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LIPIcs, Volume 173
28th Annual European Symposium on Algorithms (ESA 2020)
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Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: European Symposium on Algorithms (ESA)
Publication Details
- published at: 2020-08-26
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-162-7
- DBLP: db/conf/esa/esa2020
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Complete Volume
LIPIcs, Volume 173, ESA 2020, Complete Volume
Abstract
LIPIcs, Volume 173, ESA 2020, Complete Volume
Cite as
28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 1-1598, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@Proceedings{grandoni_et_al:LIPIcs.ESA.2020,
title = {{LIPIcs, Volume 173, ESA 2020, Complete Volume}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {1--1598},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020},
URN = {urn:nbn:de:0030-drops-128651},
doi = {10.4230/LIPIcs.ESA.2020},
annote = {Keywords: LIPIcs, Volume 173, ESA 2020, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{grandoni_et_al:LIPIcs.ESA.2020.0,
author = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {0:i--0:xx},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.0},
URN = {urn:nbn:de:0030-drops-128669},
doi = {10.4230/LIPIcs.ESA.2020.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Planar Bichromatic Bottleneck Spanning Trees
Abstract
Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8√2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck λ can be converted to a planar bichromatic spanning tree of bottleneck at most 8√2 λ.
Cite as
A. Karim Abu-Affash, Sujoy Bhore, Paz Carmi, and Joseph S. B. Mitchell. Planar Bichromatic Bottleneck Spanning Trees. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{abuaffash_et_al:LIPIcs.ESA.2020.1,
author = {Abu-Affash, A. Karim and Bhore, Sujoy and Carmi, Paz and Mitchell, Joseph S. B.},
title = {{Planar Bichromatic Bottleneck Spanning Trees}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {1:1--1:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.1},
URN = {urn:nbn:de:0030-drops-128670},
doi = {10.4230/LIPIcs.ESA.2020.1},
annote = {Keywords: Approximation Algorithms, Bottleneck Spanning Tree, NP-Hardness}
}
Document
Parallel Batch-Dynamic Trees via Change Propagation
Abstract
The dynamic trees problem is to maintain a forest subject to edge insertions and deletions while facilitating queries such as connectivity, path weights, and subtree weights. Dynamic trees are a fundamental building block of a large number of graph algorithms. Although traditionally studied in the single-update setting, dynamic algorithms capable of supporting batches of updates are increasingly relevant today due to the emergence of rapidly evolving dynamic datasets. Since processing updates on a single processor is often unrealistic for large batches of updates, designing parallel batch-dynamic algorithms that achieve provably low span is important for many applications.
In this work, we design the first work-efficient parallel batch-dynamic algorithm for dynamic trees that is capable of supporting both path queries and subtree queries, as well as a variety of nonlocal queries. Previous work-efficient dynamic trees of Tseng et al. were only capable of handling subtree queries [ALENEX'19, (2019), pp. 92 - 106]. To achieve this, we propose a framework for algorithmically dynamizing static round-synchronous algorithms to obtain parallel batch-dynamic algorithms. In our framework, the algorithm designer can apply the technique to any suitably defined static algorithm. We then obtain theoretical guarantees for algorithms in our framework by defining the notion of a computation distance between two executions of the underlying algorithm.
Our dynamic trees algorithm is obtained by applying our dynamization framework to the parallel tree contraction algorithm of Miller and Reif [FOCS'85, (1985), pp. 478 - 489], and then performing a novel analysis of the computation distance of this algorithm under batch updates. We show that k updates can be performed in O(klog(1+n/k)) work in expectation, which matches the algorithm of Tseng et al. while providing support for a substantially larger number of queries and applications.
Cite as
Umut A. Acar, Daniel Anderson, Guy E. Blelloch, Laxman Dhulipala, and Sam Westrick. Parallel Batch-Dynamic Trees via Change Propagation. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{acar_et_al:LIPIcs.ESA.2020.2,
author = {Acar, Umut A. and Anderson, Daniel and Blelloch, Guy E. and Dhulipala, Laxman and Westrick, Sam},
title = {{Parallel Batch-Dynamic Trees via Change Propagation}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {2:1--2:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.2},
URN = {urn:nbn:de:0030-drops-128686},
doi = {10.4230/LIPIcs.ESA.2020.2},
annote = {Keywords: Dynamic trees, Graph algorithms, Parallel algorithms, Dynamic algorithms}
}
Document
Reconstructing Biological and Digital Phylogenetic Trees in Parallel
Abstract
In this paper, we study the parallel query complexity of reconstructing biological and digital phylogenetic trees from simple queries involving their nodes. This is motivated from computational biology, data protection, and computer security settings, which can be abstracted in terms of two parties, a responder, Alice, who must correctly answer queries of a given type regarding a degree-d tree, T, and a querier, Bob, who issues batches of queries, with each query in a batch being independent of the others, so as to eventually infer the structure of T. We show that a querier can efficiently reconstruct an n-node degree-d tree, T, with a logarithmic number of rounds and quasilinear number of queries, with high probability, for various types of queries, including relative-distance queries and path queries. Our results are all asymptotically optimal and improve the asymptotic (sequential) query complexity for one of the problems we study. Moreover, through an experimental analysis using both real-world and synthetic data, we provide empirical evidence that our algorithms provide significant parallel speedups while also improving the total query complexities for the problems we study.
