Geometric applications of a matrix searching algorithm

Aggarwal, Alok; Klawe, Maria; Moran, Shlomo; Shor, Peter W.; Wilber, Robert

doi:10.1145/10515.10546

Cited by 107 publications

(260 citation statements)

References 14 publications

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“…, for the two edges between c and d. However, Lemma 4 now requires that the line segments ab and cd cross each other. This proves(2). Finally,…”

supporting

confidence: 55%

“…claims (8) and (9) above), aO <-ab'. Therefore, aa'<_ max{ac, ab'}, which yields the desired contradiction because aa'> ac and aa'> ab', by (2). This finishes the proof.…”

Section: Lemma 3 Let G = { V E W: E ~ R+} Be a Weighted Graph And mentioning

confidence: 51%

“…[12] or Lee [10] for furthest-point Voronoi diagrams and Preparata and Shamos [11] for pointlocation algorithms.) Fact 2 follows from a recent result of Aggarwal et al [2].…”

Section: Algorithm Maximum-spanning-treementioning

confidence: 86%

“…LEMMA 5. (1) Both endpoints of a spine of FNG(P) lie on CH(P). (2) Each pair of spines of FNG( P) properly intersects. We have seen that the furthest-neighbor graph of P, FNG(P), has out-degree one, and each (weak) component of this graph has exactly one strongly connected component (of size two), namely the component having the two endpoints of its spine.…”

Section: Lemma 3 Let G = { V E W: E ~ R+} Be a Weighted Graph And mentioning

confidence: 99%

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Computing euclidean maximum spanning trees

et al. 1990

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An algorithm is presented for finding a maximum-weight spanning tree of a set of n points in the Euclidean plane, where the weight of an edge (p,, pj) equals the Euclidean distance between the points Pl and pj. The algorithm runs in O(n log h) time and requires O(n) space; h denotes the number of points on the convex hull of the given set. If the points are vertices of a convex polygon (given in order along the boundary), then our algorithm requires only a linear amount of time and space. These bounds are the best possible in the algebraic computation-tree model. We also establish various properties of maximum spanning trees that can be exploited to solve other geometric problems.

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“…, for the two edges between c and d. However, Lemma 4 now requires that the line segments ab and cd cross each other. This proves(2). Finally,…”

supporting

confidence: 55%

“…claims (8) and (9) above), aO <-ab'. Therefore, aa'<_ max{ac, ab'}, which yields the desired contradiction because aa'> ac and aa'> ab', by (2). This finishes the proof.…”

Section: Lemma 3 Let G = { V E W: E ~ R+} Be a Weighted Graph And mentioning

confidence: 51%

“…[12] or Lee [10] for furthest-point Voronoi diagrams and Preparata and Shamos [11] for pointlocation algorithms.) Fact 2 follows from a recent result of Aggarwal et al [2].…”

Section: Algorithm Maximum-spanning-treementioning

confidence: 86%

Section: Lemma 3 Let G = { V E W: E ~ R+} Be a Weighted Graph And mentioning

confidence: 99%

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Computing euclidean maximum spanning trees

et al. 1990

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“…Observe that the output of the OQ problem can be represented as a k-node list. Thus, the OQ problem can be modeled as a k-link shortest path problem on a DAG (directed acyclic graph) [33]; further, the length matrices of the paths in the DAG for this problem satisfy the Monge property [1,2,26], and hence can be computed very efficiently [1-3, 29, 33]. In contrast, the FBT output can be represented as a k-node forest; thus, the FBT problem can be modeled as a k-MST problem [6-8, 18, 21, 24, 25, 28, 34] on a geometric graph.…”

Section: Previous and Related Workmentioning

confidence: 99%

Shape Rectangularization Problems in Intensity-Modulated Radiation Therapy

et al. 2009

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In this paper, we present a theoretical study of several shape approximation problems, called shape rectangularization (SR), which arise in intensity-modulated Partial preliminary versions of this work appeared in Proc. of the 9th Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) and in Proc. of the 17th International Symposium on Algorithms and Computation (ISAAC). 422Algorithmica (2011) 60: 421-450 radiation therapy (IMRT). Given a piecewise linear function f such that f (x) ≥ 0 for any x ∈ R, the SR problems seek an optimal set of constant window functions to approximate f under a certain error criterion, such that the sum of the resulting constant window functions equals (or well approximates) f . A constant window function W (·) is defined on an interval I such that W (x) is a fixed value h > 0 for any x ∈ I and is 0 otherwise. A constant window function can be viewed as a rectangle (or a block) geometrically, or as a vector with the consecutive a's property combinatorially. The SR problems find applications in setup time and beam-on time minimization and dose simplification of the IMRT treatment planning process. We show that the SR problems are APX-Hard, and thus we aim to develop theoretically efficient and provably good quality approximation SR algorithms. Our main contribution is to present algorithms for a key SR problem that achieve approximation ratios better than 2. For the general case, we give a 24 13 -approximation algorithm. For unimodal input curves, we give a 9 7 -approximation algorithm. We also consider other variants for which better approximation ratios are possible. We show that an important SR case that has been studied in medical literature can be formulated as a k-MST (k-minimum-spanning-tree) problem on a certain geometric graph G; based on a set of geometric observations and a non-trivial dynamic programming scheme, we are able to compute an optimal k-MST in G efficiently.

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Vehicle routing problems with regular objective functions on a path

Liu

2013

Naval Research Logistics

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We investigate the problem of scheduling a fleet of vehicles to visit the customers located on a path to minimize some regular function of the visiting times of the customers. For the single-vehicle problem, we prove that it is pseudopolynomially solvable for any minsum objective and polynomially solvable for any minmax objective. Also, we establish the NP-hardness of minimizing the weighted number of tardy customers and the total weighted tardiness, and present polynomial algorithms for their special cases with a common due date. For the multivehicle problem involving n customers, we show that an optimal solution can be found by solving O(n 2 ) or O(n) single-vehicle problems.

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Geometric applications of a matrix searching algorithm

Cited by 107 publications

References 14 publications

Computing euclidean maximum spanning trees

Computing euclidean maximum spanning trees

Shape Rectangularization Problems in Intensity-Modulated Radiation Therapy

Vehicle routing problems with regular objective functions on a path

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