Dynamic Dominators and Low-High Orders in DAGs

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

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Loukas Georgiadis
  • Department of Computer Science & Engineering, University of Ioannina, Greece
Konstantinos Giannis
  • Department of Computer Science & Engineering, University of Ioannina, Greece
Giuseppe F. Italiano
  • LUISS University, Rome, Italy
Aikaterini Karanasiou
  • Università di Roma "Tor Vergata", Italy
Luigi Laura
  • LUISS University, Rome, Italy


We thank the anonymous reviewers for several comments that helped us improve the presentation of our paper.

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Connectivity
  • dominators
  • low-high orders


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