Cite as
Ramtin Afshar, Michael T. Goodrich, Pedro Matias, and Martha C. Osegueda. Reconstructing Biological and Digital Phylogenetic Trees in Parallel. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 3:1-3:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{afshar_et_al:LIPIcs.ESA.2020.3,
author = {Afshar, Ramtin and Goodrich, Michael T. and Matias, Pedro and Osegueda, Martha C.},
title = {{Reconstructing Biological and Digital Phylogenetic Trees in Parallel}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {3:1--3:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.3},
URN = {urn:nbn:de:0030-drops-128696},
doi = {10.4230/LIPIcs.ESA.2020.3},
annote = {Keywords: Tree Reconstruction, Parallel Algorithms, Privacy, Phylogenetic Trees, Data Structures, Hierarchical Clustering}
}
Document
Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem
Abstract
We study the multi-level Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals T require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree containing edges of varying rates such that any two terminals u, v with priorities P(u), P(v) are connected using edges of rate min{P(u),P(v)} or better. The case where edge costs are proportional to their rate is approximable to within a constant factor of the optimal solution. For the more general case of non-proportional costs, this problem is hard to approximate with ratio c log log n, where n is the number of vertices in the graph. A simple greedy algorithm by Charikar et al., however, provides a min{2(ln |T|+1), 𝓁 ρ}-approximation in this setting, where ρ is an approximation ratio for a heuristic solver for the Steiner tree problem and 𝓁 is the number of priorities or levels (Byrka et al. give a Steiner tree algorithm with ρ≈1.39, for example).
In this paper, we describe a natural generalization to the multi-level case of the classical (single-level) Steiner tree approximation algorithm based on Kruskal’s minimum spanning tree algorithm. We prove that this algorithm achieves an approximation ratio at least as good as Charikar et al., and experimentally performs better with respect to the optimum solution. We develop an integer linear programming formulation to compute an exact solution for the multi-level Steiner tree problem with non-proportional edge costs and use it to evaluate the performance of our algorithm on both random graphs and multi-level instances derived from SteinLib.
Cite as
Reyan Ahmed, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence. Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ahmed_et_al:LIPIcs.ESA.2020.4,
author = {Ahmed, Reyan and Sahneh, Faryad Darabi and Hamm, Keaton and Kobourov, Stephen and Spence, Richard},
title = {{Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {4:1--4:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.4},
URN = {urn:nbn:de:0030-drops-128709},
doi = {10.4230/LIPIcs.ESA.2020.4},
annote = {Keywords: multi-level, Steiner tree, approximation algorithms}
}
Document
Analysis of the Period Recovery Error Bound
Abstract
The recovery problem is the problem whose input is a corrupted text T that was originally periodic, and where one wishes to recover its original period. The algorithm’s input is T without any information about either the period’s length or the period itself. An algorithm that solves this problem is called a recovery algorithm. In order to make recovery possible, there must be some assumption that not "too many" errors corrupted the initial periodic string. This is called the error bound. In previous recovery algorithms, it was shown that a given error bound of n/((2+ε)p) can lead to O(log_{1+ε} n) period candidates, that are guaranteed to include the original period, where p is the length of the original period (unknown by the algorithm) and ε > 0 is an arbitrary constant.
This paper provides the first analysis of the relationship between the error bound and the number of candidates, as well as identification of the error parameters that still guarantee recovery. We improve the previously known upper error bound on the number of corruptions, n/((2+ε)p), that outputs O(log_{1+ε} n) period candidates. We show how to (1) remove ε from the bound, (2) relax the error bound to allow more errors while keeping the candidates set of size O(log n). It turns out that this relaxation on the previously known upper bound is quite challenging.
To achieve this result we provide what, to our knowledge, is the first known non-trivial lower bound on the Hamming distance between two periodic strings. This proof leads to an error bound, that produces a family of period candidates of size 2log₃ n. We show that this result is tight and further provide a compact representation of the period candidates. We call this representation the canonic period seed.
In addition to providing less restrictive error bounds that guarantee a smaller candidate set, we also provide a hierarchy of more restrictive upper error bounds that asymptotically reduces the size of the potential period candidate set.
