Computing shortest paths amid non-overlapping weighted disks

TL;DR

An approximation algorithm for solving the Weighted Region Problem amidst a set of n non-overlapping weighted disks in the plane using Dijkstra's algorithm for computing a shortest path in the geometric graph obtained in (pseudo-)polynomial time.

Abstract

In this article, we present an approximation algorithm for solving the Weighted Region Problem amidst a set of $ n $ non-overlapping weighted disks in the plane. For a given parameter $ \varepsilon \in (0,1]$, the length of the approximate path is at most $ (1 +\varepsilon) $ times larger than the length of the actual shortest path. The algorithm is based on the discretization of the space by placing points on the boundary of the disks. Using such a discretization we can use Dijkstra's algorithm for computing a shortest path in the geometric graph obtained in (pseudo-)polynomial time.

Figures (14)

• Figure 1: The two possible types of shortest paths between $p$ and $q$ on the boundary of a disk are depicted in red and blue.
• Figure 2: Distinguished points on $D$ are depicted as white disks.
• Figure 3: The path $u' \rightarrow u \rightarrow v \rightarrow v"$ is forward going. The distinguished points $u', u", v'$ and $v"$ are depicted as white disks.
• Figure 4: The path $\delta(u,v)$ is represented in blue.
• Figure 5: Illustrating the definitions of $\ell$, $b"$, $B_r$ and $B_\ell$.

Theorems & Definitions (23)

• Lemma 3
• proof
• Lemma 4
• proof
• Proposition 5
• Lemma 6
• proof
• Corollary 7
• Definition 8
• Proposition 9