On Low-High Orders of Directed Graphs: Incremental Algorithms and Applications
Loukas Georgiadis, Aikaterini Karanasiou, Giannis Konstantinos, Luigi, Laura

TL;DR
This paper introduces efficient incremental algorithms for maintaining low-high orders and dominator trees in flow graphs, with applications to connectivity problems and practical performance demonstrated on real-world graphs.
Contribution
It presents the first incremental certifying algorithms for dominator trees with improved $O(mn)$ total time complexity and applies low-high orders to develop a linear-time approximation for 2-vertex-connected spanning subgraphs.
Findings
Algorithms run efficiently on real-world graphs
Significant improvement over previous $O(m^2)$ methods
Practical implementations show strong performance
Abstract
A flow graph is a directed graph with a distinguished start vertex . The dominator tree of is a tree rooted at , such that a vertex is an ancestor of a vertex if and only if all paths from to include . 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 is a preorder of that certifies the correctness of and has further applications in connectivity and path-determination problems. In this paper, we first consider how to maintain efficiently a low-high order of a flow graph incrementally under edge insertions. We present algorithms that run in total time for a sequence of edge insertions in an initially…
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