Nearly Linear Time

The notion of linear-time computability is very sensitive to machine model. In this connection, we introduce a class NLT of functions computable in nearly linear time n(log n)O(1) on random access computers. NLT is very robust and does not depend on the particular choice of random access computers. Kolmogorov machines, Schonhage machines, random access Turing machines, etc. also compute exactly NLT functions in nearly linear time. It is not known whether usual multitape Turing machines are able to compute all NLT functions in nearly linear time. We do not believe they are and do not consider them necessarily appropriate for this relatively low complexity level. It turns out, however, that nondeterministic Turing machines accept exactly the languages in the nondeterministic version of NLT. We give also a machine-independent de nition of NLT and a natural problem complete for NLT. Springer LNCS 363, 1989, 108{118. Partially supported by an NSF grant and a grant from Binational US-Israel Science Foundation. A substantial portion of the work was done during a week in Fall 1985 when both authors visited Rutgers University; during the last stage of the work, the rst author was with Stanford University and IBM Almaden Research Center (on a sabbatical leave from the University of Michigan).