Dynamic Geometric Independent Set

Bhore, Sujoy; Cardinal, Jean; Iacono, John; Koumoutsos, Grigorios

doi:10.48550/arxiv.2007.08643

Cited by 4 publications

(7 citation statements)

References 29 publications

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“…This result improves the best-known time complexities [BCIK20,HNW20]. Unfortunately, it does not immediately generalize well to the weighted variant.…”

Section: Dynamic Unweighted Interval Scheduling On a Single Machinementioning

confidence: 70%

“…Our first result, given in Section 3, focuses on designing an efficient dynamic algorithm for unweighted interval scheduling on a single machine. Prior to our work, the state-of-the-art result for this problem was due to [BCIK20], who design an algorithm with O( log n /ε 2 ) update and query time. We provide an improvement in the dependence on ε.…”

Section: Our Resultsmentioning

confidence: 99%

“…The closest prior work to ours is that of Henzinger et al [HNW20] and of Sujoy et al [BCIK20]. [HNW20] studies (1 + ε)-approximate dynamic interval scheduling in both the weighted and unweighted setting.…”

Section: Related Workmentioning

confidence: 99%

“…The authors of [BCIK20] primarily focus on the unweighted case of approximating maximum independent set of a set of cubes. For the 1-dimensional case, which equals interval scheduling, they obtain O( log n /ε 2 ) update time, which is slower by a factor of 1/ε than our approach.…”

Section: Related Workmentioning

confidence: 99%

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New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

Compton¹,

Mitrović²,

Rubinfeld³

2020

Preprint

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Interval scheduling is a basic problem in the theory of algorithms and a classical task in combinatorial optimization. We develop a set of techniques for partitioning and grouping jobs based on their starting and ending times, that enable us to view an instance of interval scheduling on many jobs as a union of multiple interval scheduling instances, each containing only a few jobs. Instantiating these techniques in dynamic and local settings of computation leads to several new results.For (1 + ε)-approximation of job scheduling of n jobs on a single machine, we obtain a fully dynamic algorithm with O( log n /ε) update and O(log n) query worst-case time. Further, we design a local computation algorithm that uses only O( log n /ε) queries. Our techniques are also applicable in a setting where jobs have rewards/weights. For this case we obtain a fully dynamic algorithm whose worst-case update and query time has only polynomial dependence on 1/ε, which is an exponential improvement over the result of Henzinger et al. [SoCG, 2020].We extend our approaches for unweighted interval scheduling on a single machine to the setting with M machines, while achieving the same approximation factor and only M times slower update time in the dynamic setting. In addition, we provide a general framework for reducing the task of interval scheduling on M machines to that of interval scheduling on a single machine. In the unweighted case this approach incurs a multiplicative approximation factor 2 − 1/M .

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“…This result improves the best-known time complexities [BCIK20,HNW20]. Unfortunately, it does not immediately generalize well to the weighted variant.…”

Section: Dynamic Unweighted Interval Scheduling On a Single Machinementioning

confidence: 70%

Section: Our Resultsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

Compton¹,

Mitrović²,

Rubinfeld³

2020

Preprint

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“…A second significant contribution of our paper is to extend the dynamic set cover data structures to weighted instances, thus providing the first nontrivial results for dynamic weighted geometric set cover. (Although there were previous results on weighted independent set for 1D intervals and other ranges by Henzinger, Neumann, and Wiese [24] and Bhore et al [8], no results on dynamic weighted geometric set cover were known even in 1D. This is in spite of the considerable work on static weighted geometric set cover [13,21,23,27,29].)…”

Section: Rangesmentioning

confidence: 93%

Dynamic Geometric Set Cover, Revisited

Chan¹,

He²,

Suri³

et al. 2021

Preprint

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Geometric set cover is a classical problem in computational geometry, which has been extensively studied in the past. In the dynamic version of the problem, points and ranges may be inserted and deleted, and our goal is to efficiently maintain a set cover solution (satisfying certain quality requirement) for the dynamic problem instance. In this paper, we give a plethora of new dynamic geometric set cover data structures in 1D and 2D, which significantly improve and extend the previous results. Our results include the following:• The first data structure for (1 + ε)-approximate dynamic interval set cover with polylogarithmic amortized update time. Specifically, we achieve an update time of O(log 3 n/ε), improving the O(n δ /ε) bound of Agarwal et al. [SoCG'20], where δ > 0 denotes an arbitrarily small constant. • A data structure for O(1)-approximate dynamic unit-square set cover with 2 O( √ log n) amortized update time, substantially improving the O(n 1/2+δ ) update time of Agarwal et al. [SoCG'20]. • A data structure for O(1)-approximate dynamic square set cover with O(n 1/2+δ ) randomized amortized update time, improving the O(n 2/3+δ ) update time of Chan and He [SoCG'21]. • A data structure for O(1)-approximate dynamic 2D halfplane set cover with O(n 17/23+δ ) randomized amortized update time. The previous solution for halfplane set cover by Chan and He [SoCG'21] is slower and can only report the size of the approximate solution. • The first sublinear results for the weighted version of dynamic geometric set cover. Specifically, we give a data structure for (3 + o(1))-approximate dynamic weighted interval set cover with 2 O( √ log n log log n) amortized update time and a data structure for O(1)-approximate dynamic weighted unit-square set cover with O(n δ ) amortized update time.

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On Streaming Algorithms for Geometric Independent Set and Clique

Bhore

Klute

Oostveen

2022

Approximation and Online Algorithms

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We study the maximum geometric independent set and clique problems in the streaming model. Given a collection of geometric objects arriving in an insertion only stream, the aim is to find a subset such that all objects in the subset are pairwise disjoint or intersect respectively.We show that no constant factor approximation algorithm exists to find a maximum set of independent segments or 2-intervals without using a linear number of bits. Interestingly, our proof only requires a set of segments whose intersection graph is also an interval graph. This reveals an interesting discrepancy between segments and intervals as there does exist a 2-approximation for finding an independent set of intervals that uses only O(α(I) log |I|) bits of memory for a set of intervals I with α(I) being the size of the largest independent set of I. On the flipside we show that for the geometric clique problem there is no constant-factor approximation algorithm using less than a linear number of bits even for unit intervals. On the positive side we show that the maximum geometric independent set in a set of axis-aligned unit-height rectangles can be 4-approximated using only O(α(R) log |R|) bits.

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Dynamic Geometric Independent Set

Cited by 4 publications

References 29 publications

New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

Dynamic Geometric Set Cover, Revisited

On Streaming Algorithms for Geometric Independent Set and Clique

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