The standard simulation of a nondeterministic Turing machine (NTM) by a deterministic one essentially searches a large bounded-degree graph whose size is exponential in the running time of the NTM. The graph is the natural one defined by the configurations of the NTM. All methods in the literature have required time linear in the size S of this graph. This paper presents a new simulation method that runs in time O(S). The search savings exploit the one-dimensional nature of Turing machine tapes. In addition, we remove most of the time dependence on nondeterministic choices of states and tape head movements. © 2011 Elsevier B.V. All rights reserved.