Finding Efficient Region in The Plane with Line segments

Let $\mathscr O$ be a set of $n$ disjoint obstacles in $\mathbb{R}^2$, $\mathscr M$ be a moving object. Let $s$ and $l$ denote the starting point and maximum path length of the moving object $\mathscr M$, respectively. Given a point $p$ in ${R}^2$, we say the point $p$ is achievable for $\mathscr M$ such that $\pi(s,p)\leq l$, where $\pi(\cdot)$ denotes the shortest path length in the presence of obstacles. One is to find a region $\mathscr R$ such that, for any point $p\in \mathbb{R}^2$, if it is achievable for $\mathscr M$, then $p\in \mathscr R$; otherwise, $p\notin \mathscr R$. In this paper, we restrict our attention to the case of line-segment obstacles. To tackle this problem, we develop three algorithms. We first present a simpler-version algorithm for the sake of intuition. Its basic idea is to reduce our problem to computing the union of a set of circular visibility regions (CVRs). This algorithm takes $O(n^3)$ time. By analysing its dominant steps, we break through its bottleneck by using the short path map (SPM) technique to obtain those circles (unavailable beforehand), yielding an $O(n^2\log n)$ algorithm. Owing to the finding above, the third algorithm also uses the SPM technique. It however, does not continue to construct the CVRs. Instead, it directly traverses each region of the SPM to trace the boundaries, the final algorithm obtains $O(n\log n)$ complexity.