Distributed Approximation Algorithms for Weighted Shortest Paths

A distributed network is modeled by a graph having $n$ nodes (processors) and diameter $D$. We study the time complexity of approximating {\em weighted} (undirected) shortest paths on distributed networks with a $O(\log n)$ {\em bandwidth restriction} on edges (the standard synchronous \congest model). The question whether approximation algorithms help speed up the shortest paths (more precisely distance computation) was raised since at least 2004 by Elkin (SIGACT News 2004). The unweighted case of this problem is well-understood while its weighted counterpart is fundamental problem in the area of distributed approximation algorithms and remains widely open. We present new algorithms for computing both single-source shortest paths (\sssp) and all-pairs shortest paths (\apsp) in the weighted case. Our main result is an algorithm for \sssp. Previous results are the classic $O(n)$-time Bellman-Ford algorithm and an $\tilde O(n^{1/2+1/2k}+D)$-time $(8k\lceil \log (k+1) \rceil -1)$-approximation algorithm, for any integer $k\geq 1$, which follows from the result of Lenzen and Patt-Shamir (STOC 2013). (Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use $\tilde O(\cdot)$ to hide the O(\poly\log n) term.) We present an $\tilde O(n^{1/2}D^{1/4}+D)$-time $(1+o(1))$-approximation algorithm for \sssp. This algorithm is {\em sublinear-time} as long as $D$ is sublinear, thus yielding a sublinear-time algorithm with almost optimal solution. When $D$ is small, our running time matches the lower bound of $\tilde \Omega(n^{1/2}+D)$ by Das Sarma et al. (SICOMP 2012), which holds even when $D=\Theta(\log n)$, up to a \poly\log n factor.Comment: Full version of STOC 201

Uniquely colourable m-dichromatic oriented graphs

AbstractThe dichromatic number dk(D) of a diagraph D is the minimum number of colours needed to colour V(D) in such a way that no monochromatic directed cycle is obtained. A digraph D is called uniquely colourable if any acyclic dk(D)-colouring of V(D) induces the same partition of V(D). In this paper we construct an infinite family of uniquely colourable m-dichromatic oriented graphs for all m ⩾ 2

Structural design studies of a supersonic cruise arrow wing configuration

Structural member cross sections were sized with a system of integrated computer programs to satisfy strength and flutter design requirements for several variants of the arrow wing supersonic cruise vehicle. The resulting structural weights provide a measure of the structural efficiency of the planform geometry, structural layout, type of construction, and type of material including composites. The material distribution was determined for a baseline metallic structure and the results indicate that an approximate fatigue constraint has an important effect on the structural weight required for strength but, in all cases, additional material had to be added to satisfy flutter requirements with lighter mass engines with minimum fuel onboard. The use of composite materials on the baseline configuration was explored and indicated increased structural efficiency. In the strength sizing, the all-composite construction provided a lower weight design than the hybrid construction which contained composites only in the wing cover skins. Subsequent flutter analyses indicated a corresponding lower flutter speed

Iterative Approximate Consensus in the presence of Byzantine Link Failures

This paper explores the problem of reaching approximate consensus in synchronous point-to-point networks, where each directed link of the underlying communication graph represents a communication channel between a pair of nodes. We adopt the transient Byzantine link failure model [15, 16], where an omniscient adversary controls a subset of the directed communication links, but the nodes are assumed to be fault-free. Recent work has addressed the problem of reaching approximate consen- sus in incomplete graphs with Byzantine nodes using a restricted class of iterative algorithms that maintain only a small amount of memory across iterations [22, 21, 23, 12]. However, to the best of our knowledge, we are the first to consider approximate consensus in the presence of Byzan- tine links. We extend our past work that provided exact characterization of graphs in which the iterative approximate consensus problem in the presence of Byzantine node failures is solvable [22, 21]. In particular, we prove a tight necessary and sufficient condition on the underlying com- munication graph for the existence of iterative approximate consensus algorithms under transient Byzantine link model. The condition answers (part of) the open problem stated in [16].Comment: arXiv admin note: text overlap with arXiv:1202.609

Troubling identities: teacher education students` constructions of class and ethnicity

Working with diverse student populations productively depends on teachers and teacher educators recognizing and valuing difference. Too often, in teacher education programs, when markers of identity such as gender, ethnicity, \u27race\u27, or social class are examined, the focus is on developing student teachers\u27 understandings of how these discourses shape learner identities and rarely on how these also shape teachers\u27 identities. This article reports on a research project that explored how student teachers understand ethnicity and socio-economic status. In a preliminary stage of the research, we asked eight Year 3 teacher education students who had attended mainly Anglo-Australian, middle class schools as students and as student teachers, to explore their own ethnic and classed identities. The complexities of identity are foregrounded in both the assumptions we made in selecting particular students for the project and in the ways they constructed their own identities around ethnicity and social class. In this article we draw on these findings to interrogate how categories of identity are fluid, shifting and ongoing processes of negotiation, troubling and complex. We also consider the implications for teacher education.<br /

Critical elements in nonsulphide Zn deposits: A reanalysis of the Kabwe Zn-Pb ores (central Zambia)

