Amortized Analysis of Smooth Quadtrees in All Dimensions

Bennett, Huck; Yap, Chee

doi:10.1007/978-3-319-08404-6_4

Cited by 6 publications

(12 citation statements)

References 13 publications

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“…Recently Sheehy [She] proposed extending results in his previous work on optimal mesh sizes [She12] to prove the efficient smoothing results presented in this paper. A reviewer of [BY14] proposed a similar proof strategy based on Ruppert's work on local feature size [Rup93]. Future work involves studying these continuous techniques, and determining whether the approach is both viable and leads to better bounds than those given by the combinatorial approach used in this paper.…”

Section: Other Resultsmentioning

confidence: 99%

Amortized analysis of smooth quadtrees in all dimensions

Bennett

Yap

2017

Computational Geometry

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Quadtrees are a well-known data structure for representing geometric data in the plane, and naturally generalize to higher dimensions. A basic operation is to expand the tree by splitting any given leaf. A quadtree is smooth if any two adjacent leaf boxes differ by at most one in depth. In this paper, we analyze quadtrees that restore smoothness after each split operation, and also maintain neighbor pointers. Our main result shows that the smooth split operation has an amortized cost of at most 2 D • (D + 1)! auxiliary split operations, which corresponds to amortized constant time in quadtrees of any fixed dimension D. We also show that the exponential dependence on the dimension is unavoidable via a lower bound construction. We additionally give a lower bound construction showing an amortized cost of Ω(log n) for splits in a related quadtree model that does not maintain smoothness.

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Section: Other Resultsmentioning

confidence: 99%

Amortized analysis of smooth quadtrees in all dimensions

Bennett

Yap

2017

Computational Geometry

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“…Suppose you want to dynamically maintain a d-dimensional quadtree T , with d > 1, where each leaf in T has pointers to adjacent leaves. Bennet and Yap [2] show that this is possible with constant update time if and only we maintain a smooth quadtree. Hoog et al [15,Theorem 20] show how to dynamically maintain a smooth compressed quadtree with constant update time.…”

Section: Basic Properties Of Quadtreesmentioning

confidence: 99%

Preprocessing Ambiguous Imprecise Points

van der Hoog,

Kostitsyna,

Löffler

et al. 2019

Preprint

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Let R = {R 1 , R 2 , . . . , R n } be a set of regions and let X = {x 1 , x 2 , . . . , x n } be an (unknown) point set with x i ∈ R i . Region R i represents the uncertainty region of x i . We consider the following question: how fast can we establish order if we are allowed to preprocess the regions in R? The preprocessing model of uncertainty uses two consecutive phases: a preprocessing phase which has access only to R followed by a reconstruction phase during which a desired structure on X is computed. Recent results in this model parametrize the reconstruction time by the ply of R, which is the maximum overlap between the regions in R. We introduce the ambiguity A(R) as a more fine-grained measure of the degree of overlap in R. We show how to preprocess a set of d-dimensional disks in O(n log n) time such that we can sort X (if d = 1) and reconstruct a quadtree on X (if d ≥ 1 but constant) in O(A(R)) time. If A(R) is sub-linear, then reporting the result dominates the running time of the reconstruction phase. However, we can still return a suitable data structure representing the result in O(A(R)) time.In one dimension, R is a set of intervals and the ambiguity is linked to interval entropy, which in turn relates to the well-studied problem of sorting under partial information. The number of comparisons necessary to find the linear order underlying a poset P is lower-bounded by the graph entropy of P . We show that if P is an interval order, then the ambiguity provides a constant-factor approximation of the graph entropy. This gives a lower bound of Ω(A(R)) in all dimensions for the reconstruction phase (sorting or any proximity structure), independent of any preprocessing; hence our result is tight. Finally, our results imply that one can approximate the entropy of interval graphs in O(n log n) time, improving the O(n 2.5

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“…We write sSplit(B) for sSplit(B; S) when S is understood. More generally, if C ⊆ S is a set of boxes, we define sSplit(C; S) as the smooth split of each B ∈ C in S. Recently, it was shown that in any sequence of smooth split operations starting with S = {B 0 }, each operation has amortized constant cost [BY14]. Our data structure for S provides the ability to retrieve the set of boxes adjacent to any B ∈ S. (see [BY14]).…”

Section: Box Subdivisions and Data Structuresmentioning

confidence: 99%

“…Our data structure for S provides the ability to retrieve the set of boxes adjacent to any B e S . (see [BY14]).…”

Section: Introductionmentioning

confidence: 99%

Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

Bennett

Papadopoulou

Yap

2016

Computer Graphics Forum

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Let X = {f1, …, fn} be a set of scalar functions of the form fi : ℝ2 → ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε‐isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi‐algebraic. We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results.

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Amortized Analysis of Smooth Quadtrees in All Dimensions

Cited by 6 publications

References 13 publications

Amortized analysis of smooth quadtrees in all dimensions

Amortized analysis of smooth quadtrees in all dimensions

Preprocessing Ambiguous Imprecise Points

Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

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