Minimum-link paths revisited

A path or a polygonal domain is C-oriented if the orientations of its edges belong to a set of C given orientations; this is a generalization of the notable rectilinear case (C = 2). We study exact and approximation algorithms for minimum-link C-oriented paths and paths with unrestricted orientations, both in C-oriented and in general domains. We obtain the following results: • An optimal algorithm for finding a minimum-link rectilinear path in a rectilinear domain; using only elementary data structures, our algorithm and its analysis are simpler than the earlier ones of [Sato, Sakanaka and Ohtsuki, 1987], [Das and Narasimhan, 1991]. • An algorithm to find a minimum-link C-oriented path in a C-oriented domain; our algorithm is simpler and more time-space efficient than the earlier one of [Adegeest, Overmars and Snoeyink, 1994]. • An extension of our techniques to find a C-oriented minimum-link path in a general (not necessarily C-oriented) domain. • 3SUM-hardness of finding a minimum-link path with unrestricted orientations (even in a C-oriented domain). This answers a question from the survey of Mitchell [Goodman and O’Rourke, eds., 1997, 2004] and Problem 22 in The Open Problems Project. We also present approximation methods for link distance, including the following: • A more time-space efficient algorithm to find a 2-approximate C-oriented minimum-link path. • A notion of “robust” paths. We show how minimum-link C-oriented paths approximate the robust paths with unrestricted orientations to within an additive error of 1. • A subquadratic-time algorithm with a non-trivial approximation guarantee for the general (unrestricted-orientation) minimum-link paths in general domains. All of our algorithms not only find minimum-link paths but also build, within the same time and space bounds, the corresponding link distance maps—exact or approximate. For instance, using our algorithms, one can construct in subquadratic time linear-size approximate (additive or multiplicative) maps for general minimum-link paths in general domains; this is in contrast with the exact link distance maps, which may have quartic complexity [Suri and O’Rourke, 1986].