Cluster computing and the power of edge recognition

• Authors: Lane A. Hemaspaandra;Christopher M. Homan;Sven Kosub
• Venue: Information and Computation
• Year: 2007

Though complexity theory already extensively studies path-cardinality-based restrictions on the power of nondeterminism, this paper is motivated by a more recent goal: To gain insight into how much of a restriction, it is of nondeterminism to limit machines to have just one contiguous (with respect to some simple order) interval of accepting paths. In particular, we study the robustness-the invariance under definition changes-of the cluster class CL#P. This class contains each #P function that is computed by a balanced Turing machine whose accepting paths always form a cluster with respect to some length-respecting total order with efficient adjacency checks. The definition of CL#P is heavily influenced by the defining paper's focus on (global) orders. In contrast, we define a cluster class, CLU#P, to capture what seems to us a more natural model of cluster computing. We prove that the naturalness is costless: CL#P=CLU#P. Then we exploit the more natural, flexible features of CLU#P to prove new robustness results for CL#P and to expand what is known about the closure properties of CL#P. The complexity of recognizing edges-of an ordered collection of computation paths or of a cluster of accepting computation paths-is central to this study. Most particularly, our proofs exploit the power of unique discovery of edges-the ability of nondeterministic functions to, in certain settings, discover on exactly one (in some cases, on at most one) computation path a critical piece of information regarding edges of orderings or clusters.