  Consider $N$ points randomly distributed along a line segment of unitary
length. A walker explores this disordered medium moving according to a
partially self-avoiding deterministic walk. The walker, with memory $\mu$,
leaves from the leftmost point and moves, at each discrete time step, to the
nearest point, which has not been visited in the preceding $\mu$ steps. We have
obtained analytically the probability $P_N(\mu) = (1 - 2^{-\mu})^{N - \mu - 1}$
that all $N$ points are visited in this open system, with $N \gg \mu \gg 1$.
The expression for $P_N(\mu)$ evaluated in the mentioned limit is valid even
for small $N$ and leads to a transition region centered at $\mu_1 = \ln N/\ln
2$ and with width $\epsilon = e/\ln2$. For $\mu < \mu_1 - \epsilon/2$, the
walker gets trapped in cycles and does not fully explore the system. For $\mu >
\mu_1 + \epsilon/2$ the walker explores the whole system. In both cases the
walker presents diffusive behavior. Nevertheless, in the intermediate regime
$\mu \sim \mu_1 \pm \epsilon/2$, the walker presents anoumalous diffusion
behavior. Since the intermediate region increases as $\ln N$ and its width is
constant, a sharp transition is obtained for one-dimensional large systems. The
walker does not need to have full memory of its trajectory to explore the whole
system, it suffices to have memory of order $\mu_1$.
