Maintenance of configurations in the plane

Overmars, Mark H.; Leeuwen, Jan van

doi:10.1016/0022-0000(81)90012-x

Cited by 398 publications

(253 citation statements)

References 15 publications

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“…1 In the pointer machine linear space dynamic data structures have been presented that support reporting all maximal points. They achieve O(m) worst case [8,9,3,13] or amortized [15] query time. They support deciding whether a given query point lies above or below the staircase (maxima emptiness query) in O(log n) [8,9,15] or O(log m) [3,13] worst case time.…”

Section: Previous Resultsmentioning

confidence: 99%

“…They achieve O(m) worst case [8,9,3,13] or amortized [15] query time. They support deciding whether a given query point lies above or below the staircase (maxima emptiness query) in O(log n) [8,9,15] or O(log m) [3,13] worst case time. Maxima contour queries are supported by the structures of [3] and [9] in O(log n + t) worst case time, where t is size of the output.…”

Section: Previous Resultsmentioning

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%

Section: Previous Resultsmentioning

confidence: 99%

Dynamic Planar Range Maxima Queries

Brodal

Tsakalidis

2011

Automata, Languages and Programming

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“…The time complexity of both algorithms is O(log 2 k) per created edge. This factor comes from the use of the dynamic convex hull data structure of Overmars and van Leeuwen [10]. Chan [3] has given a data structure that allows dynamic maintenance of the convex hull of n points in O(lg 1+ε n) amortized time.…”

Section: Resultsmentioning

confidence: 99%

“…1 for an illustration). (8,9) e 8,10,12 (11,6) g (6,8,10,12) From now on, any oriented edge g(P ∪{s})g(P ∪{t}) of g k (V ) will be denoted by e P (s, t).…”

Section: Counting K-sets Of Convex Inclusion Chainsmentioning

confidence: 99%

“…Using results given by Overmars and van Leeuwen [10] (see also Overmars [9]), this structure needs O(h) size to store the convex hull of h points of the plane, allows to get the predeccessor and the successor of any edge in constant time, and can be updated in O(log 2 h) time after inserting or deleting a point.…”

Section: Theorem 2 C Sv Is the Sequence Of Polygonal Linesmentioning

confidence: 99%

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k-Sets of Convex Inclusion Chains of Planar Point Sets

Oraiby

Schmitt

2006

Lecture Notes in Computer Science

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Abstract. Given a set V of n points in the plane, we introduce a new number of k-sets that is an invariant of V : the number of k-sets of a convex inclusion chain of V . A convex inclusion chain of V is an ordering (v1, v2, ..., vn) of the points of V such that no point of the ordering belongs to the convex hull of its predecessors. The k-sets of such a chain are then the distinct k-sets of all the subsets {v1, ..., vi}, for all i in {k + 1, ..., n}. We show that the number of these k-sets depends only on V and not on the chosen convex inclusion chain. Moreover, this number is surprisingly equal to the number of regions of the order-k Voronoi diagram of V . As an application, we give an efficient on-line algorithm to compute the k-sets of the vertices of a simple polygonal line, no vertex of which belonging to the convex hull of its predecessors on the line.

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Improved algorithms for a lot‐sizing problem with inventory bounds and backlogging

Hwang

Heuvel

2012

Naval Research Logistics

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This paper considers a dynamic lot-sizing problem with storage capacity limitation in which backlogging is allowed. For general concave production and inventory costs, we present an O(T 2 ) dynamic programming algorithm where is the length of the planning horizon. Furthermore, for fixed-charge and nonspeculative costs, we provide O(Tlog T) and O(T) algorithms, respectively. This paper therefore concludes that the time complexity to solve the bounded inventory lot-sizing problem with backlogging is the same as the complexity to solve the uncapacitated lot-sizing problem for the commonly used cost structures.

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Maintenance of configurations in the plane

Cited by 398 publications

References 15 publications

Dynamic Planar Range Maxima Queries

Dynamic Planar Range Maxima Queries

k-Sets of Convex Inclusion Chains of Planar Point Sets

Improved algorithms for a lot‐sizing problem with inventory bounds and backlogging

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