  We consider the problem of computing an approximate weighted shortest path in
a weighted subdivision, with weights assigned from the set $\{0, 1, \infty\}$.
We present a data structure $B$, which stores a set of convex, non-overlapping
regions. These include zero-cost regions (0-regions) with a weight of $0$ and
obstacles with a weight of $\infty$, all embedded in a plane with a weight of
$1$. The data structure $B$ can be constructed in expected time $O(N +
(n/\varepsilon^3)(\log(n/\varepsilon) + \log N))$, where $n$ is the total
number of regions, $N$ represents the total complexity of the regions, and $1 +
\varepsilon$ is the approximation factor, for any $0 < \varepsilon < 1$. Using
$B$, one can compute an approximate weighted shortest path from any point $s$
to any point $t$ in $O(N + n/\varepsilon^3 + (n/\varepsilon^2)
\log(n/\varepsilon) + (\log N)/\varepsilon)$ time. In the special case where
the 0-regions and obstacles are polygons (not necessarily convex), $B$ contains
a $(1 + \varepsilon)$-spanner of the input vertices.
