Reversal complexity revisited

Abstract

We study a generalized version of reversal bounded Turing machines where, apart from several tapes on which the number of head reversals is bounded by r ( n ) , there are several further tapes on which head reversals remain unrestricted, but size is bounded by s ( n ) (where n denotes the input length). Recently [M. Grohe, C. Koch, N. Schweikardt, Tight lower bounds for query processing on streaming and external memory data, Theoretical Computer Science 380 (1–2) (2007) 199–217; M. Grohe, N. Schweikardt, Lower bounds for sorting with few random accesses to external memory, in: Proc. PODS’05, ACM Press, 2005, pp. 238–249], such machines were introduced as a formalization of a computation model that restricts random access to external memory and internal memory space. Here, each of the tapes with a restriction on the head reversals corresponds to an external memory device, and the tapes of restricted size model internal memory. We use ST ( r ( n ) , s ( n ) , O ( 1 ) ) to denote the class of all problems that can be solved by deterministic Turing machines that comply to the above resource bounds. Similarly, NST ( ⋯ ) and RST ( ⋯ ) , respectively, are used for the corresponding nondeterministic and randomized classes. While previous papers focused on lower bounds for particular problems, including sorting, the set equality problem, and several query evaluation problems, the present paper addresses the relations between the (R,N) ST ( ⋯ ) -classes and classical complexity classes and investigates the structural complexity of the (R,N) ST ( ⋯ ) -classes. Our main results are (1) a trade-off between internal memory space and external memory head reversals, (2) correspondences between the (R,N) ST ( ⋯ ) classes and “classical” time-bounded, space-bounded, reversal-bounded, and circuit complexity classes, and (3) hierarchies of (R) ST ( ⋯ ) -classes in terms of increasing numbers of head reversals on external memory tapes.