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DOI: 10.4230/LIPIcs.ESA.2019.50

Dynamic Dominators and Low-High Orders in DAGs

Authors: Loukas Georgiadis, Konstantinos Giannis, Giuseppe F. Italiano, Aikaterini Karanasiou, and Luigi Laura

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

Abstract

We consider practical algorithms for maintaining the dominator tree and a low-high order in directed acyclic graphs (DAGs) subject to dynamic operations. Let G be a directed graph with a distinguished start vertex s. The dominator tree D of G is a tree rooted at s, such that a vertex v is an ancestor of a vertex w if and only if all paths from s to w in G include v. The dominator tree is a central tool in program optimization and code generation, and has many applications in other diverse areas including constraint programming, circuit testing, biology, and in algorithms for graph connectivity problems. A low-high order of G is a preorder of D that certifies the correctness of D, and has further applications in connectivity and path-determination problems. We first provide a practical and carefully engineered version of a recent algorithm [ICALP 2017] for maintaining the dominator tree of a DAG through a sequence of edge deletions. The algorithm runs in O(mn) total time and O(m) space, where n is the number of vertices and m is the number of edges before any deletion. In addition, we present a new algorithm that maintains a low-high order of a DAG under edge deletions within the same bounds. Both results extend to the case of reducible graphs (a class that includes DAGs). Furthermore, we present a fully dynamic algorithm for maintaining the dominator tree of a DAG under an intermixed sequence of edge insertions and deletions. Although it does not maintain the O(mn) worst-case bound of the decremental algorithm, our experiments highlight that the fully dynamic algorithm performs very well in practice. Finally, we study the practical efficiency of all our algorithms by conducting an extensive experimental study on real-world and synthetic graphs.

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Loukas Georgiadis, Konstantinos Giannis, Giuseppe F. Italiano, Aikaterini Karanasiou, and Luigi Laura. Dynamic Dominators and Low-High Orders in DAGs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{georgiadis_et_al:LIPIcs.ESA.2019.50,
  author =	{Georgiadis, Loukas and Giannis, Konstantinos and Italiano, Giuseppe F. and Karanasiou, Aikaterini and Laura, Luigi},
  title =	{{Dynamic Dominators and Low-High Orders in DAGs}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{50:1--50:18},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.50},
  URN =		{urn:nbn:de:0030-drops-111715},
  doi =		{10.4230/LIPIcs.ESA.2019.50},
  annote =	{Keywords: Connectivity, dominators, low-high orders}
}

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DOI: 10.4230/LIPIcs.SEA.2017.9

Approximating the Smallest 2-Vertex-Connected Spanning Subgraph via Low-High Orders

Authors: Loukas Georgiadis, Giuseppe F. Italiano, and Aikaterini Karanasiou

Published in: LIPIcs, Volume 75, 16th International Symposium on Experimental Algorithms (SEA 2017)

Abstract

Let G = (V, E) be a 2-vertex-connected directed graph with m edges and n vertices. We consider the problem of approximating the smallest 2-vertex connected spanning subgraph (2VCSS) of G, and provide new efficient algorithms for this problem based on a clever use of low-high orders. The best previously known algorithms were able to compute a 3/2-approximation in O(m n+n 2) time, or a 3-approximation faster in linear time. In this paper, we present a linear-time algorithm that achieves a better approximation ratio of 2, and another algorithm that matches the previous 3/2-approximation in O(m n + n 2 ) time. We conducted a thorough experimental evaluation of all the above algorithms on a variety of input graphs. The experimental results show that both our two new algorithms perform well in practice. In particular, in our experiments the new 3/2-approximation algorithm was always faster than the previous 3/2-approximation algorithm, while their two approximation ratios were close. On the other side, our new linear-time algorithm yielded consistently better approximation ratios than the previously known linear-time algorithm, at the price of a small overhead in the running time.

