# On bends and lengths of rectilinear paths: A graph-theoretic approach

## Abstract

We consider the problem of finding a rectilinear path between two designated points in the presence of rectilinear obstacles subject to various optimization functions in terms of the number of bends and the total length of the path. Specifically we are interested in finding a minimum bend shortest path, a shortest minimum bend path or a least-cost path where the cost is defined as a function of both the length and the number of bends of the path. We provide a unified approach by constructing a *path-preserving graph* guaranteed to preserve all these three kinds of paths and give an *O*(*K*+*e* log *e*) algorithm to find them, where *e* is the total number of obstacle edges, and *K* is the number of intersections between *tracks* from *extreme point* and other tracks (defined in the text). *K* is bounded by *O*(*et*), where *t* is the number of *extreme edges*. In particular, if the obstacles are rectilinearly convex, then *K* is *O*(*ne*), where *n* is the number of obstacles. Extensions are made to find a shortest path with a bounded number of bends and a minimum-bend path with a bounded length. When a source point and obstacles are pre-given, queries for the assorted paths from the source to given points can be handled in *O*(log *n*+*k*) time after *O*(*K*+*e* log *e*) preprocessing, where *k* is the size of the goal path. The *trans-dichotomous* algorithm of Fredman and Willard[8] and the running time for these problems are also discussed.

## Keywords

Short Path Projection Point Linear Time Algorithm Simple Polygon Edge Cost## Preview

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