On Minimum Link Monotone Path Problems

The problem of finding monotone paths between two given points has useful applications in path planning, and in particular, it is useful to look for minimum link paths. We are given a simple polygon P or a polygonal domain D with n vertices and a triplet of input parameters: (s, t, d), where s and t are two points in the plane and d is any direction. We show how to answer a query for the existence of a d-monotone path between s and t inside P (or D) in logarithmic time after preprocessing P in O(En) time, or D in O(En + ERlogR) time, where E is the size of the visibility graph of P (or D), and R is the number of reflex vertices in D. Our approach is based on the novel idea utilizing the dual graph of the trapezoidal map of P (or D). For polygonal domains, our approach uses a trapezoidal map associated with each visibility edge of D, and we show how to compute this large set of trapezoidal maps efficiently. Furthermore, we show how to output a minimum linkd-monotone path between points s and t, for an arbitrary input triplet (s, t, d).

THE SHUFFLING BUFFER

The complexity of randomized incremental algorithms is analyzed with the assumption of a random order of the input. To guarantee this hypothesis, the n data have to be known in advance in order to be mixed what contradicts with the on-line nature of the algorithm. We present the shuffling buffer technique to introduce sufficient randomness to guarantee an improvement on the worst case complexity by knowing only k data in advance. Typically, an algorithm with O(n2) worst-case complexity and O(n) or O(n log n) randomized complexity has an [Formula: see text] complexity for the shuffling buffer. We illustrate this with binary search trees, the number of Delaunay triangles or the number of trapezoids in a trapezoidal map created during an incremental construction.