Dominator Tree Certification and Independent Spanning Trees

How does one verify that the output of a complicated program is correct? One can formally prove that the program is correct, but this may be beyond the power of existing methods. Alternatively one can check that the output produced for a particular input satisfies the desired input-output relation, by running a checker on the input-output pair. Then one only needs to prove the correctness of the checker. But for some problems even such a checker may be too complicated to formally verify. There is a third alternative: augment the original program to produce not only an output but also a correctness certificate, with the property that a very simple program (whose correctness is easy to prove) can use the certificate to verify that the input-output pair satisfies the desired input-output relation. We consider the following important instance of this general question: How does one verify that the dominator tree of a flow graph is correct? Existing fast algorithms for finding dominators are complicated, and even verifying the correctness of a dominator tree in the absence of additional information seems complicated. We define a correctness certificate for a dominator tree, show how to use it to easily verify the correctness of the tree, and show how to augment fast dominator-finding algorithms so that they produce a correctness certificate. We also relate the dominator certificate problem to the problem of finding independent spanning trees in a flow graph, and we develop algorithms to find such trees. All our algorithms run in linear time. Previous algorithms apply just to the special case of only trivial dominators, and they take at least quadratic time. ∗This work is a rewritten and expanded combination of two conference papers, “Dominator Tree Verification and Vertex-Disjoint Paths,” Proc. 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 433–442, 2005, and “Dominators, Directed Bipolar Orders, and Independent Spanning Trees,” Proc. 39th International Colloquium on Automata, Languages and Programming, pp. 375–386, 2012. Department of Computer Science, University of Ioannina, Greece. E-mail: [email protected]. Department of Computer Science, Princeton University, 35 Olden Street, Princeton, NJ, 08540, and Hewlett-Packard Laboratories. E-mail: [email protected]. Research at Princeton University partially supported by NSF grants CCF-0830676 and CCF-0832797. Research while visiting Stanford University partially supported by an AFOSR MURI grant. The information contained herein does not necessarily reflect the opinion or policy of the federal government and no official endorsement should be inferred.