Long cycles and paths in distance graphs

AbstractFor n∈N and D⊆N, the distance graph PnD has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant cD such that the greatest common divisor of the integers in D is 1 if and only if for every n, PnD has a component of order at least n−cD if and only if for every n≥cD+3, PnD has a cycle of order at least n−cD. Furthermore, we discuss some consequences and variants of this result

Connectivity and diameter in distance graphs

For $n\in \mathbb{N}$ and $D\subseteq \mathbb{N}$, the distance graph $P_n^D$ has vertex set $\{ 0,1,\ldots,n-1\}$ and edge set $\{ ij\mid 0\leq i,j\leq n-1, |j-i|\in D\}$. The class of distance graphs generalizes the important and very well-studied class of circulant graphs which have been proposed for numerous network applications. In view of fault tolerance and delay issues in these applications, the connectivity and diameter of circulant graphs have been studied in great detail. Our main contributions are hardness results concerning computational problems related to the connectivity and diameter of distance graphs and a number-theoretic characterization of the connected distance graphs $P_n^D$ for $|D|=2$

Transforming Asynchronous Systems with Crash-Stop Failures and Failure Detectors to the General Omission Model

This paper studies the impact of omission failures on asynchronous distributed s ystems with crash-stop failures. For the large group of problem specifications that are restricted to correct processes, we show how to transform a crash-stop related problem specification into an equivalent omission one. For that, we provide transformations for algorithms and failure detectors, such that if and only if an algorithm using a failure detector satisfies a problem specification, then the transformed algorithm using the transformed failure detector satisfies the transformed problem specification. Our transformed problem specification is ensured to be non-trivial, and moreover, the transformation reveals itself to be in a reasonable sense weakest failure detector preserving. Our results help to use the power of the well-understood crash-stop model to aut omatically derive solutions for the general omission model, which has recently raised interest for being noticeably applicable for security problems in distributed environments equipped with security modules such as smartcards

Optimal Randomized Fair Exchange with Secret Shared Coins

In the fair exchange problem, mutually untrusting parties must securely exchange digital goods. A fair exchange protocol must ensure that no combination of cheating or failures will result in some goods being delivered but not others, and that all goods will be delivered in the absence of cheating and failures

Tight bounds for k-set agreement with limited scope accuracy failure detectors

In the k-set agreement problem [1], each process in a group starts with a private input value, communicates with the others, and then halts after choosing a private output value. Each process is required to choose some process's input, and at most k distinct values may be chosen. We consider this problem in an asynchronous message-passing system of n + 1 processes, of which at most f may fail by halting. A failure detector- [5] is an unreliable oracle that informs each process when it suspects other processes to have failed. In practice, however, a process can typically detect some failures more easily than others. For exam-ple, timeouts may reliably detect failures on the same local area network, but less reliably over a wide-area network. A limited-scope failure detector model, the set of processes encompasses sets X0,..., Xq _ 1, such that some correct pro

Efficient Reductions for Wait-Free Termination Detection in Faulty Distributed Systems

We investigate the problem of detecting termination of a distributed computation in asynchronous systems where processes can fail by crashing. More specifically, for both fully and arbitrarily connected communication topologies, we describe efficient ways to transform any fault-sensitive termination detection algorithm that has been designed for a failure-free environment , into a wait-free that tolerates up to any number of process crashes. The transformations are such that a competitive fault-sensitive termination detection algorithm results in a competitive wait-free termination detection algorithm B