Cite as
Amihood Amir, Itai Boneh, Michael Itzhaki, and Eitan Kondratovsky. Analysis of the Period Recovery Error Bound. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{amir_et_al:LIPIcs.ESA.2020.5,
author = {Amir, Amihood and Boneh, Itai and Itzhaki, Michael and Kondratovsky, Eitan},
title = {{Analysis of the Period Recovery Error Bound}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {5:1--5:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.5},
URN = {urn:nbn:de:0030-drops-128717},
doi = {10.4230/LIPIcs.ESA.2020.5},
annote = {Keywords: Period Recovery, Period Recovery Hierarchy, Hamming Distance}
}
Document
Approximation of the Diagonal of a Laplacian’s Pseudoinverse for Complex Network Analysis
Abstract
The ubiquity of massive graph data sets in numerous applications requires fast algorithms for extracting knowledge from these data. We are motivated here by three electrical measures for the analysis of large small-world graphs G = (V, E) - i. e., graphs with diameter in O(log |V|), which are abundant in complex network analysis. From a computational point of view, the three measures have in common that their crucial component is the diagonal of the graph Laplacian’s pseudoinverse, L^+. Computing diag(L^+) exactly by pseudoinversion, however, is as expensive as dense matrix multiplication - and the standard tools in practice even require cubic time. Moreover, the pseudoinverse requires quadratic space - hardly feasible for large graphs. Resorting to approximation by, e. g., using the Johnson-Lindenstrauss transform, requires the solution of O(log |V| / ε²) Laplacian linear systems to guarantee a relative error, which is still very expensive for large inputs.
In this paper, we present a novel approximation algorithm that requires the solution of only one Laplacian linear system. The remaining parts are purely combinatorial - mainly sampling uniform spanning trees, which we relate to diag(L^+) via effective resistances. For small-world networks, our algorithm obtains a ± ε-approximation with high probability, in a time that is nearly-linear in |E| and quadratic in 1 / ε. Another positive aspect of our algorithm is its parallel nature due to independent sampling. We thus provide two parallel implementations of our algorithm: one using OpenMP, one MPI + OpenMP. In our experiments against the state of the art, our algorithm (i) yields more accurate approximation results for diag(L^+), (ii) is much faster and more memory-efficient, and (iii) obtains good parallel speedups, in particular in the distributed setting.
Cite as
Eugenio Angriman, Maria Predari, Alexander van der Grinten, and Henning Meyerhenke. Approximation of the Diagonal of a Laplacian’s Pseudoinverse for Complex Network Analysis. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{angriman_et_al:LIPIcs.ESA.2020.6,
author = {Angriman, Eugenio and Predari, Maria and van der Grinten, Alexander and Meyerhenke, Henning},
title = {{Approximation of the Diagonal of a Laplacian’s Pseudoinverse for Complex Network Analysis}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {6:1--6:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.6},
URN = {urn:nbn:de:0030-drops-128723},
doi = {10.4230/LIPIcs.ESA.2020.6},
annote = {Keywords: Laplacian pseudoinverse, electrical centrality measures, uniform spanning tree, effective resistance, parallel sampling}
}
Document
Cutting Polygons into Small Pieces with Chords: Laser-Based Localization
Abstract
Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem.
Cite as
Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, and Csaba D. Tóth. Cutting Polygons into Small Pieces with Chords: Laser-Based Localization. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{arkin_et_al:LIPIcs.ESA.2020.7,
author = {Arkin, Esther M. and Das, Rathish and Gao, Jie and Goswami, Mayank and Mitchell, Joseph S. B. and Polishchuk, Valentin and T\'{o}th, Csaba D.},
title = {{Cutting Polygons into Small Pieces with Chords: Laser-Based Localization}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {7:1--7:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.7},
URN = {urn:nbn:de:0030-drops-128736},
doi = {10.4230/LIPIcs.ESA.2020.7},
annote = {Keywords: Polygon partition, Arrangements, Visibility, Localization}
}
Document
Set Cover with Delay - Clairvoyance Is Not Required
Abstract
In most online problems with delay, clairvoyance (i.e. knowing the future delay of a request upon its arrival) is required for polylogarithmic competitiveness. In this paper, we show that this is not the case for set cover with delay (SCD) - specifically, we present the first non-clairvoyant algorithm, which is O(log n log m)-competitive, where n is the number of elements and m is the number of sets. This matches the best known result for the classic online set cover (a special case of non-clairvoyant SCD). Moreover, clairvoyance does not allow for significant improvement - we present lower bounds of Ω(√{log n}) and Ω(√{log m}) for SCD which apply for the clairvoyant case.
In addition, the competitiveness of our algorithm does not depend on the number of requests. Such a guarantee on the size of the universe alone was not previously known even for the clairvoyant case - the only previously-known algorithm (due to Carrasco et al.) is clairvoyant, with competitiveness that grows with the number of requests.
For the special case of vertex cover with delay, we show a simpler, deterministic algorithm which is 3-competitive (and also non-clairvoyant).
