Optimal Search in Planar Subdivisions

Kirkpatrick, David G.

doi:10.1137/0212002

Cited by 662 publications

(362 citation statements)

References 21 publications

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“…Together with the segments in Γ 0 , they form a planar subdivision Q with O(n/ε 7/6 ) edges, which we preprocess for efficient point location. Using Kirkpatrick's algorithm [24], this can be done with O(n/ε 7/6 ) preprocessing time and storage, and a query can be answered in O(log(n/ε)) time. We also store (with no extra asymptotic cost) a count N e with each edge e of Q.…”

Section: Fast Query Timementioning

confidence: 99%

Relative (p,ε)-Approximations in Geometry

Har-Peled

Sharir

2010

Discrete Comput Geom

141

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We re-examine the notion of relative (p, ε)-approximations, recently introduced in Cohen et al. (Manuscript, 2006), and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in Li et al. (J. Comput. Syst. Sci. 62:516-527, 2001) and in several earlier studies (Pollard in Manuscript, 1986; Haussler in Inf. Comput. 100:78-150, 1992; Talagrand in Ann. Probab. 22:28-76, 1994). We also survey the different notions of sampling, used in computational geometry, learning, and other areas, and show how they relate to each other. We then give constructions of smaller-size relative (p, ε)-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure-spanning trees with small relative crossing number, which we believe to be of independent interest. Relative (p, ε)-approximations arise in several geometric problems, such as approximate range counting, and we apply our new structures to obtain efficient solutions for approximate range counting in three dimensions. We also present a simple solution for the planar case.

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Section: Fast Query Timementioning

confidence: 99%

Relative (p,ε)-Approximations in Geometry

Har-Peled

Sharir

2010

Discrete Comput Geom

141

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“…When "locally" means "nearest by great-circle distance", we repeat the process of building the rotational separation diagram, this time using the Voronoi vertices of the first diagram as the sites. This requires the same preprocessing time and space as before, after which we can apply a point-location algorithm [15,19], again with the same preprocessing time and space requirements. The resulting diagram can be used to find the nearest site (the locally best viewpoint) in O(log IS D time.…”

Section: (Dmentioning

confidence: 99%

Finding the best viewpoints for three-dimensional graph drawings

Eades¹,

Houle²,

Webber³

1997

Graph Drawing

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Abstract. In this paper we address the problem of finding the best viewpoints for three-dimensional straight-line graph drawings. We define goodness in terms of preserving the relational structure of the graph, and develop two continuous measures of goodness under orthographic parallel projection. We develop Voronoi variants to find the best viewpoints under these measures, and present results on the complexity of these diagrams.

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“…Then we can use any of the known linear-time algorithms (e.g., [9]) to process this partitioning into a structure supporting queries taking D(log n) time to determine into which triangle a point falls. Then of course the shortest path from any point x inside P to scan be constrncted by determining the triangle into which x faIls and concatenating the line from x to the appropriate vertex y of the triangle, as determined when the triangle was constructed, plus the path from y to s.…”

Section: Shortest Path Tree Algorithrnmentioning

confidence: 99%

Computing the external geodesic diameter of a simple polygon

Samuel

Toussaint

1990

Computing

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Given a simple polygon P of n vertices, we present an algorithm that finds the pair of points on the boundary of P that maximize the externat shortest path between them. This path is defined as the externat geodesic diameter of P. The algorithm takes O(n 2 ) time and requires D(n) space. Surprisingly, this problem is quite different from that of computing the internai geodeslc diameter of P. WhiIe the internai diameter le; determined by a pair of verticls of P, this is not the case for the external diameter.Finally, we show how this algo~ithm can be extended to solve the ail external geodesicfurthest neighbours problem.-- R~suméCe travail présente un a1gorithme qui trouve une paire de points sur la frontière d'un polygone simple P de n coins, qui maximise le trajectoire externe le plus court entre les deux points. Ce trajectoire est définit par le diamètre géodésique externe de P. L'algorithrrle prend un temps O(n 2 ) et requiert un champ O(n). Fait surprennant, ce ploblème n'est pas semblable au prlJblème de calculer le diamètre géodésique interne. Bien que le diamètre interne est delerminé par une paire de coins de P, ce n'est pas le cas pour le diamètre externe. En fin, ce travail demontre comment cet algorithme peut être étendu pour résoudre le problème de tous les voisins à la plus grande distance géodésique externe.

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Optimal Search in Planar Subdivisions

Cited by 662 publications

References 21 publications

Relative (p,ε)-Approximations in Geometry

Relative (p,ε)-Approximations in Geometry

Finding the best viewpoints for three-dimensional graph drawings

Computing the external geodesic diameter of a simple polygon

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