An Optimal Algorithm for Minimum-Link Rectilinear Paths in Triangulated Rectilinear Domains

Abstract

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.

Bicriteria Rectilinear Shortest Paths Among Rectilinear Obstacles in the Plane

“…Imai and Asano [17] presented an O(n log n) time and space algorithm for finding a minimum-link s-t path in P, and the space was reduced to O(n) [14,21,25]. Recently, Mitchell et al [22] proposed an O(n + h log h) time and O(n) space algorithm for the problem, after P is triangulated (which can be done in O(n log n) time or O(n + h log 1+ǫ h) time for any ǫ > 0 [1]). The algorithms in [14,21,22] also construct an O(n) size data structure that can answer each one-point minimum-link path query in O(log n) time.…”

“…To further reduce the running time (for small h), our main idea is to use a reduced graph G r of size O(h log h) instead of G. We show that G r contains an s-t path π Gr (s, t) that is homotopic to an optimal s-t path π(s, t) in P with the same length, and further, π(s, t) can be obtained from π Gr (s, t) by performing the dragging operations as in [29] and a new kind of operations, called through-corridor-path generating operations. The graph G r is built based on a corridor structure of P, which was used to find minimum-link paths in [22]. More specifically, we decompose P into O(h) junction rectangles and O(h) corridors.…”

“…Hence, similar to the minimum-link shortest paths, we can find a minimum-link path in O(n log 3/2 n) time and O(n log n) space. As discussed in Section 1, the problem of finding a single minimum-link path and its one-point query problem have been solved optimally [22] (after P is triangulated). We discussed the above result mainly because we will use it to answer the two-point queries in Section 6.…”

“…In this section, we improve our algorithm proposed in Section 3, so that in addition to O(n), the complexities of our improved algorithm only depend on h, i.e., the number of holes of P. We first review the corridor structure of P [22] and the histogram partitions of rectilinear simple polygons [26].…”

“…The corridor structure of P has been introduced before, e.g., see [22]. Let G vd be the dual graph of VD(P) (e.g., see Fig.…”