Cite as
Yossi Azar, Ashish Chiplunkar, Shay Kutten, and Noam Touitou. Set Cover with Delay - Clairvoyance Is Not Required. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{azar_et_al:LIPIcs.ESA.2020.8,
author = {Azar, Yossi and Chiplunkar, Ashish and Kutten, Shay and Touitou, Noam},
title = {{Set Cover with Delay - Clairvoyance Is Not Required}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {8:1--8:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.8},
URN = {urn:nbn:de:0030-drops-128749},
doi = {10.4230/LIPIcs.ESA.2020.8},
annote = {Keywords: Set Cover, Delay, Clairvoyant}
}
Document
Improved Bounds for Metric Capacitated Covering Problems
Abstract
In the Metric Capacitated Covering (MCC) problem, given a set of balls ℬ in a metric space P with metric d and a capacity parameter U, the goal is to find a minimum sized subset ℬ' ⊆ ℬ and an assignment of the points in P to the balls in ℬ' such that each point is assigned to a ball that contains it and each ball is assigned with at most U points. MCC achieves an O(log |P|)-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of o(log |P|) even with β < 3 factor expansion of the balls. Bandyapadhyay et al. [SoCG 2018, DCG 2019] showed that one can obtain an O(1)-approximation for the problem with 6.47 factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound 3 and the upper bound 6.47. In this current work, we show that it is possible to obtain an O(1)-approximation with only 4.24 factor expansion of the balls. We also show a similar upper bound of 5 for a more generalized version of MCC for which the best previously known bound was 9.
Cite as
Sayan Bandyapadhyay. Improved Bounds for Metric Capacitated Covering Problems. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bandyapadhyay:LIPIcs.ESA.2020.9,
author = {Bandyapadhyay, Sayan},
title = {{Improved Bounds for Metric Capacitated Covering Problems}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {9:1--9:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.9},
URN = {urn:nbn:de:0030-drops-128759},
doi = {10.4230/LIPIcs.ESA.2020.9},
annote = {Keywords: Capacitated covering, approximation algorithms, bicriteria approximation, LP rounding}
}
Document
Minimum Neighboring Degree Realization in Graphs and Trees
Abstract
We study a graph realization problem that pertains to degrees in vertex neighborhoods. The classical problem of degree sequence realizability asks whether or not a given sequence of n positive integers is equal to the degree sequence of some n-vertex undirected simple graph. While the realizability problem of degree sequences has been well studied for different classes of graphs, there has been relatively little work concerning the realizability of other types of information profiles, such as the vertex neighborhood profiles.
In this paper we introduce and explore the minimum degrees in vertex neighborhood profile as it is one of the most natural extensions of the classical degree profile to vertex neighboring degree profiles. Given a graph G = (V,E), the min-degree of a vertex v ∈ V, namely MinND(v), is given by min{deg(w) ∣ w ∈ N[v]}. Our input is a sequence σ = (d_𝓁^{n_𝓁}, ⋯ , d₁^{n₁}), where d_{i+1} > d_i and each n_i is a positive integer. We provide some necessary and sufficient conditions for σ to be realizable. Furthermore, under the restriction that the realization is acyclic, i.e., a tree or a forest, we provide a full characterization of realizable sequences, along with a corresponding constructive algorithm.
We believe our results are a crucial step towards understanding extremal neighborhood degree relations in graphs.
Cite as
Amotz Bar-Noy, Keerti Choudhary, Avi Cohen, David Peleg, and Dror Rawitz. Minimum Neighboring Degree Realization in Graphs and Trees. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{barnoy_et_al:LIPIcs.ESA.2020.10,
author = {Bar-Noy, Amotz and Choudhary, Keerti and Cohen, Avi and Peleg, David and Rawitz, Dror},
title = {{Minimum Neighboring Degree Realization in Graphs and Trees}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {10:1--10:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.10},
URN = {urn:nbn:de:0030-drops-128765},
doi = {10.4230/LIPIcs.ESA.2020.10},
annote = {Keywords: Graph realization, neighborhood profile, graph algorithms, degree sequences}
}
Document
Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations
Abstract
We develop polynomial-time algorithms for the fair and efficient allocation of indivisible goods among n agents that have subadditive valuations over the goods. We first consider the Nash social welfare as our objective and design a polynomial-time algorithm that, in the value oracle model, finds an 8n-approximation to the Nash optimal allocation. Subadditive valuations include XOS (fractionally subadditive) and submodular valuations as special cases. Our result, even for the special case of submodular valuations, improves upon the previously best known O(n log n)-approximation ratio of Garg et al. (2020).
More generally, we study maximization of p-mean welfare. The p-mean welfare is parameterized by an exponent term p ∈ (-∞, 1] and encompasses a range of welfare functions, such as social welfare (p = 1), Nash social welfare (p → 0), and egalitarian welfare (p → -∞). We give an algorithm that, for subadditive valuations and any given p ∈ (-∞, 1], computes (in the value oracle model and in polynomial time) an allocation with p-mean welfare at least 1/(8n) times the optimal.
