Homotopic Rectilinear Routing with Few Links and Thick Edges

Abstract

We study the problem of finding non-crossing thick minimum-link rectilinear paths homotopic to a set of input paths in an environment with rectangular obstacles. This problem occurs in the context of map schematization under geometric embedding restrictions, for example, when schematizing a highway network for use as a thematic layer. We present a 2-approximation algorithm that runs in O(n 3 + k in log n + k out ) time, where n is the total number of input paths and obstacles and k in and k out are the total complexities of the input and output paths, respectively. Our algorithm not only approximates the minimum number of links, but also minimizes the total length of the paths. An approximation factor of 2 is optimal when using smallest paths as lower bound.

Homotopic $\mathcal{C}$ -Oriented Routing

“…Results. In this paper we generalize our previous result [14] to C-oriented paths. To this end we introduce a special type of C-oriented paths-smooth paths-which are a generalization of rectilinear paths that retain several important properties of rectilinear paths.…”

“…Proof. The first part follows from the discussion above and the result in [14]. Consider a minimum-link C-oriented path π with L links.…”

“…Their paths are not C-oriented. In a previous paper [14] we presented a 2-approximation algorithm for the rectilinear version of our problem.…”

“…4). Now we can apply the incremental untangling algorithm of [14], and untangle the staircase chains (of this type) using at most twice the number of links. We do this for all types of staircase chains.…”

“…But how many links are necessary? For smooth paths with L links, a similar construction as for rectilinear paths [14] shows that 2L links can be necessary. For general C-oriented paths, we give a lower bound that is based on a construction from [8].…”