Dominator Tree Certification and Divergent Spanning Trees

Abstract

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. For some problems, however, 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 divergent 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.

Vaccination allocation in large dynamic networks

“…A dominator tree will yield sub-trees that show the relative connection between nodes, thus can yield maximum solutions based on those sub-trees [11]. If the structure of a dominator tree is correct [22] a new relational mapping of a network can be built quickly to determine which parent nodes singularly expose their children to infection. A dominator tree shows that if there is a non-unique cycle between two nodes x and y then x and y will be the only common nodes to all paths.…”

2-Edge Connectivity in Directed Graphs

“…To tackle this challenge, we provide some key observations regarding edges and paths that connect different subtrees T (r). We will use the parent property of dominator trees [13], that we state next. Now we prove some structural properties for paths that connect vertices in different subtrees.…”

“…As in Section 3 we can assume without loss of generality that G is strongly connected, in which case subgraph C(G) will also be strongly connected (see the proof of Lemma 4.1 below). The certificate uses the concept of independent spanning trees [13]. In this context, a spanning tree T of a flow graph G(s) is a tree with root s that contains a path from s to v for all vertices v. Two spanning trees B and R rooted at s are independent if for all vertices v, the paths from s to v in B and R share only the dominators of v. Every flow graph G(s) has two such spanning trees, computable in linear time [13].…”

“…The certificate uses the concept of independent spanning trees [13]. In this context, a spanning tree T of a flow graph G(s) is a tree with root s that contains a path from s to v for all vertices v. Two spanning trees B and R rooted at s are independent if for all vertices v, the paths from s to v in B and R share only the dominators of v. Every flow graph G(s) has two such spanning trees, computable in linear time [13]. Moreover, the computed spanning trees are maximally edge-disjoint, meaning that the only edges they have in common are the bridges of G(s).…”

Validating Dominator Trees for a Fast, Verified Dominance Test

“…Georgiadis et al [11] propose a linear-time checker of the dominator tree, based on the notions of headers and loop nesting forests. Georgiadis et al [9] propose a linear-time certifying algorithm, producing a certificate (a preorder of the vertices of the dominator tree, with a so-called property low-high), that helps simplifying the checking process.…”