG representation of G then we are associating to each vertex x of G a point p(x) in P. Given a graph G, the minimum positive integer d such that G has a d-dimensional SIG representation (with respect to the metric ρ) is called the SIG dimension of G (with respect to the metric ρ) and is denoted by SIG ρ (G). It is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have an SIG representation under L ∞ metric in some space of finite dimension [1]. The SIG dimension under L ∞ metric of a graph G without isolated vertices is defined to be the minimum positive integer d such that G has a d-dimensional SIG representation under the L ∞ metric.…”

Section: Sig-representation and Sig Dimensionsmentioning

confidence: 99%

“…Toussaint has used the sphere-of-influence graphs under L 2 -metric to capture low-level perceptual information in certain dot patterns. It is argued in [1] that sphere-of-influence graphs under the L ∞ -metric perform better for this purpose. Also, several results regarding SIG ∞ dimension are proved in [1].…”

Section: Literature Surveymentioning

confidence: 99%

“…It is argued in [1] that sphere-of-influence graphs under the L ∞ -metric perform better for this purpose. Also, several results regarding SIG ∞ dimension are proved in [1]. Bounds for the SIG ∞ dimension of complete multipartite graphs are considered in [10].…”

Section: Literature Surveymentioning

confidence: 99%

See 2 more Smart Citations

On the SIG-Dimension of Trees Under the L ∞-Metric

Chandran

Chitnis

Kumar

2012

Graphs and Combinatorics

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Let P, where |P| ≥ 2, be a set of points in d-dimensional space with a given metric ρ. For a point p ∈ P, let rp be the distance of p with respect to ρ from its nearest neighbor in P. Let B(p, rp) be the open ball with respect to ρ centered at p and having the radius rp. We define the sphere-of-influence graph (SIG) of P as the intersection graph of the family of sets {B(p, rp) | p ∈ P}. Given a graph G, a set of pointsIt is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have an SIG representation under L∞ metric in some space of finite dimension. The SIG dimension under L∞ metric of a graph G without isolated vertices is defined to be the minimum positive integer d such that G has a d-dimensional SIG representation under the L∞ metric. It is denoted as SIG∞(G).We study the SIG dimension of trees under L∞ metric and almost completely answer an open problem posed by Michael and Quint (Discrete Applied Mathematics: 127, pages 447-460, 2003). Let T be a tree with at least two vertices. For each v ∈ V (T ), let leaf-degree(v) denote the number of neighbors of v that are leaves. We define the maximum leafdegree as α(T ) = max x∈V (T ) leaf-degree(x). Let S = {v ∈ V (T ) | leafdegree(v) = α}. If |S| = 1, we define β(T ) = α(T ) − 1. Otherwise define β(T ) = α(T ). We show that for a tree T , SIG∞(T ) = log 2 (β + 2) where β = β(T ), provided β is not of the form 2 k − 1, for some positive integer k ≥ 1. If β = 2 k − 1, then SIG∞(T ) ∈ {k, k + 1}. We show that both values are possible.

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Section: Sig-representation and Sig Dimensionsmentioning

confidence: 99%

Section: Literature Surveymentioning

confidence: 99%

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On the SIG-Dimension of Trees Under the L ∞-Metric

Chandran

Chitnis

Kumar

2012

Graphs and Combinatorics

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“…Many proximity graphs have been studied, including the Delaunay and nearest neighbour graphs, the α-shape and the sphere-of-influence graph, or SIG [10]. The SIG has shown the most promise and has been studied from various points of view, often in the field of optical character recognition [4], [17] and also in the development of surfaces from point clouds [11]. With the SIG method, each data point is the centre of an open sphere of radius equal to the distance between that point and its nearest neighbour, and two points are adjacent if their spheres intersect.…”

Section: Related Workmentioning

confidence: 99%

“…Klein [11] found the sphere-of-influence graph (SIG), although less used than others, to be the best performing proximity graph for implicit surface fitting. The SIG method [4], [17] is used to determine if a sphere of influence centred at each data point intersects with the sphere of other data points, and if so the two points are connected by an edge. Mathematically, for each point P i , the distance d i to the nearest neighbour is determined by calculating the distance between that point and all other points P j , and then an edge is connected between any two points P i and P j if…”

Section: Sig Modelmentioning

confidence: 99%

Weighted Least Squares for Visualization of Scanned Point Clouds

Goldstein¹

2006

Computer-Aided Design and Applications

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This paper describes several improvements to implicit surface fitting over point clouds. Noisy stamped part point cloud data gathered by both a laser digitizer scanning system and a stereo vision sensor is processed using a method based on the weighted least squares algorithm. Two kernel functions are proposed: one using an enhanced sphere-of-influence graph (SIG) method to determine nearest neighbours; and a second using an improved Gaussian kernel. Automatic bandwidth adjustment is implemented for both methods. A method of incorporating a scalar attribute at each data point, which can represent any desired quantity and relate it to the surface at that point, is introduced. In this case surface strain data was gathered and processed by the scanning system, with a resulting thickness strain measurement calculated at each point. This strain data was used to introduce a temperature plot style colour map over the surface, allowing for fast and simple analysis of the strain of the formed part.

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On the Cubicity of AT-Free Graphs and Circular-Arc Graphs

Chandran

Francis

Sivadasan³

2009

Graph Theory, Computational Intelligence and Thought

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A unit cube in k dimensions (k-cube) is defined as the the Cartesian product R1 × R2 × · · · × R k where Ri(for 1 ≤ i ≤ k) is a closed interval of the form [ai, ai + 1] on the real line. A graph G on n nodes is said to be representable as the intersection of k-cubes (cube representation in k dimensions) if each vertex of G can be mapped to a k-cube such that two vertices are adjacent in G if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G denoted as cub(G) is the minimum k for which G can be represented as the intersection of k-cubes. An interesting aspect about cubicity is that many problems known to be NP-complete for general graphs have polynomial time deterministic algorithms or have good approximation ratios in graphs of low cubicity. In most of these algorithms, computing a low dimensional cube representation of the given graph is usually the first step. We give an O(bw · n) algorithm to compute the cube representation of a general graph G in bw + 1 dimensions given a bandwidth ordering of the vertices of G, where bw is the bandwidth of G. As a consequence, we get O(∆) upper bounds on the cubicity of many well-known graph classes such as AT-free graphs, circular-arc graphs and co-comparability graphs which have O(∆) bandwidth. Thus we have:1. cub(G) ≤ 3∆ − 1, if G is an AT-free graph. 2. cub(G) ≤ 2∆ + 1, if G is a circular-arc graph. 3. cub(G) ≤ 2∆, if G is a co-comparability graph. Also for these graph classes, there are constant factor approximation algorithms for bandwidth computation that generate orderings of vertices with O(∆) width. We can thus generate the cube representation of such graphs in O(∆) dimensions in polynomial time.

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Sphere of influence graphs and the L∞-metric

Cited by 11 publications

References 10 publications

On the SIG-Dimension of Trees Under the L ∞-Metric

On the SIG-Dimension of Trees Under the L ∞-Metric

Weighted Least Squares for Visualization of Scanned Point Clouds

On the Cubicity of AT-Free Graphs and Circular-Arc Graphs

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