5 Search Results for "Fagerberg, Rolf"

Document
Invited Talk
DOI: 10.4230/LIPIcs.STACS.2023.2
Amortised Analysis of Dynamic Data Structures (Invited Talk)

Authors: Eva Rotenberg

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
In dynamic data structures, one is interested in efficiently facilitating queries to a data set, while being able to efficiently perform updates as the data set undergoes changes. Often, relaxing the efficiency measure to the amortised setting allows for simpler algorithms. A well-known example of a data structure with amortised guarantees is the splay tree by Sleator and Tarjan [Daniel D. Sleator and Robert E. Tarjan, 1985]. Similarly, in data structures for dynamic graphs, one is interested in efficiently maintaining some information about the graph, or facilitating queries, as the graph undergoes changes in the form of insertion and deletion of edges. Examples of such information include connectivity, planarity, and approximate sparsity of the graph: is the graph presently connected? Is it planar? Has its arboricity grossly exceeded some specified number α̃? The related queries could be: is a connected to b? Are the edges uv and uw consecutive in the ordering around u in its current planar embedding? Or, report the O(α) out-edges of vertex x. In this talk, we will see Brodal and Fagerberg’s amortised algorithm for orienting sparse graphs (i.e. of arboricity ≤ α), so that each vertex has O(α) out-edges [Gerth Stølting Brodal and Rolf Fagerberg, 1999]. The algorithm itself is extremely simple, and uses an elegant amortised argument in its analysis. Then, we will visit the problem of dynamic planarity testing: is the graph presently planar? Here, we will see an elegant amortised reduction to the seemingly easier problem, where planarity-violating edges may be detected and rejected [Eppstein et al., 1996]. We will see a sketch of how the current state-of-the-art algorithm for efficient planarity testing [Jacob Holm and Eva Rotenberg, 2020] uses ideas similar to those in [Gerth Stølting Brodal and Rolf Fagerberg, 1999] to analyse the behaviour of a greedy algorithm via a possibly inefficient algorithm with provably low recourse [Jacob Holm and Eva Rotenberg, 2020]. If time permits, we will touch upon a recent simple amortised data structure for maintaining information in dynamic forests [Jacob Holm et al., 2023], which builds on ideas from splay trees. The talk concludes with some open questions in the area.
Cite as

Eva Rotenberg. Amortised Analysis of Dynamic Data Structures (Invited Talk). In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{rotenberg:LIPIcs.STACS.2023.2,
  author =	{Rotenberg, Eva},
  title =	{{Amortised Analysis of Dynamic Data Structures}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{2:1--2:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.2},
  URN =		{urn:nbn:de:0030-drops-176547},
  doi =		{10.4230/LIPIcs.STACS.2023.2},
  annote =	{Keywords: Amortised analysis, splaying, dynamic graphs, planarity testing}
}
Document
DOI: 10.4230/LIPIcs.FUN.2022.8
Priority Queues with Decreasing Keys

Authors: Gerth Stølting Brodal

Published in: LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

Abstract
A priority queue stores a set of items with associated keys and supports the insertion of a new item and extraction of an item with minimum key. In applications like Dijkstra’s single source shortest path algorithm and Prim-Jarník’s minimum spanning tree algorithm, the key of an item can decrease over time. Usually this is handled by either using a priority queue supporting the deletion of an arbitrary item or a dedicated DecreaseKey operation, or by inserting the same item multiple times but with decreasing keys. In this paper we study what happens if the keys associated with items in a priority queue can decrease over time without informing the priority queue, and how such a priority queue can be used in Dijkstra’s algorithm. We show that binary heaps with bottom-up insertions fail to report items with unchanged keys in correct order, while binary heaps with top-down insertions report items with unchanged keys in correct order. Furthermore, we show that skew heaps, leftist heaps, and priority queues based on linking roots of heap-ordered trees, like pairing heaps, binomial queues and Fibonacci heaps, work correctly with decreasing keys without any modifications. Finally, we show that the post-order heap by Harvey and Zatloukal, a variant of a binary heap with amortized constant time insertions and amortized logarithmic time deletions, works correctly with decreasing keys and is a strong contender for an implicit priority queue supporting decreasing keys in practice.
Cite as

Gerth Stølting Brodal. Priority Queues with Decreasing Keys. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brodal:LIPIcs.FUN.2022.8,
  author =	{Brodal, Gerth St{\o}lting},
  title =	{{Priority Queues with Decreasing Keys}},
  booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
  pages =	{8:1--8:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-232-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{226},
  editor =	{Fraigniaud, Pierre and Uno, Yushi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.8},
  URN =		{urn:nbn:de:0030-drops-159787},
  doi =		{10.4230/LIPIcs.FUN.2022.8},
  annote =	{Keywords: priority queue, decreasing keys, post-order heap, Dijkstra’s algorithm}
}
Document
DOI: 10.4230/LIPIcs.SEA.2021.23
An Experimental Study of External Memory Algorithms for Connected Components

Authors: Gerth Stølting Brodal, Rolf Fagerberg, David Hammer, Ulrich Meyer, Manuel Penschuck, and Hung Tran

