🤖 AI Summary
This work addresses the problem of efficient routing between arbitrary pairs of vertices in a planar domain containing multiple disjoint convex polygonal obstacles. The authors propose a recursive routing scheme that relies solely on local information, achieved through geometric recursive partitioning, compact label encoding, and the construction of localized routing tables. The scheme guarantees a path stretch factor of $(7+\varepsilon)\log h$, where $h$ is the number of obstacles, while using vertex labels of size $O(\sqrt{h}\log h\log n)$. The routing table size nearly matches the known theoretical lower bound, and the preprocessing time is $O(n^2\log n)$. This approach strikes a favorable balance among label overhead, routing table size, and path quality.
📝 Abstract
Given a polygonal domain $\cal P$ comprising $h$ pairwise disjoint convex polygonal obstacles in the plane, together defined with $n$ vertices, this paper presents an algorithm to preprocess $\cal P$ to compute routing tables at the vertices of $\cal P$ so that a data packet from any vertex of $\cal P$ is routed to any other vertex belonging to $\cal P$. At every vertex $v$ of $\cal P$ along the routing path, until the packet reaches its destination, the next hop is determined using the routing tables at $v$ and the information stored in the packet header. In $O(n^2(\lg{n}))$ time, our preprocessing algorithm assigns a unique label of size $O(\sqrt{h} (\lg{h}) \lg{n})$ to each vertex of $\cal P$ and computes routing tables of size $O(h\lg{n} + \sqrt{h}(\lg{h})(\min((\frac{1}ε)^{O( \lg α)},n))$ $\lg {n})$ at each vertex of $\cal P$. The routing path output has a $(7 + ε)(\lg{h})$ multiplicative stretch. Here, $ε> 0$ is an input parameter and $α> 1$ is a geometric parameter.