  In online makespan minimization a sequence of jobs $\sigma = J_1,..., J_n$
has to be scheduled on $m$ identical parallel machines so as to minimize the
maximum completion time of any job. We investigate the problem with an
essentially new model of resource augmentation. Here, an online algorithm is
allowed to build several schedules in parallel while processing $\sigma$. At
the end of the scheduling process the best schedule is selected. This model can
be viewed as providing an online algorithm with extra space, which is invested
to maintain multiple solutions. The setting is of particular interest in
parallel processing environments where each processor can maintain a single or
a small set of solutions.
  We develop a $(4/3+\eps)$-competitive algorithm, for any $0<\eps\leq 1$, that
uses a number of $1/\eps^{O(\log (1/\eps))}$ schedules. We also give a
$(1+\eps)$-competitive algorithm, for any $0<\eps\leq 1$, that builds a
polynomial number of $(m/\eps)^{O(\log (1/\eps) / \eps)}$ schedules. This value
depends on $m$ but is independent of the input $\sigma$. The performance
guarantees are nearly best possible. We show that any algorithm that achieves a
competitiveness smaller than 4/3 must construct $\Omega(m)$ schedules. Our
algorithms make use of novel guessing schemes that (1) predict the optimum
makespan of a job sequence $\sigma$ to within a factor of $1+\eps$ and (2)
guess the job processing times and their frequencies in $\sigma$. In (2) we
have to sparsify the universe of all guesses so as to reduce the number of
schedules to a constant.
  The competitive ratios achieved using parallel schedules are considerably
smaller than those in the standard problem without resource augmentation.