Published in: LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

Abstract
We empirically investigate algorithms for solving Connected Components in the external memory model. In particular, we study whether the randomized O(Sort(E)) algorithm by Karger, Klein, and Tarjan can be implemented to compete with practically promising and simpler algorithms having only slightly worse theoretical cost, namely Borůvka’s algorithm and the algorithm by Sibeyn and collaborators. For all algorithms, we develop and test a number of tuning options. Our experiments are executed on a large set of different graph classes including random graphs, grids, geometric graphs, and hyperbolic graphs. Among our findings are: The Sibeyn algorithm is a very strong contender due to its simplicity and due to an added degree of freedom in its internal workings when used in the Connected Components setting. With the right tunings, the Karger-Klein-Tarjan algorithm can be implemented to be competitive in many cases. Higher graph density seems to benefit Karger-Klein-Tarjan relative to Sibeyn. Borůvka’s algorithm is not competitive with the two others.
Cite as

Gerth Stølting Brodal, Rolf Fagerberg, David Hammer, Ulrich Meyer, Manuel Penschuck, and Hung Tran. An Experimental Study of External Memory Algorithms for Connected Components. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 23:1-23:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brodal_et_al:LIPIcs.SEA.2021.23,
  author =	{Brodal, Gerth St{\o}lting and Fagerberg, Rolf and Hammer, David and Meyer, Ulrich and Penschuck, Manuel and Tran, Hung},
  title =	{{An Experimental Study of External Memory Algorithms for Connected Components}},
  booktitle =	{19th International Symposium on Experimental Algorithms (SEA 2021)},
  pages =	{23:1--23:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-185-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{190},
  editor =	{Coudert, David and Natale, Emanuele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.23},
  URN =		{urn:nbn:de:0030-drops-137958},
  doi =		{10.4230/LIPIcs.SEA.2021.23},
  annote =	{Keywords: Connected Components, Experimental Evaluation, External Memory, Graph Algorithms, Randomization}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2019.35
On Optimal Balance in B-Trees: What Does It Cost to Stay in Perfect Shape?

Authors: Rolf Fagerberg, David Hammer, and Ulrich Meyer

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract
Any B-tree has height at least ceil[log_B(n)]. Static B-trees achieving this height are easy to build. In the dynamic case, however, standard B-tree rebalancing algorithms only maintain a height within a constant factor of this optimum. We investigate exactly how close to ceil[log_B(n)] the height of dynamic B-trees can be maintained as a function of the rebalancing cost. In this paper, we prove a lower bound on the cost of maintaining optimal height ceil[log_B(n)], which shows that this cost must increase from Omega(1/B) to Omega(n/B) rebalancing per update as n grows from one power of B to the next. We also provide an almost matching upper bound, demonstrating this lower bound to be essentially tight. We then give a variant upper bound which can maintain near-optimal height at low cost. As two special cases, we can maintain optimal height for all but a vanishing fraction of values of n using Theta(log_B(n)) amortized rebalancing cost per update and we can maintain a height of optimal plus one using O(1/B) amortized rebalancing cost per update. More generally, for any rebalancing budget, we can maintain (as n grows from one power of B to the next) optimal height essentially up to the point where the lower bound requires the budget to be exceeded, after which optimal height plus one is maintained. Finally, we prove that this balancing scheme gives B-trees with very good storage utilization.
Cite as

Rolf Fagerberg, David Hammer, and Ulrich Meyer. On Optimal Balance in B-Trees: What Does It Cost to Stay in Perfect Shape?. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fagerberg_et_al:LIPIcs.ISAAC.2019.35,
  author =	{Fagerberg, Rolf and Hammer, David and Meyer, Ulrich},
  title =	{{On Optimal Balance in B-Trees: What Does It Cost to Stay in Perfect Shape?}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{35:1--35:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.35},
  URN =		{urn:nbn:de:0030-drops-115313},
  doi =		{10.4230/LIPIcs.ISAAC.2019.35},
  annote =	{Keywords: B-trees, Data structures, Lower bounds, Complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.2
Fragile Complexity of Comparison-Based Algorithms

Authors: Peyman Afshani, Rolf Fagerberg, David Hammer, Riko Jacob, Irina Kostitsyna, Ulrich Meyer, Manuel Penschuck, and Nodari Sitchinava

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes part in. We give a number of upper and lower bounds on the fragile complexity for fundamental problems, including Minimum, Selection, Sorting and Heap Construction. The results include both deterministic and randomized upper and lower bounds, and demonstrate a separation between the two settings for a number of problems. The depth of a comparator network is a straight-forward upper bound on the worst case fragile complexity of the corresponding fragile algorithm. We prove that fragile complexity is a different and strictly easier property than the depth of comparator networks, in the sense that for some problems a fragile complexity equal to the best network depth can be achieved with less total work and that with randomization, even a lower fragile complexity is possible.
Cite as

Peyman Afshani, Rolf Fagerberg, David Hammer, Riko Jacob, Irina Kostitsyna, Ulrich Meyer, Manuel Penschuck, and Nodari Sitchinava. Fragile Complexity of Comparison-Based Algorithms. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{afshani_et_al:LIPIcs.ESA.2019.2,
  author =	{Afshani, Peyman and Fagerberg, Rolf and Hammer, David and Jacob, Riko and Kostitsyna, Irina and Meyer, Ulrich and Penschuck, Manuel and Sitchinava, Nodari},
  title =	{{Fragile Complexity of Comparison-Based Algorithms}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{2:1--2:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.2},
  URN =		{urn:nbn:de:0030-drops-111235},
  doi =		{10.4230/LIPIcs.ESA.2019.2},
  annote =	{Keywords: Algorithms, comparison based algorithms, lower bounds}
}
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