Colouring random graphs and maximising local diversity

We study a variation of the graph colouring problem on random graphs of finite average connectivity. Given the number of colours, we aim to maximise the number of different colours at neighbouring vertices (i.e. one edge distance) of any vertex. Two efficient algorithms, belief propagation and Walksat are adapted to carry out this task. We present experimental results based on two types of random graphs for different system sizes and identify the critical value of the connectivity for the algorithms to find a perfect solution. The problem and the suggested algorithms have practical relevance since various applications, such as distributed storage, can be mapped onto this problem.Comment: 10 pages, 10 figure

Minimizing Unsatisfaction in Colourful Neighbourhoods

Colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance. Here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods, given a predetermined number of colours. In the analytical framework of a tree approximation, carried out at both zero and finite temperatures, solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition, which are consistent with those of existing algorithms. The nature of the transition as well as the validity of the tree approximation are investigated.Comment: 28 pages, 12 figures, substantially revised with additional explanatio

Simulating chemistry efficiently on fault-tolerant quantum computers

Quantum computers can in principle simulate quantum physics exponentially faster than their classical counterparts, but some technical hurdles remain. Here we consider methods to make proposed chemical simulation algorithms computationally fast on fault-tolerant quantum computers in the circuit model. Fault tolerance constrains the choice of available gates, so that arbitrary gates required for a simulation algorithm must be constructed from sequences of fundamental operations. We examine techniques for constructing arbitrary gates which perform substantially faster than circuits based on the conventional Solovay-Kitaev algorithm [C.M. Dawson and M.A. Nielsen, \emph{Quantum Inf. Comput.}, \textbf{6}:81, 2006]. For a given approximation error $\epsilon$, arbitrary single-qubit gates can be produced fault-tolerantly and using a limited set of gates in time which is $O(\log \epsilon)$ or $O(\log \log \epsilon)$; with sufficient parallel preparation of ancillas, constant average depth is possible using a method we call programmable ancilla rotations. Moreover, we construct and analyze efficient implementations of first- and second-quantized simulation algorithms using the fault-tolerant arbitrary gates and other techniques, such as implementing various subroutines in constant time. A specific example we analyze is the ground-state energy calculation for Lithium hydride.Comment: 33 pages, 18 figure

Hybrid Approaches for Distributed Storage Systems

International audienceDistributed or peer-to-peer storage solutions rely on the introduction of redundant data to be fault-tolerant and to achieve high reliability. One way to introduce redundancy is by simple replication. This strategy allows an easy and fast access to data, and a good bandwidth e ciency to repair the missing redundancy when a peer leaves or fails in high churn systems. However, it is known that erasure codes, like Reed-Solomon, are an e - cient solution in terms of storage space to obtain high durability when compared to replication. Recently, the Regenerating Codes were proposed as an improvement of erasure codes to better use the available bandwidth when reconstructing the missing information. In this work, we compare these codes with two hybrid approaches. The rst was already proposed and mixes erasure codes and replication. The second one is a new proposal that we call Double Coding. We compare these approaches with the traditional Reed-Solomon code and also Regenerating Codes from the point of view of availability, durability and storage space. This comparison uses Markov Chain Models that take into account the reconstruction time of the systems

Synchronous Byzantine Agreement with Expected O(1)O(1) Rounds, Expected O(n2)O(n^2) Communication, and Optimal Resilience

We present new protocols for Byzantine agreement in the synchronous and authenticated setting, tolerating the optimal number of $f$ faults among $n=2f+1$ parties. Our protocols achieve an expected $O(1)$ round complexity and an expected $O(n^2)$ communication complexity. The exact round complexity in expectation is 10 for a static adversary and 16 for a strongly rushing adaptive adversary. For comparison, previous protocols in the same setting require expected 29 rounds

Competitive Hill-Climbing Strategies for Replica Placement in a Distributed File System

1 Introduction This paper analyzes algorithms for automated placement of file replicas in the Farsite [3] system, using both theory and simulation. In the Farsite distributed file system, multiple replicas of files are stored on multiple machines, so that files can be accessed even if some of the machines are down or inaccessible. The purpose of the placement algorithm is to determine an assignment of file replicas to machines that maximally exploits the availability provided by machines. The file placement algorithm is given a fixed value, R, for the number of replicas of each file. For systems reasons, we are most interested in a value of R = 3 [9]. However, to ensure that our results are not excessively sensitive to the file replication factor, we also provide tight bounds for R = 2 and lower bounds for all R (tight at different values of R)