Constant Space and Non-Constant Time in Distributed Computing

While the relationship of time and space is an established topic in traditional centralised com- plexity theory, this is not the case in distributed computing. We aim to remedy this by studying the time and space complexity of algorithms in a weak message-passing model of distributed com- puting. While a constant number of communication rounds implies a constant number of states visited during the execution, the other direction is not clear at all. We show that indeed, there exist non-trivial graph problems that are solvable by constant-space algorithms but that require a non-constant running time. Somewhat surprisingly, this holds even when restricted to the class of only cycle and path graphs. Our work provides us with a new complexity class for distributed computing and raises interesting questions about the existence of further combinations of time and space complexity

Distributed algorithms for edge dominating sets

An edge dominating set for a graph G is a set D of edges such that each edge of G is in D or adjacent to at least one edge in D. This work studies deterministic distributed approximation algorithms for finding minimum-size edge dominating sets. The focus is on anonymous port-numbered networks: there are no unique identifiers, but a node of degree d can refer to its neighbours by integers 1, 2, ..., d. The present work shows that in the port-numbering model, edge dominating sets can be approximated as follows: in d-regular graphs, to within 4 − 6/(d + 1) for an odd d and to within 4 − 2/d for an even d; and in graphs with maximum degree Δ, to within 4 − 2/(Δ − 1) for an odd Δ and to within 4 − 2/Δ for an even Δ. These approximation ratios are tight for all values of d and Δ: there are matching lower bounds.Peer reviewe

Towards Optimal Synchronous Counting

Consider a complete communication network of $n$ nodes, where the nodes receive a common clock pulse. We study the synchronous $c$-counting problem: given any starting state and up to $f$ faulty nodes with arbitrary behaviour, the task is to eventually have all correct nodes counting modulo $c$ in agreement. Thus, we are considering algorithms that are self-stabilizing despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have linear stabilisation time in $f$, (3) use a small number of states, and (4) achieve almost-optimal resilience. Prior algorithms either resort to randomisation, use a large number of states, or have poor resilience. In particular, we achieve an exponential improvement in the space complexity of deterministic algorithms, while still achieving linear stabilisation time and almost-linear resilience.Comment: 17 pages, 2 figure

Effects of Dietary Fat Oxidation Products and Flavonols on Lipoprotein Oxidation

Various studies suggest that oxidative modifications of low density lipoprotein (LDL), and also other lipoproteins, have an important role in the development of atherosclerosis. In addition to the oxidation products formed endogenously, oxidised triacylglycerols (TAG) and oxysterols in the diet contribute to the oxidised lipoproteins found in circulation. However, studies on both the effect of oxidised dietary lipids on lipoprotein lipid oxidation and the reactions that modify oxidised fat after ingestion have been scarce. Studies on the effects of dietary antioxidants on the lipid oxidation in vivo and the risk of atherosclerosis have been inconclusive. More clinical trials are needed to test the importance of lipoprotein oxidation as a cardiovascular risk factor in humans. In the recent years, various methods have been optimised and applied to the analysis of lipid oxidation products in vivo, and information on the molecular structures of oxidised lipids in plasma, lipoproteins and atherosclerotic plaques has started to accumulate. However, specific structures of oxidised TAG molecules present in these tissues and lipoprotein fractions have not been investigated earlier. In the orginal research in this thesis, an approach based on highperformance liquid chromatographyelectrospray ionisationmass spectrometry (HPLCESIMS) and baseline diene conjugation (BDC) methods was used in order to investigate lipid oxidation level and oxidised TAG molecular structures in pig and human lipoproteins after dietary interventions. The approach was optimised with human LDL samples, which contained various oxidation products of TAG. LDL particles of hyperlipidaemic subjects contained an elevated amount of conjugated dienes. In the pig studies, several oxidised TAG structures with hydroxy, keto, epoxy or aldehydic groups were found in chylomicrons and VLDL after diets rich in sunflower seed oil. Also, the results showed that oxidised sunflower seed oil increased the oxidation of lipoprotein lipids and their TAG molecules. TAG hydroperoxides could be detected neither in the small intestinal mucosa of the pigs fed on the oxidised oil nor in their chylomicrons or VLDL.6 In the clinical studies, dietary flavonol aglycones extracted from sea buckthorn berries did not have an effect on lipoprotein lipid oxidation and other potential risk factors of atherosclerosis, but their absorption was demonstrated. Oil supplementation seemed to increase the bioavailability of the flavonols. Oxidised TAG molecules were detected in LDL particles of the subjects after both flavonol and control diets.Siirretty Doriast