Cite as

Loukas Georgiadis, Giuseppe F. Italiano, and Aikaterini Karanasiou. Approximating the Smallest 2-Vertex-Connected Spanning Subgraph via Low-High Orders. In 16th International Symposium on Experimental Algorithms (SEA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 75, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{georgiadis_et_al:LIPIcs.SEA.2017.9,
  author =	{Georgiadis, Loukas and Italiano, Giuseppe F. and Karanasiou, Aikaterini},
  title =	{{Approximating the Smallest 2-Vertex-Connected Spanning Subgraph via Low-High Orders}},
  booktitle =	{16th International Symposium on Experimental Algorithms (SEA 2017)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-036-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{75},
  editor =	{Iliopoulos, Costas S. and Pissis, Solon P. and Puglisi, Simon J. and Raman, Rajeev},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2017.9},
  URN =		{urn:nbn:de:0030-drops-76299},
  doi =		{10.4230/LIPIcs.SEA.2017.9},
  annote =	{Keywords: 2-vertex connectivity, approximation algorithms, directed graphs}
}

Document

DOI: 10.4230/LIPIcs.SEA.2017.27

Incremental Low-High Orders of Directed Graphs and Applications

Authors: Loukas Georgiadis, Konstantinos Giannis, Aikaterini Karanasiou, and Luigi Laura

Published in: LIPIcs, Volume 75, 16th International Symposium on Experimental Algorithms (SEA 2017)

Abstract

A flow graph G = (V, E, s) is a directed graph with a distinguished start vertex s. The dominator tree D of G is a tree rooted at s, such that a vertex v is an ancestor of a vertex w if and only if all paths from s to w include v. The dominator tree is a central tool in program optimization and code generation, and has many applications in other diverse areas including constraint programming, circuit testing, biology, and in algorithms for graph connectivity problems. A low-high order of G is a preorder d of D that certifies the correctness of D, and has further applications in connectivity and path-determination problems. In this paper we consider how to maintain efficiently a low-high order of a flow graph incrementally under edge insertions. We present algorithms that run in O(mn) total time for a sequence of edge insertions in a flow graph with n vertices, where m is the total number of edges after all insertions. These immediately provide the first incremental certifying algorithms for maintaining the dominator tree in O(mn) total time, and also imply incremental algorithms for other problems. Hence, we provide a substantial improvement over the O(m^2) straightforward algorithms, which recompute the solution from scratch after each edge insertion. Furthermore, we provide efficient implementations of our algorithms and conduct an extensive experimental study on real-world graphs taken from a variety of application areas. The experimental results show that our algorithms perform very well in practice.

Cite as

Loukas Georgiadis, Konstantinos Giannis, Aikaterini Karanasiou, and Luigi Laura. Incremental Low-High Orders of Directed Graphs and Applications. In 16th International Symposium on Experimental Algorithms (SEA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 75, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{georgiadis_et_al:LIPIcs.SEA.2017.27,
  author =	{Georgiadis, Loukas and Giannis, Konstantinos and Karanasiou, Aikaterini and Laura, Luigi},
  title =	{{Incremental Low-High Orders of Directed Graphs and Applications}},
  booktitle =	{16th International Symposium on Experimental Algorithms (SEA 2017)},
  pages =	{27:1--27:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-036-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{75},
  editor =	{Iliopoulos, Costas S. and Pissis, Solon P. and Puglisi, Simon J. and Raman, Rajeev},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2017.27},
  URN =		{urn:nbn:de:0030-drops-76319},
  doi =		{10.4230/LIPIcs.SEA.2017.27},
  annote =	{Keywords: connectivity, directed graphs, dominators, dynamic algorithms}
}

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Documents authored by Karanasiou, Aikaterini

Dynamic Dominators and Low-High Orders in DAGs

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Approximating the Smallest 2-Vertex-Connected Spanning Subgraph via Low-High Orders

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Incremental Low-High Orders of Directed Graphs and Applications

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