  The greedy and nearest-neighbor TSP heuristics can both have $\log n$
approximation factors from optimal in worst case, even just for $n$ points in
Euclidean space. In this note, we show that this approximation factor is only
realized when the optimal tour is unusually short. In particular, for points
from any fixed $d$-Ahlfor's regular metric space (which includes any
$d$-manifold like the $d$-cube $[0,1]^d$ in the case $d$ is an integer but also
fractals of dimension $d$ when $d$ is real-valued), our results imply that the
greedy and nearest-neighbor heuristics have \emph{additive} errors from optimal
on the order of the \emph{optimal} tour length through \emph{random} points in
the same space, for $d>1$.
