  We develop a method of constructing percolation clusters that allows us to
build very large clusters using very little computer memory by limiting the
maximum number of sites for which we maintain state information to a number of
the order of the number of sites in the largest chemical shell of the cluster
being created. The memory required to grow a cluster of mass s is of the order
of $s^\theta$ bytes where $\theta$ ranges from 0.4 for 2-dimensional lattices
to 0.5 for 6- (or higher)-dimensional lattices. We use this method to estimate
$d_{\scriptsize min}$, the exponent relating the minimum path $\ell$ to the
Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing both site
and bond percolation, we find $d_{\scriptsize min}=1.607\pm 0.005$ (4D) and
$d_{\scriptsize min}=1.812\pm 0.006$ (5D). In order to determine
$d_{\scriptsize min}$ to high precision, and without bias, it was necessary to
first find precise values for the percolation threshold, $p_c$:
$p_c=0.196889\pm 0.000003$ (4D) and $p_c=0.14081\pm 0.00001$ (5D) for site and
$p_c=0.160130\pm 0.000003$ (4D) and $p_c=0.118174\pm 0.000004$ (5D) for bond
percolation. We also calculate the Fisher exponent, $\tau$, determined in the
course of calculating the values of $p_c$: $\tau=2.313\pm 0.003$ (4D) and
$\tau=2.412\pm 0.004$ (5D).
