Dynamic algorithms for monotonic interval scheduling problem

Gavruskin, Alexander; Khoussainov, Bakhadyr; Kokho, Mikhail; Liu, Jiamou

doi:10.1016/j.tcs.2014.09.046

Cited by 14 publications

(13 citation statements)

References 11 publications

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“…To handle this bottleneck, we require another data structure. The dynamic monotone interval data structure [17] reports whether there exist m − 1 non-overlapping y-intervals, which represent our m − 1 non-overlapping monotone paths, without explicitly computing them. See Figure 5, right.…”

Section: Key Insight 3: Handling Additional Critical Points and Refer...mentioning

confidence: 99%

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Cubic upper and lower bounds for subtrajectory clustering under the continuous Fréchet distance

Gudmundsson¹,

Wong²

2021

Preprint

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Detecting commuting patterns or migration patterns in movement data is an important problem in computational movement analysis. Given a trajectory, or set of trajectories, this corresponds to clustering similar subtrajectories.We study subtrajectory clustering under the continuous and discrete Fréchet distances. The most relevant theoretical result is by Buchin et al. (2011). They provide, in the continuous case, an O(n 5 ) time algorithm 1 and a 3SUM-hardness lower bound, and in the discrete case, an O(n 3 ) time algorithm. We show, in the continuous case, an O(n 3 log 2 n) time algorithm and a 3OV-hardness lower bound, and in the discrete case, an O(n 2 log n) time algorithm and a quadratic lower bound. Our bounds are almost tight unless SETH fails.1 Buchin et al. [8] show an O(|S| • n 2 ) time algorithm, where S is the set of (internal) critical points. In this paper, we show that |S| = Θ(n 3 ), yielding an overall running time of O(n 5 ).

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Section: Key Insight 3: Handling Additional Critical Points and Refer...mentioning

confidence: 99%

“…Fact 26 (Theorem 16 in [17]). A dynamic monotonic interval data structure maintains a set of monotonic intervals.…”

Section: Reference Subtrajectory Is Arbitrarymentioning

confidence: 99%

Cubic upper and lower bounds for subtrajectory clustering under the continuous Fréchet distance

Gudmundsson¹,

Wong²

2021

Preprint

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“…The worst-case update time of their algorithm is O(log(n) + d), where d refers to the depth of what they call idle intervals; they define an idle interval to be the period of time in a schedule between two consecutive jobs in a given machine. The same set of authors, in [GKKL15], study dynamic algorithms for the monotone case as well, in which no interval completely contains another one. For this setup they obtain an algorithm with O(log(n)) update and query time.…”

Section: Related Workmentioning

confidence: 99%

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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“…Most previous work dates from 2020 onwards. The only exception is the work of Gavruskin et al [31] who considered the very special case of intervals where no interval is fully contained in any other interval. The study of this area was revitalized by a paper of Henzinger, Neumann and Wiese in SoCG'20 [35].…”

Section: Introductionmentioning

confidence: 99%

Worst-Case Efficient Dynamic Geometric Independent Set

Cardinal¹,

Iacono²,

Koumoutsos³

2021

Preprint

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We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present data structures that maintain a constant-factor approximate maximum independent set for broad classes of fat objects in d dimensions, where d is assumed to be a constant, in sublinear worst-case update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. For axis-aligned squares and hypercubes, our result improves upon all (recently announced) previous works. We obtain, in particular, a dynamic (4 + ǫ)-approximation for squares, with O(log 4 n) worstcase update time.Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with amortized update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem.Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane.

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Dynamic algorithms for monotonic interval scheduling problem

Cited by 14 publications

References 11 publications

Cubic upper and lower bounds for subtrajectory clustering under the continuous Fréchet distance

Cubic upper and lower bounds for subtrajectory clustering under the continuous Fréchet distance

New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

Worst-Case Efficient Dynamic Geometric Independent Set

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