Volker Turau

Abstract: It is known for some time that a random graph G(n, p) con- tains w.h.p. a Hamiltonian cycle if p is larger than the critical value pcrit = (log n+log log n+ωn)/n. The determination of a concrete Hamil- tonian cycle is even for values much larger than pcrit a nontrivial task. In this paper we consider random graphs G(n, p) with p in Ω ̃(1/√n), where Ω ̃ hides poly-logarithmic factors in n. For this range of p we present a distributed algorithm AHC that finds w.h.p. a Hamiltonian cycle in O(log n) rounds. The algorithm works in the synchronous model and uses messages of size O(log n) and O(log n) memory per node.

Abstract: We introduce a new scheduling problem in distributed computing that we call the DSMS problem. We are given a set of k ≥ 1 mobile identical servers that are initially hosted by some processors of the given network. Further, we are given a set of requests that are issued by the processors possibly at any time in an online fashion, and the servers must serve them. A request must be scheduled before the request is served. The delay for scheduling a request is the time taken since the request is issued until it is scheduled. The goal is to minimize the average delay. Navigating mobile servers in a large-scale distributed system needs a scalable location service. We devise the distributed GNN protocol, a novel linked-reversal-based protocol for the DSMS problem that works on overlay trees. We prove that GNN is a starvation-free protocol that correctly integrates locating the servers and synchronizing the concurrent access to the servers despite asynchrony. Further, we analyze the GNN protocol for one-shot executions where all requests are simultaneously issued. We show that when running the GNN protocol on top of a special family of tree topologies---known as hierarchically well-separated trees (HSTs)---we obtain a randomized distributed protocol with an expected competitive ratio of O(log n) on general network topologies for the DSMS problem where n is the number of processors in the given network. From a technical point of view, the main result of our paper shows that the GNN protocol optimally solves the DSMS problem on HSTs for one-shot executions. Our results hold even if communication is asynchronous.