we give an algorithm, called t*, for finding the k shortest simplepaths connecting a certain pair of nodes, s and t, in a acyclic digraph.first the nodes of the graph are labeled according to the topologicalordering. then for node i an ordered list of simple s − i paths iscreated. the length of the list is at most k and it is created by usingtournament trees. we prove the correctness of t* and ...

Suppose one has a line arrangement in which many lines are parallel, so that the number of diierent line orientations is k, and one wants to nd a shortest path from one point on a line in the arrangement to another such point. Using known techniques one can nd the shortest path in time and space O(n 2). We present an algorithm that can nd the shortest path in time and space O(n + k 2).

A bstract We construct the boundary phase space in D -dimensional Einstein gravity with a generic given co-dimension one null surface $$ \mathcal{N} N as boundary. The associated symmetry algebra is semi-direct sum of diffeomorphisms and Weyl rescalings. It generated by towers charges that are functions over ...

Based on the fundamental theories of null curves in Minkowski 3-space, Darboux mate a curve are defined which can be regarded as kind extension for Bertrand and Mannheim 3-space. The relationships pairs explored their expression forms presented explicitly.