The Kabwe Zn-Pb deposit (central Zambia) consists of a cluster of mixed sulfide and non-sulfide orebodies. The sulfide ores comprise sphalerite, galena, pyrite, chalcopyrite and accessory Ge-sulfides (±Ga and In). The non-sulfide ores comprise: (1) willemite-dominated zones encasing massive sulfide orebodies and (2) oxide-dominated alteration bands, overlying both the sulfide and Zn-silicate orebodies. This study focuses on the Ge, In and Ga distribution in the non-sulfide mineralization, and was carried out on a suite of Kabwe specimens, housed in the Natural History Museum Ore Collection (London). Petrography confirmed that the original sulfides were overprinted by at least two contrasting oxidation stages dominated by the formation of willemite (W1 and W2), and a further event characterized by weathering-related processes. Oxygen isotopic analyses have shown that W1 and W2 are unrelated genetically and furthermore not related to supergene Zn-Pb-carbonates in the oxide-dominated assemblage. The δ18O composition of 13.9-15.7‰ V-SMOW strongly supports a hydrothermal origin for W1. The δ18O composition of W2 (-3.5‰ to 0‰ V-SMOW) indicates that it precipitated from groundwaters of meteoric origin in either a supergene or a low-T hydrothermal environment. Gallium and Ge show a diversity of distribution among the range of Zn-bearing minerals. Gallium has been detected at the ppm level in W1, sphalerite, goethite and hematite. Germanium occurs at ppm levels in W1 and W2, and in scarcely detectable amounts in hemimorphite, goethite and hematite. Indium has low concentrations in goethite and hematite. These different deportments among the various phases are probably due to the different initial Ga, In and Ge abundances in the mineralization, to the different solubilities of the three elements at different temperatures and pH values, and finally to their variable affinities with the various minerals formed

Valence-bond states in dynamical Jahn-Teller molecular systems

We discuss a hopping model of electrons between idealized molecular sites with local orbital degeneracy and dynamical Jahn-Teller effect, for crystal field environments of sufficiently high symmetry. For the Mott-insulating case (one electron per site and large Coulomb repulsions), in the simplest two-fold degenerate situation, we are led to consider a particular exchange hamiltonian, describing two isotropic spin-1/2 Heisenberg problems coupled by a quartic term on equivalent bonds. This twin-exchange hamiltonian applies to a physical regime in which the inter-orbital singlet is the lowest-energy intermediate state available for hopping. This regime is favored by a relatively strong electron-phonon coupling. Using variational arguments, a large-N limit, and exact diagonalization data, we find that the ground state, in the one dimensional case, is a solid valence bond state. The situation in the two dimensional case is less clear. Finally, the behavior of the system upon hole doping is studied in one dimension.Comment: 11 pages, 5 figure

Fast Partial Distance Estimation and Applications

We study approximate distributed solutions to the weighted {\it all-pairs-shortest-paths} (APSP) problem in the CONGEST model. We obtain the following results. $1.$ A deterministic $(1+o(1))$-approximation to APSP in $\tilde{O}(n)$ rounds. This improves over the best previously known algorithm, by both derandomizing it and by reducing the running time by a $\Theta(\log n)$ factor. In many cases, routing schemes involve relabeling, i.e., assigning new names to nodes and require that these names are used in distance and routing queries. It is known that relabeling is necessary to achieve running times of $o(n/\log n)$. In the relabeling model, we obtain the following results. $2.$ A randomized $O(k)$-approximation to APSP, for any integer $k>1$, running in $\tilde{O}(n^{1/2+1/k}+D)$ rounds, where $D$ is the hop diameter of the network. This algorithm simplifies the best previously known result and reduces its approximation ratio from $O(k\log k)$ to $O(k)$. Also, the new algorithm uses uses labels of asymptotically optimal size, namely $O(\log n)$ bits. $3.$ A randomized $O(k)$-approximation to APSP, for any integer $k>1$, running in time $\tilde{O}((nD)^{1/2}\cdot n^{1/k}+D)$ and producing {\it compact routing tables} of size $\tilde{O}(n^{1/k})$. The node lables consist of $O(k\log n)$ bits. This improves on the approximation ratio of $\Theta(k^2)$ for tables of that size achieved by the best previously known algorithm, which terminates faster, in $\tilde{O}(n^{1/2+1/k}+D)$ rounds

Interplay of Orbital Degeneracy and Superconductivity in a Molecular Conductor

We study electron propagation in a molecular lattice model. Each molecular site involves doubly degenerate electronic states coupled to doubly degenerate molecular vibration, leading to a so--called E-e type of Jahn-Teller Hamiltonian. For weak electron-phonon coupling and in the anti-adiabatic limit we find that the orbital degeneracy induces an intersite pairing mechanism which is absent in the standard non-degenerate polaronic model. In this limit we analyse the model in the presence of an additional on-site repulsion and we determine, within BCS mean field theory, the region of stability of superconductivity. In one dimension, where powerful analytical techniques are available, we are able to calculate the phase diagram of the model both for weak and for strong electron-phonon coupling.Comment: 11 pages, REVTEX style, 3 compressed figures adde