The complexity of matrix transposition on one-tape off-line Turing machines

Abstract

This paper contains the first concrete lower bound argument for Turing machines with one worktape and a two-way input tape (“one-tape off-line Turing machines”): an optimal lower bound of Ω(n·l/⌈( log(l) p ) 1 2 ⌉) for transposing an I × l-matrix with elements of bit length p on such machines is proved. (The length of the input is denoted by n.) A special case is a lower bound of Ω( n 3 2 (log n) 1 2 ) for transposing Boolean l × l-matrices ( n = l 2) on such Turing machines. The proof of the matching upper bound (which is nontrivial for p<log l) uses the fact that one-tape off-line Turing machines can copy strings slightly faster than if the straightforward method is used. As a corollary of the lower bound it is shown that sorting n (3 log n) strings of 3 log n bits each takes Ω( n 3 2 (log n) 1 2 ) steps on one-tape off-line Turing machines. Further corollaries give the first non-linear lower bound for the version of the two-tapes-versus-one problem concerning one-tape off-line Turing machines, and separate one-tape off-line Turing machines from those Turing machines with one input tape, one worktape, and an additional write-only output tape.