Further, we show that our approximation guarantees are essentially tight for XOS and, hence, subadditive valuations. We adapt a result of Dobzinski et al. (2010) to show that, under XOS valuations, an O (n^{1-ε}) approximation for the p-mean welfare for any p ∈ (-∞,1] (including the Nash social welfare) requires exponentially many value queries; here, ε > 0 is any fixed constant.
Cite as
Siddharth Barman, Umang Bhaskar, Anand Krishna, and Ranjani G. Sundaram. Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{barman_et_al:LIPIcs.ESA.2020.11,
author = {Barman, Siddharth and Bhaskar, Umang and Krishna, Anand and Sundaram, Ranjani G.},
title = {{Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {11:1--11:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.11},
URN = {urn:nbn:de:0030-drops-128775},
doi = {10.4230/LIPIcs.ESA.2020.11},
annote = {Keywords: Discrete Fair Division, Nash Social Welfare, Subadditive Valuations, Submodular Valuations}
}
Document
Mincut Sensitivity Data Structures for the Insertion of an Edge
Abstract
Let G = (V,E) be an undirected graph on n vertices with non-negative capacities on its edges. The mincut sensitivity problem for the insertion of an edge is defined as follows.
Build a compact data structure for G and a given set S ⊆ V of vertices that, on receiving any edge (x,y) ∈ S×S of positive capacity as query input, can efficiently report the set of all pairs from S× S whose mincut value increases upon insertion of the edge (x,y) to G.
The only result that exists for this problem is for a single pair of vertices (Picard and Queyranne, Mathematical Programming Study, 13 (1980), 8-16). We present the following results for the single source and the all-pairs versions of this problem.
1) Single source: Given any designated source vertex s, there exists a data structure of size 𝒪(|S|) that can output all those vertices from S whose mincut value to s increases upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(|S|).
2) All-pairs: There exists an 𝒪(|S|²) size data structure that can output all those pairs of vertices from S× S whose mincut value gets increased upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(k), where k is the number of pairs of vertices whose mincut increases.
For both these versions, we also address the problem of reporting the values of the mincuts upon insertion of any given edge. To derive our results, we use interesting insights into the nearest and the farthest mincuts for a pair of vertices. In addition, a crucial result, that we establish and use in our data structures, is that there exists a directed acyclic graph of 𝒪(n) size that compactly stores the farthest mincuts from all vertices of V to a designated vertex s in the graph. We believe that this result is of independent interest, especially, because it also complements a previously existing result by Hariharan et al. (STOC 2007) that the nearest mincuts from all vertices of V to s is a laminar family, and hence, can be stored compactly in a tree of 𝒪(n) size.
Cite as
Surender Baswana, Shiv Gupta, and Till Knollmann. Mincut Sensitivity Data Structures for the Insertion of an Edge. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{baswana_et_al:LIPIcs.ESA.2020.12,
author = {Baswana, Surender and Gupta, Shiv and Knollmann, Till},
title = {{Mincut Sensitivity Data Structures for the Insertion of an Edge}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {12:1--12:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.12},
URN = {urn:nbn:de:0030-drops-128781},
doi = {10.4230/LIPIcs.ESA.2020.12},
annote = {Keywords: Mincut, Sensitivity, Data Structure}
}
Document
Linear Time LexDFS on Chordal Graphs
Abstract
Lexicographic Depth First Search (LexDFS) is a special variant of a Depth First Search (DFS), which was introduced by Corneil and Krueger in 2008. While this search has been used in various applications, in contrast to other graph searches, no general linear time implementation is known to date. In 2014, Köhler and Mouatadid achieved linear running time to compute some special LexDFS orderings for cocomparability graphs. In this paper, we present a linear time implementation of LexDFS for chordal graphs. Our algorithm even implements the extended version LexDFS^+ and is, therefore, able to find any LexDFS ordering for this graph class. To the best of our knowledge this is the first unrestricted linear time implementation of LexDFS on a non-trivial graph class. In the algorithm we use a search tree computed by Lexicographic Breadth First Search (LexBFS).
Cite as
Jesse Beisegel, Ekkehard Köhler, Robert Scheffler, and Martin Strehler. Linear Time LexDFS on Chordal Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{beisegel_et_al:LIPIcs.ESA.2020.13,
author = {Beisegel, Jesse and K\"{o}hler, Ekkehard and Scheffler, Robert and Strehler, Martin},
title = {{Linear Time LexDFS on Chordal Graphs}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {13:1--13:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.13},
URN = {urn:nbn:de:0030-drops-128790},
doi = {10.4230/LIPIcs.ESA.2020.13},
annote = {Keywords: LexDFS, chordal graphs, linear time implementation, search trees, LexBFS}
}
Document
Grundy Distinguishes Treewidth from Pathwidth
Abstract
Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable to intractable? This type of question is by now very well-studied, but, somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes "for free".
Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy First-Fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width.
Cite as
Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Yota Otachi. Grundy Distinguishes Treewidth from Pathwidth. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{belmonte_et_al:LIPIcs.ESA.2020.14,
author = {Belmonte, R\'{e}my and Kim, Eun Jung and Lampis, Michael and Mitsou, Valia and Otachi, Yota},
title = {{Grundy Distinguishes Treewidth from Pathwidth}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {14:1--14:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.14},
URN = {urn:nbn:de:0030-drops-128803},
doi = {10.4230/LIPIcs.ESA.2020.14},
annote = {Keywords: Treewidth, Pathwidth, Clique-width, Grundy Coloring}
}
Document
On the Complexity of BWT-Runs Minimization via Alphabet Reordering
Abstract
The Burrows-Wheeler Transform (BWT) has been an essential tool in text compression and indexing. First introduced in 1994, it went on to provide the backbone for the first encoding of the classic suffix tree data structure in space close to entropy-based lower bound. Within the last decade, it has seen its role further enhanced with the development of compact suffix trees in space proportional to "r", the number of runs in the BWT. While r would superficially appear to be only a measure of space complexity, it is actually appearing increasingly often in the time complexity of new algorithms as well. This makes having the smallest value of r of growing importance. Interestingly, unlike other popular measures of compression, the parameter r is sensitive to the lexicographic ordering given to the text’s alphabet. Despite several past attempts to exploit this fact, a provably efficient algorithm for finding, or approximating, an alphabet ordering which minimizes r has been open for years.
We help to explain this lack of progress by presenting the first set of results on the computational complexity of minimizing BWT-runs via alphabet reordering. We prove that the decision version of this problem is NP-complete and cannot be solved in time poly(n)⋅ 2^o(σ) unless the Exponential Time Hypothesis fails, where σ is the size of the alphabet and n is the length of the text. Moreover, we show that the optimization variant is APX-hard. In doing so, we relate two previously disparate topics: the optimal traveling salesperson path of a graph and the number of runs in the BWT of a text. In addition, by relating recent results in the field of dictionary compression, we illustrate that an arbitrary alphabet ordering provides an O(log² n)-approximation. Lastly, we provide an optimal linear-time algorithm for a more restricted problem of finding an optimal ordering on a subset of symbols (occurring only once) under ordering constraints.
Cite as
Jason W. Bentley, Daniel Gibney, and Sharma V. Thankachan. On the Complexity of BWT-Runs Minimization via Alphabet Reordering. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bentley_et_al:LIPIcs.ESA.2020.15,
author = {Bentley, Jason W. and Gibney, Daniel and Thankachan, Sharma V.},
title = {{On the Complexity of BWT-Runs Minimization via Alphabet Reordering}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {15:1--15:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.15},
URN = {urn:nbn:de:0030-drops-128819},
doi = {10.4230/LIPIcs.ESA.2020.15},
annote = {Keywords: BWT, NP-hardness, APX-hardness}
}
Document
Simulating Population Protocols in Sub-Constant Time per Interaction
Abstract
We consider the efficient simulation of population protocols. In the population model, we are given a system of n agents modeled as identical finite-state machines. In each step, two agents are selected uniformly at random to interact by updating their states according to a common transition function. We empirically and analytically analyze two classes of simulators for this model. First, we consider sequential simulators executing one interaction after the other. Key to the performance of these simulators is the data structure storing the agents' states. For our analysis, we consider plain arrays, binary search trees, and a novel Dynamic Alias Table data structure. Secondly, we consider batch processing to efficiently update the states of multiple independent agents in one step. For many protocols considered in literature, our simulator requires amortized sub-constant time per interaction and is fast in practice: given a fixed time budget, the implementation of our batched simulator is able to simulate population protocols several orders of magnitude larger compared to the sequential competitors, and can carry out 2^50 interactions among the same number of agents in less than 400s.
Cite as
Petra Berenbrink, David Hammer, Dominik Kaaser, Ulrich Meyer, Manuel Penschuck, and Hung Tran. Simulating Population Protocols in Sub-Constant Time per Interaction. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 16:1-16:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{berenbrink_et_al:LIPIcs.ESA.2020.16,
author = {Berenbrink, Petra and Hammer, David and Kaaser, Dominik and Meyer, Ulrich and Penschuck, Manuel and Tran, Hung},
title = {{Simulating Population Protocols in Sub-Constant Time per Interaction}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {16:1--16:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.16},
URN = {urn:nbn:de:0030-drops-128827},
doi = {10.4230/LIPIcs.ESA.2020.16},
annote = {Keywords: Population Protocols, Simulation, Random Sampling, Dynamic Alias Table}
}
Document
An Optimal Decentralized (Δ + 1)-Coloring Algorithm
Abstract
Consider the following simple coloring algorithm for a graph on n vertices. Each vertex chooses a color from {1, ..., Δ(G) + 1} uniformly at random. While there exists a conflicted vertex choose one such vertex uniformly at random and recolor it with a randomly chosen color. This algorithm was introduced by Bhartia et al. [MOBIHOC'16] for channel selection in WIFI-networks. We show that this algorithm always converges to a proper coloring in expected O(n log Δ) steps, which is optimal and proves a conjecture of Chakrabarty and de Supinski [SOSA'20].
