On Finding the Maxima of a Set of Vectors

Kung, H. T.; Luccio, Fabrizio; Preparata, Franco P.

doi:10.1145/321906.321910

Cited by 777 publications

(529 citation statements)

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“…The maximal points of a static set of two-dimensional points can be computed in optimal O(n log n) time [2]. 1 In the pointer machine linear space dynamic data structures have been presented that support reporting all maximal points.…”

Section: Previous Resultsmentioning

confidence: 99%

“…. , p[k] among the points p y−max (u) in R with y-coordinate larger than y b using [2]. Let u i be the node of R such that p y−max (u i ) = p[i].…”

Section: -Sided Range Maxima Queriesmentioning

confidence: 99%

“…. , k} in O(log n) time [2]. Let p[i] = p y−max (u i ) for a particular node u i of R, and let there be t i maximal points to be reported in the subtree T ui .…”

Section: -Sided Range Maxima Queriesmentioning

confidence: 99%

“…For each interval we identify the child c with maximum p y−max (c) using S in O(1) time. These elements define the set R from which we compute the maximal points using [2] in O( log n log log n ) worst case time. We apply the above modified algorithm to this set, ignoring the maximal points p [j] outside the x-range of the query.…”

Section: -Sided Range Maxima In the Ram Modelmentioning

confidence: 99%

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Dynamic Planar Range Maxima Queries

Brodal

Tsakalidis

2011

Automata, Languages and Programming

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Abstract. We consider the dynamic two-dimensional maxima query problem. Let P be a set of n points in the plane. A point is maximal if it is not dominated by any other point in P . We describe two data structures that support the reporting of the t maximal points that dominate a given query point, and allow for insertions and deletions of points in P .In the pointer machine model we present a linear space data structure with O(log n + t) worst case query time and O(log n) worst case update time. This is the first dynamic data structure for the planar maxima dominance query problem that achieves these bounds in the worst case. The data structure also supports the more general query of reporting the maximal points among the points that lie in a given 3-sided orthogonal range unbounded from above in the same complexity. We can support 4-sided queries in O(log 2 n + t) worst case time, and O(log 2 n) worst case update time, using O(n log n) space, where t is the size of the output. This improves the worst case deletion time of the dynamic rectangular visibility query problem from O(log 3 n) to O(log 2 n). We adapt the data structure to the RAM model with word size w, where the coordinates of the points are integers in the range U ={0, . . . , 2 w −1}. We present a linear space data structure that supports 3-sided range maxima queries in O( log n log log n +t) worst case time and updates in O( log n log log n ) worst case time. These are the first sublogarithmic worst case bounds for all operations in the RAM model.

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Section: Previous Resultsmentioning

confidence: 99%

“…. , p[k] among the points p y−max (u) in R with y-coordinate larger than y b using [2]. Let u i be the node of R such that p y−max (u i ) = p[i].…”

Section: -Sided Range Maxima Queriesmentioning

confidence: 99%

“…. , k} in O(log n) time [2]. Let p[i] = p y−max (u i ) for a particular node u i of R, and let there be t i maximal points to be reported in the subtree T ui .…”

Section: -Sided Range Maxima Queriesmentioning

confidence: 99%

Section: -Sided Range Maxima In the Ram Modelmentioning

confidence: 99%

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Dynamic Planar Range Maxima Queries

Brodal

Tsakalidis

2011

Automata, Languages and Programming

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“…The problem of determining whether a figure is contained in another is of principal interest in computational geometry; thus, an answer to the above questions would be of immediate practical relevance due to the existence of efficient computational methods for determining dominance relationships among vectors [8,9]. Furthermore, reduction to vector dominance has already been successfully employed to solve other basic geometric problems [4,10,17].…”

Section: Given a Class Of Geometric Figures How Many Real Variables mentioning

confidence: 99%

Geometric Containment and Partial Orders

Santoro

Sidney

et al. 1989

SIAM J. Discrete Math.

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Given two solid geometric figures on the plane (eg. rectangles) we say that A fits in B (denoted A < B) if there is a translation, a rotation and (if needed) a reflection that maps A into B. Given a family F of geometric figures, we say that the relation "<" on the elements of F is reducible to vector dominance if there exists an n and a mapping f:F AER n such that for A,BOEF A < B iff f(A) < f(B) coordinate by coordinate. A recent result states that if F is the set of all rectangles, "<" is not reducible to a vector dominance relation regardless of the finite value of n. In this paper we extend this result to other families of geometric figures and to a partial order obtained from quadratic polynomials.

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