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

Approximating the smallest 2-vertex connected spanning subgraph of a directed graph

We consider the problem of approximating the smallest 2-vertex connected spanning subgraph (2VCSS) of a 2-vertex connected directed graph, and provide new efficient algorithms. We provide two linear-time algorithms, the first based on a linear-time test for 2-vertex connectivity and divergent spanning trees, and the second based on low-high orders, that correspondingly give 3- and 2-approximations. Then we show that these linear-time algorithms can be combined with an algorithm of Cheriyan and Thurimella that achieves a 3/2-approximation. The combined algorithms preserve the 3/2 approximation guarantee of the Cheriyan-Thurimella algorithm and improve its running time from O(m2) to O(mn+n2), for a digraph with n vertices and m edges. Finally, we present an experimental evaluation of the above algorithms for a variety of input data. The experimental results show that our linear-time algorithms perform very well in practice. Furthermore, the experiments show that the combined algorithms not only improve the running time of the Cheriyan-Thurimella algorithm, but it may also compute a better solution

Approximating the smallest 2-vertex connected spanning subgraph of a directed graph

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