Cite as
Daniel Bertschinger, Johannes Lengler, Anders Martinsson, Robert Meier, Angelika Steger, Miloš Trujić, and Emo Welzl. An Optimal Decentralized (Δ + 1)-Coloring Algorithm. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 17:1-17:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bertschinger_et_al:LIPIcs.ESA.2020.17,
author = {Bertschinger, Daniel and Lengler, Johannes and Martinsson, Anders and Meier, Robert and Steger, Angelika and Truji\'{c}, Milo\v{s} and Welzl, Emo},
title = {{An Optimal Decentralized (\Delta + 1)-Coloring Algorithm}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {17:1--17:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.17},
URN = {urn:nbn:de:0030-drops-128837},
doi = {10.4230/LIPIcs.ESA.2020.17},
annote = {Keywords: Decentralized Algorithm, Distributed Computing, Graph Coloring, Randomized Algorithms}
}
Document
Noisy, Greedy and Not so Greedy k-Means++
Abstract
The k-means++ algorithm due to Arthur and Vassilvitskii [David Arthur and Sergei Vassilvitskii, 2007] has become the most popular seeding method for Lloyd’s algorithm. It samples the first center uniformly at random from the data set and the other k-1 centers iteratively according to D²-sampling, i.e., the probability that a data point becomes the next center is proportional to its squared distance to the closest center chosen so far. k-means++ is known to achieve an approximation factor of 𝒪(log k) in expectation.
Already in the original paper on k-means++, Arthur and Vassilvitskii suggested a variation called greedy k-means++ algorithm in which in each iteration multiple possible centers are sampled according to D²-sampling and only the one that decreases the objective the most is chosen as a center for that iteration. It is stated as an open question whether this also leads to an 𝒪(log k)-approximation (or even better). We show that this is not the case by presenting a family of instances on which greedy k-means++ yields only an Ω(𝓁⋅log k)-approximation in expectation where 𝓁 is the number of possible centers that are sampled in each iteration.
Inspired by the negative results, we study a variation of greedy k-means++ which we call noisy k-means++ algorithm. In this variation only one center is sampled in every iteration but not exactly by D²-sampling. Instead in each iteration an adversary is allowed to change the probabilities arising from D²-sampling individually for each point by a factor between 1-ε₁ and 1+ε₂ for parameters ε₁ ∈ [0,1) and ε₂ ≥ 0. We prove that noisy k-means++ computes an 𝒪(log² k)-approximation in expectation. We use the analysis of noisy k-means++ to design a moderately greedy k-means++ algorithm.
Cite as
Anup Bhattacharya, Jan Eube, Heiko Röglin, and Melanie Schmidt. Noisy, Greedy and Not so Greedy k-Means++. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bhattacharya_et_al:LIPIcs.ESA.2020.18,
author = {Bhattacharya, Anup and Eube, Jan and R\"{o}glin, Heiko and Schmidt, Melanie},
title = {{Noisy, Greedy and Not so Greedy k-Means++}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {18:1--18:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.18},
URN = {urn:nbn:de:0030-drops-128848},
doi = {10.4230/LIPIcs.ESA.2020.18},
annote = {Keywords: k-means++, greedy, adaptive sampling}
}
Document
An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling
Abstract
Map labeling is a classical problem in cartography and geographic information systems (GIS) that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent, i.e., overlap-free, labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each time step, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sub-linear update time. Motivated by this, we study the maximal independent set (MIS) and maximum independent set (Max-IS) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types - (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs.
We present the first deterministic algorithm for maintaining a MIS (and thus a 4-approximate Max-IS) of a dynamic set of uniform rectangles with amortized sub-logarithmic update time. This breaks the natural barrier of Ω(Δ) update time (where Δ is the maximum degree in the graph) for vertex updates presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes. For uniform rectangles, we first give an algorithm that maintains a 4-approximate Max-IS with O(1) update time. In a subsequent algorithm, we establish the trade-off between approximation quality 2(1+1/k) and update time O(k²log n), for k ∈ ℕ. We conclude with an algorithm that maintains a 2-approximate Max-IS for dynamic sets of unit-height and arbitrary-width rectangles with O(ω log n) update time, where ω is the maximum size of an independent set of rectangles stabbed by any horizontal line. We have implemented our algorithms and report the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances.
