Balanced Aspect Ratio Trees: Combining the Advantages of k-d Trees and Octrees

Duncan, Christian A.; Goodrich, Michael T.; Kobourov, Stephen G.

doi:10.1006/jagm.2000.1135

Cited by 60 publications

(64 citation statements)

References 31 publications

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“…To compute an SSPD for a given point set S, we use a bounded aspect ratio (BAR) tree, as introduced by Duncan et al [11]. A BAR-tree for a point set S is a binary space partition (BSP) tree with the following properties: Let T be a BAR-tree on the point set S. For a node ν, we use pa(ν) to denote the parent of ν, and we use S(ν) to denote the subset of points from S that are stored in the leaves of the subtree T ν rooted at ν.…”

Section: Computing An Sspd In R Dmentioning

confidence: 99%

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Geometric Spanners for Weighted Point Sets

et al. 2010

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Let (S, d) be a finite metric space, where each element p ∈ S has a nonnegative weight w(p). We study spanners for the set S with respect to the following weighted distance function: Algorithmica (2011) 61:207-225 We present a general method for turning spanners with respect to the d-metric into spanners with respect to the d ω -metric. For any given ε > 0, we can apply our method to obtain (5 + ε)-spanners with a linear number of edges for three cases: points in Euclidean space R d , points in spaces of bounded doubling dimension, and points on the boundary of a convex body in R d where d is the geodesic distance function.We also describe an alternative method that leads to (2 + ε)-spanners for weighted point points in R d and for points on the boundary of a convex body in R d . The number of edges in these spanners is O(n log n). This bound on the stretch factor is nearly optimal: in any finite metric space and for any ε > 0, it is possible to assign weights to the elements such that any non-complete graph has stretch factor larger than 2 − ε.

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Section: Computing An Sspd In R Dmentioning

confidence: 99%

“…According to Lemma 16 summation (6) Proof The proof is the same as the proof of Lemma 3.10 from [2] where the dependency on the dimension follows from plugging in Corollary 3 and the fact that the BAR tree T requires O(d2 d n log n) time to compute [11].…”

Section: D(r(pa( ν)) R(pa( μ)))mentioning

confidence: 99%

Geometric Spanners for Weighted Point Sets

et al. 2010

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“…In this paper we present the BAR tree in IR 2 . The generalized BAR tree in IR d is presented in [9]. The construction of the BAR tree is very similar to that of a k-d tree, except for two important differences:…”

Section: The Balanced Aspect Ratio (Bar) Treementioning

confidence: 99%

Balanced Aspect Ratio Trees and Their Use for Drawing Large Graphs

Duncan¹,

Goodrich²,

Kobourov³

2000

JGAA

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We describe a new approach for cluster-based drawing of large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSP-type decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounded aspect ratios. In addition, the tree depth is O(log n), and its construction takes O(n log n) time, where n is the number of points. We show that the BAR tree can be used to recursively divide a graph embedded in the plane into subgraphs of roughly equal size, such that the drawing of each subgraph has a balanced aspect ratio. As a result, we obtain a representation of a graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. The overall running time of the algorithm is O(n log n+m+D0(G)), where n and m are the number of vertices and edges of the graph G, and D0(G) is the time it takes to obtain an initial embedding of G in the plane. In particular, if the graph is planar each layer is a graph drawn with straight lines and without crossings on the n×n grid and the running time reduces to O(n log n).

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“…Approximate nearest neighbor queries in spaces of fixed dimension have been widely studied. Data structures with O(n) storage space and query times no better than O(log n+1/ε d−1 ) have been proposed by several authors [8,9,11,15]. In subsequent papers, it was shown that query times could be reduced, at the expense of greater space [10,19,13,23].…”

Section: Introductionmentioning

confidence: 99%

A Unified Approach to Approximate Proximity Searching

Arya

Fonseca

Mount

2010

Algorithms – ESA 2010

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Abstract. The inability to answer proximity queries efficiently for spaces of dimension d > 2 has led to the study of approximation to proximity problems. Several techniques have been proposed to address different approximate proximity problems. In this paper, we present a new and unified approach to proximity searching, which provides efficient solutions for several problems: spherical range queries, idempotent spherical range queries, spherical emptiness queries, and nearest neighbor queries. In contrast to previous data structures, our approach is simple and easy to analyze, providing a clear picture of how to exploit the particular characteristics of each of these problems. As applications of our approach, we provide simple and practical data structures that match the best previous results up to logarithmic factors, as well as advanced data structures that improve over the best previous results for all aforementioned proximity problems.

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Balanced Aspect Ratio Trees: Combining the Advantages of k-d Trees and Octrees

Cited by 60 publications

References 31 publications

Geometric Spanners for Weighted Point Sets

Geometric Spanners for Weighted Point Sets

Balanced Aspect Ratio Trees and Their Use for Drawing Large Graphs

A Unified Approach to Approximate Proximity Searching

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