Dynamic Dominators and Low-High Orders in DAGs

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

Towards Automated Generation of Time-Predictable Code

Knowledge of the worst-case execution time of software components is essential in safety-critical hard real-time systems. The analysis thereof is not trivial as the execution time depends on many factors, including the underlying hardware platform, the program structure, and the code produced by the compiler. Often, the execution time is variable and highly sensitive to the input data the program has to process. This paper presents a code transformation applicable in a compiler backend that produces time-predictable code. The resulting code contains a single input-data independent execution path, in order to obtain programs of stable timing behaviour. The transformation technique has been validated by applying it on a number of benchmarks. Experiments show a reduction of execution time variability, at acceptable costs for the single execution path

Dominators in Directed Graphs: A Survey of Recent Results, Applications, and Open Problems

The computation of dominators is a central tool in program optimization and code generation, and it has applications in other diverse areas includingconstraint programming, circuit testing, and biology. In this paper we survey recent results, applications, and open problems related to the notion of dominators in directed graphs,including dominator verification and certification, computing independent spanning trees, and connectivity and path-determination problems in directed graphs

Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3-SAT

In directed graphs, we investigate the problems of finding: 1) a minimum feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2) a feedback vertex set inducing an acyclic graph (also called the Vertex 2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS). We show that these problems are strongly related to (variants of) Monotone 3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in positive form. As a consequence, we deduce several NP-completeness results on restricted versions of these problems. In particular, we define the 2-Choice version of an optimization problem to be its restriction where the optimum value is known to be either D or D+1 for some integer D, and the problem is reduced to decide which of D or D+1 is the optimum value. We show that the 2-Choice versions of MFVS, Acyclic MFVS, Min Ones Monotone 3-SAT and Min Ones Monotone NAE 3-SAT are NP-complete. The two latter problems are the variants of Monotone 3-SAT and respectively Monotone NAE 3-SAT requiring that the truth assignment minimize the number of variables set to true. Finally, we propose two classes of directed graphs for which Acyclic FVS is polynomially solvable, namely flow reducible graphs (for which MFVS is already known to be polynomially solvable) and C1P-digraphs (defined by an adjacency matrix with the Consecutive Ones Property)

Finding Dominators via Disjoint Set Union

The problem of finding dominators in a directed graph has many important applications, notably in global optimization of computer code. Although linear and near-linear-time algorithms exist, they use sophisticated data structures. We develop an algorithm for finding dominators that uses only a "static tree" disjoint set data structure in addition to simple lists and maps. The algorithm runs in near-linear or linear time, depending on the implementation of the disjoint set data structure. We give several versions of the algorithm, including one that computes loop nesting information (needed in many kinds of global code optimization) and that can be made self-certifying, so that the correctness of the computed dominators is very easy to verify