A near optimal algorithm for finding Euclidean shortest path in polygonal domain

We present an algorithm to find an {\it Euclidean Shortest Path} from a source vertex $s$ to a sink vertex $t$ in the presence of obstacles in ${\mathrm{\Re }}^2$. Our algorithm takes $O(T+m(\lg m)(\lg n))$ time and $O(n)$ space. Here, $O(T)$ is the time to triangulate the polygonal region, $m$ is the number of obstacles, and $n$ is the number of vertices. This bound is close to the known lower bound of $O(n+m\lg m)$ time and $O(n)$ space. Our approach involve progressing shortest path wavefront as in continuous Dijkstra-type method, and confining its expansion to regions of interest.