Cite as
Sujoy Bhore, Guangping Li, and Martin Nöllenburg. An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bhore_et_al:LIPIcs.ESA.2020.19,
author = {Bhore, Sujoy and Li, Guangping and N\"{o}llenburg, Martin},
title = {{An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {19:1--19:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.19},
URN = {urn:nbn:de:0030-drops-128856},
doi = {10.4230/LIPIcs.ESA.2020.19},
annote = {Keywords: Independent Sets, Dynamic Algorithms, Rectangle Intersection Graphs, Approximation Algorithms, Experimental Evaluation}
}
Document
Lower Bounds and Approximation Algorithms for Search Space Sizes in Contraction Hierarchies
Abstract
Contraction hierarchies (CH) is a prominent preprocessing-based technique that accelerates the computation of shortest paths in road networks by reducing the search space size of a bidirectional Dijkstra run. To explain the practical success of CH, several theoretical upper bounds for the maximum search space size were derived in previous work. For example, it was shown that in minor-closed graph families search space sizes in 𝒪(√n) can be achieved (with n denoting the number of nodes in the graph), and search space sizes in 𝒪(h log D) in graphs of highway dimension h and diameter D. In this paper, we primarily focus on lower bounds. We prove that the average search space size in a so called weak CH is in Ω(b_α) for α ≥ 2/3 where b_α is the size of a smallest α-balanced node separator. This discovery allows us to describe the first approximation algorithm for the average search space size. Our new lower bound also shows that the 𝒪(√n) bound for minor-closed graph families is tight. Furthermore, we deeper investigate the relationship of CH and the highway dimension and skeleton dimension of the graph, and prove new lower bound and incomparability results. Finally, we discuss how lower bounds for strong CH can be obtained from solving a HittingSet problem defined on a set of carefully chosen subgraphs of the input network.
Cite as
Johannes Blum and Sabine Storandt. Lower Bounds and Approximation Algorithms for Search Space Sizes in Contraction Hierarchies. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{blum_et_al:LIPIcs.ESA.2020.20,
author = {Blum, Johannes and Storandt, Sabine},
title = {{Lower Bounds and Approximation Algorithms for Search Space Sizes in Contraction Hierarchies}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {20:1--20:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.20},
URN = {urn:nbn:de:0030-drops-128861},
doi = {10.4230/LIPIcs.ESA.2020.20},
annote = {Keywords: contraction hierarchies, search space size, balanced separator, tree decomposition}
}
Document
The Minimization of Random Hypergraphs
Abstract
We investigate the maximum-entropy model B_{n,m,p} for random n-vertex, m-edge multi-hypergraphs with expected edge size pn. We show that the expected size of the minimization min(B_{n,m,p}), i.e., the number of inclusion-wise minimal edges of B_{n,m,p}, undergoes a phase transition with respect to m. If m is at most 1/(1-p)^{(1-p)n}, then E[|min(B_{n,m,p})|] is of order Θ(m), while for m ≥ 1/(1-p)^{(1-p+ε)n} for any ε > 0, it is Θ(2^{(H(α) + (1-α) log₂ p) n}/√n). Here, H denotes the binary entropy function and α = - (log_{1-p} m)/n. The result implies that the maximum expected number of minimal edges over all m is Θ((1+p)ⁿ/√n). Our structural findings have algorithmic implications for minimizing an input hypergraph. This has applications in the profiling of relational databases as well as for the Orthogonal Vectors problem studied in fine-grained complexity. We make several technical contributions that are of independent interest in probability. First, we improve the Chernoff-Hoeffding theorem on the tail of the binomial distribution. In detail, we show that for a binomial variable Y ∼ Bin(n,p) and any 0 < x < p, it holds that P[Y ≤ xn] = Θ(2^{-D(x‖p) n}/√n), where D is the binary Kullback-Leibler divergence between Bernoulli distributions. We give explicit upper and lower bounds on the constants hidden in the big-O notation that hold for all n. Secondly, we establish the fact that the probability of a set of cardinality i being minimal after m i.i.d. maximum-entropy trials exhibits a sharp threshold behavior at i^* = n + log_{1-p} m.
Cite as
Thomas Bläsius, Tobias Friedrich, and Martin Schirneck. The Minimization of Random Hypergraphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{blasius_et_al:LIPIcs.ESA.2020.21,
author = {Bl\"{a}sius, Thomas and Friedrich, Tobias and Schirneck, Martin},
title = {{The Minimization of Random Hypergraphs}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {21:1--21:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.21},
URN = {urn:nbn:de:0030-drops-128871},
doi = {10.4230/LIPIcs.ESA.2020.21},
annote = {Keywords: Chernoff-Hoeffding theorem, maximum entropy, maximization, minimization, phase transition, random hypergraphs}
}