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.Our two main algorithms are as follows:A subquadratic-time algorithm with a non-trivial approximation guarantee for general (unrestricted-orientation) minimum-link paths in general domains.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 prior algorithm.We also obtain several related results:• 3SUM-hardness of determining the link distance with unrestricted orientations (even in a rectilinear domain).• An optimal algorithm for finding a minimum-link rectilinear path in a rectilinear domain. The algorithm and its analysis are simpler than the existing ones.• An extension of our methods to find a C-oriented minimum-link path in a general (not necessarily C-oriented) domain.• A more efficient algorithm to compute 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.
In boundary labeling, features on a map are connected to a stack of labels on the map boundary, using simple polylines called leaders. We consider the setting that the labels are axis-aligned non-overlapping rectangles placed on one side of the map, and leaders are rectilinear polylines with at most one bend. The goal is to find a labeling that minimizes the total length of the leaders.We introduce three extensions of the one-sided boundary labeling problem: (i) a dynamic setting for continuous scale changes, (ii) a clustered setting for multiple label stacks, and (iii) a combined dynamic clustered setting. We obtain the following results:• Optimal label placement as a function of map scale can be computed in O(n log n + σ log n) time, where σ is the number of "combinatorially different" labelings that occur during zooming. • In a map with fixed scale, an optimal clustered label placement can be found in O(n log n) time. • In O(n log 2 n + γ log n) time one can build a structure of size O(γ) representing the optimal clustered label placement for all possible map scales; here γ is, again, the number of combinatorially different labelings.We further extend our basic model to the case where labeled features enter or leave the viewport due to map panning and zooming. Our algorithms are based on combining standard computational-geometry tools and have been implemented in a Java applet (available online), which indicates that the algorithms are fast enough for interactive use without delays.
Abstract. We consider the problem of finding minimum-link rectilinear paths in rectilinear polygonal domains in the plane. A path or a polygon is rectilinear if all its edges are axis-parallel. Given a set P of h pairwise-disjoint rectilinear polygonal obstacles with a total of n vertices in the plane, a minimumlink rectilinear path between two points is a rectilinear path that avoids all obstacles with the minimum number of edges. In this paper, we present a new algorithm for finding minimum-link rectilinear paths among P. After the plane is triangulated, with respect to any source point s, our algorithm builds an O(n)-size data structure in O(n + h log h) time, such that given any query point t, the number of edges of a minimum-link rectilinear path from s to t can be computed in O(log n) time and the actual path can be output in additional time linear in the number of the edges of the path. The previously best algorithm computes such a data structure in O(n log n) time.
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