Line Segment Disk Cover

Basappa, Manjanna

doi:10.1007/978-3-319-74180-2_7

Cited by 5 publications

(5 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In a similar line, there is another variant of the DUDC problem, a Rectangular Region Cover (RRC) problem, in which all the continuous set of points lying in a rectangular region represent wireless clients. All the available results for the DUDC problem also extend to the LSDC and RRC problems [1], with slightly different running time. We also extend our results for the k-CDUDC problem to solve the colorable variants of the LSDC and RRC problems, namely, the k-CLSDC and k-CRRC problems.…”

Section: Related Workmentioning

confidence: 99%

“…6(a)). Therefore, the edge of each subgrid cell would be τ 2 , i.e., it would be between 1 2 to 1, since 1 ≤ τ ≤ 2. We consider a subset D ′ of disks that can cover points in 2 diagonally opposite pair of subgrid cells such that the center of these disks lies in the respective quadrant of the subgrid cells, assuming that C is placed such that the center of C is at the origin of the x, y-coordinate system.…”

Section: Improved Algorithm For a Grid Cellmentioning

confidence: 99%

See 1 more Smart Citation

The $k$-Colorable Unit Disk Cover Problem

Reyunuru¹,

Jethlia²,

Basappa³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this article, we consider colorable variations of the Unit Disk Cover (UDC) problem as follows.k-Colorable Discrete Unit Disk Cover (k-CDUDC): Given a set P of n points, and a set D of m unit disks (of radius=1), both lying in the plane, and a parameter k, the objective is to compute a set D ′ ⊆ D such that every point in P is covered by at least one disk in D ′ and there exists a function χ : D ′ → C that assigns colors to disks in D ′ such that for any d and, where C denotes a set containing k distinct colors.For the k-CDUDC problem, our proposed algorithms approximate the number of colors used in the coloring if there exists a k-colorable cover. We first propose a 4-approximation algorithm in O(m 7k n log k) time for this problem and then show that the running time can be improved by a multiplicative factor of m k , where a positive integer k denotes the cardinality of a color-set. The previous best known result for the problem when k = 3 is due to the recent work of Biedl et al., (2021)[3], who proposed a 2-approximation algorithm in O(m 25 n) time. For k = 3, our algorithm runs in O(m 18 n) time, faster than the previous best algorithm, but gives a 4-approximate result. We then generalize our approach to yield a family of ρ-approximation algorithms in O(m αk n log k) time, where (ρ, α) ∈ {(4, 7), (6, 5), (7, 5), (9, 4)}. We further generalize this to exhibit a O( 1 τ )-approximation algorithm in O(m αk n log k) time for a given 1 ≤ τ ≤ 2, where α = O(τ 2

show abstract

Section: Related Workmentioning

confidence: 99%

Section: Improved Algorithm For a Grid Cellmentioning

confidence: 99%

The $k$-Colorable Unit Disk Cover Problem

Reyunuru¹,

Jethlia²,

Basappa³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…3. there is a 34 + 44 6 11 + ν -approximation if G is any subgraph of a Delaunay triangulation; in addition, for outerplane G an 24 + 28 5 7 + νapproximation exists.…”

Section: Special Segment Configurationsmentioning

confidence: 99%

“…Design of approximation algorithms is a hot topic for many NP-hard combinatorial optimization problems related to geometric coverage and intersection (see e.g. works [5], [6], [12], [13], [17]). Some problems from this class can be formulated in the following general form.…”

Section: Introductionmentioning

confidence: 99%

Efficient constant factor approximation algorithms for stabbing line segments with equal disks

Kobylkin

2018

Preprint

View full text Add to dashboard Cite

Fast constant factor approximation algorithms are devised for an NP-and W[1]hard problem of intersecting a set of n straight line segments with the smallest cardinality set of disks of fixed radii r > 0, where the set of segments forms a straight line drawing G = (V, E) of a planar graph without edge crossings. Exploiting tough connection of the problem with the geometric Hitting Set problem, an 50 + 52 12 13 + ε -approximate O n 4 log n -time and O n 2 log nspace algorithm is given based on the modified Agarwal-Pan algorithm. More accurate (34 + 24 √ 2 + ε)-and 34 + 44 6 11 + ε -approximate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where ε > 0 is an arbitrary small constant. Moreover, an O(n 2 log n)-time and O(n 2 )-space 18-approximation is designed for the case where G is any subgraph of a Gabriel graph. To the best of our knowledge, related work only tackles the case where E consists of axis-parallel segments, resulting in an O(n log n)-time and O(n log n)-space 8-approximation.

show abstract

“…In the Discrete Unit Disk Cover problem, S is finite, and the disks are restricted to a finite set of possible centers [9]; the discretized version admits a PTAS via local search [28,29]. These results extend to the cases where S is confined to a narrow strip [21], or S is a finite union of line segments [8]. Finding the minimum k ∈ N such that δ station (k, P, Q) ≤ 1 can be considered as a variant of UDC, where P (resp., Q) must be covered by disks centered at Q (resp., P ), and each disk can cover at most one contiguous arc of a curve.…”

Section: Introductionmentioning

confidence: 99%

Rock Climber Distance: Frogs versus Dogs

Akitaya,

Ryvkin,

Tóth

2019

Preprint

View full text Add to dashboard Cite

The classical measure of similarity between two polygonal chains in Euclidean space is the Fréchet distance, which corresponds to the coordinated motion of two mobile agents along the chains while minimizing their maximum distance. As computing the Fréchet distance takes near-quadratic time under the Strong Exponential Time Hypothesis (SETH), we explore two new distance measures, called rock climber distance and k-station distance, in which the agents move alternately in their coordinated motion that traverses the polygonal chains. We show that the new variants are equivalent to the Fréchet or the Hausdorff distance if the number of moves is unlimited. When the number of moves is limited to a given parameter k, we show that it is NP-hard to determine the distance between two curves. We also describe a 2-approximation algorithm to find the minimum k for which the distance drops below a given threshold.

show abstract

Line Segment Disk Cover

Cited by 5 publications

References 12 publications

The $k$-Colorable Unit Disk Cover Problem

The $k$-Colorable Unit Disk Cover Problem

Efficient constant factor approximation algorithms for stabbing line segments with equal disks

Rock Climber Distance: Frogs versus Dogs

Contact Info

Product

Resources

About