[$L_1$] Shortest Paths with Isothetic Roads

축에 평행한 도로들이 놓여 있을 때의 $L_1$ 최단 경로

We present a nearly optimal ($O(\nu\;min(\nu,\;n)n\;log\;n)$ time and O(n) srace) algorithm that constructs a shortest path map with n isothetic roads of speed $\nu$ under the $L_1$ metric. The algorithm uses the continuous Dijkstra method and its efficiency is based on a new geometric insight; the minimum in-degree of any nearest neighbor graph for points with roads of speed $\nu$ is $\Theta(\nu\;min(\nu,\;n))$, which is first shown in this paper. Also, this algorithm naturally extends to the multi-source case so that the Voronoi diagram for m sites can be computed in $O(\nu\;min(\nu,\;n)(n+m)log(n+m))$ time and O(n+m) space, which is also nearly optimal.