Let T be a tree that is embedded in the plane and let ∆ > 0 be a real number. The aim is to preprocess T into a data structure, such that, for any query polygonal path Q, we can decide if T contains a path P whose Fréchet distance δF (P,Q) to Q is less than ∆. For any real number ε > 0, we present an efficient data structure that solves an approximate version of this problem, for the case when ...

Abs t rac t Much of the focus of the research effort in path planning for mobile robots has centred on the problem of finding a path from a start location to a goal location, while minimising one or more parameters such as length of path, energy consumption or journey time. A path of complete coverage is a planned path in which a robot sweeps all areas of free space in an environment in a syste...

We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V (G), a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The kPC problem is to find a k-fixed-endpoi...

The algorithms for computing a shortest path on a polyhedral surface are slow, complicated, and numerically unstable. We have developed and implemented a robust and efficient algorithm for computing approximate shortest paths on a convex polyhedral surface. Given a convex polyhedral surface P in R3, two points s, t ∈ P , and a parameter ε > 0, it computes a path between s and t on P whose lengt...

We show that the number of t-ary trees with path length equal to p is exp ( h(t−1)t log t p log p (1 + o(1)) ) , where h(x)=−x log x−(1−x) log(1−x) is the binary entropy function. Besides its intrinsic combinatorial interest, the question recently arose in the context of information theory, where the number of t-ary trees with path length p estimates the number of universal types, or, equivalen...

Applying a space-time transformation discussed in a previous paper, the Feynman path integral for the Coulomb potential is calculated exactly by reducing it to gaussian form. The resulting Green function leads to a simple derivation of the energy spectrum and the complete (normalized) wavefunctions of the hydrogen atom. In a previous paper [ 1 ] I derived a transformation formula for a class of...

A many-to-many k-disjoint path cover of a graph joining two disjoint vertex sets S and T of equal size k is a set of k vertex-disjoint paths between S and T that altogether cover every vertex of the graph. The many-to-many k-disjoint path cover is classified as paired if each source in S is further required to be paired with a specific sink in T , or unpaired otherwise. In this paper, we first ...

p-Adic generalization of the Feynman path integrals in quantum mechanics is considered. The probability amplitude Kv(x, t′′; x′, t′) (v = ∞, 2, 3, · · · , p, · · ·) for a particle in a constant field is calculated. Path integrals over Qp have the same form as those over R.