arXiv subjects
Publications and source records attributed to David Pritchard.
Ultracold sodium atoms have been trapped inside a hollow-core optical fiber. The atoms are transferred from a free space optical dipole trap into a trap formed by a red-detuned gaussian light mode confined to the core of the fiber. We show that at least 5% of the atoms held initially in the free space trap can be loaded into the core of the fiber and retrieved outside.
The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G=(V,E), a set of terminals R, and non-negative costs c_e for all edges e in E. Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The nodes V R are called Steiner nodes. The best approximation algorithm known for the Steiner tree problem is due to Robins and Zelikovsky (SIAM J. Discrete Math, 2005); their greedy algorithm achieves a performance guarantee of 1+(ln 3)/2 ~ 1.55. The best known linear (LP)-based algorithm, on the other hand, is due to Goemans and Bertsimas (Math. Programming, 1993) and achieves an approximation ratio of 2-2/|R|. In this paper we establish a link between greedy and LP-based approaches by showing that Robins and Zelikovsky's algorithm has a natural primal-dual interpretation with respect to a novel partition-based linear programming relaxation. We also exhibit surprising connections between the new formulation and existing LPs and we show that the new LP is stronger than the bidirected cut formulation. An instance is b-quasi-bipartite if each connected component of G R has at most b vertices. We show that Robins' and Zelikovsky's algorithm has an approximation ratio better than 1+(ln 3)/2 for such instances, and we prove that the integrality gap of our LP is between 8/7 and (2b+1)/(b+1).
A mobile agent in a network wants to visit every node of an n-node network, using a small number of steps. We investigate the performance of the following ``nearest neighbor'' heuristic: always go to the nearest unvisited node. If the network graph never changes, then from (Rosenkrantz, Stearns and Lewis, 1977) and (Hurkens and Woeginger, 2004) it follows that Theta(n log n) steps are necessary and sufficient in the worst case. We give a simpler proof of the upper bound and an example that improves the best known lower bound. We investigate how the performance of this heuristic changes when it is distributively implemented in a network. Even if network edges are allow to fail over time, we show that the nearest neighbor strategy never runs for more than O(n^2) iterations. We also show that any strategy can be forced to take at least n(n-1)/2 steps before all nodes are visited, if the edges of the network are deleted in an adversarial way.
We describe a synchronous distributed algorithm which identifies the edge-biconnected components of a connected network. It requires a leader, and uses messages of size O(log |V|). The main idea is to preorder a BFS spanning tree, and then to efficiently compute least common ancestors so as to mark cycle edges. This algorithm takes O(Diam) time and uses O(|E|) messages. Furthermore, we show that no correct singly-initiated edge-biconnectivity algorithm can beat either bound on any graph by more than a constant factor. We also describe a near-optimal local algorithm for edge-biconnectivity.