Optimisation Problems in Wireless Sensor Networks : Local Algorithms and Local Graphs

This thesis studies optimisation problems related to modern large-scale distributed systems, such as wireless sensor networks and wireless ad-hoc networks. The concrete tasks that we use as motivating examples are the following: (i) maximising the lifetime of a battery-powered wireless sensor network, (ii) maximising the capacity of a wireless communication network, and (iii) minimising the number of sensors in a surveillance application. A sensor node consumes energy both when it is transmitting or forwarding data, and when it is performing measurements. Hence task (i), lifetime maximisation, can be approached from two different perspectives. First, we can seek for optimal data flows that make the most out of the energy resources available in the network; such optimisation problems are examples of so-called max-min linear programs. Second, we can conserve energy by putting redundant sensors into sleep mode; we arrive at the sleep scheduling problem, in which the objective is to find an optimal schedule that determines when each sensor node is asleep and when it is awake. In a wireless network simultaneous radio transmissions may interfere with each other. Task (ii), capacity maximisation, therefore gives rise to another scheduling problem, the activity scheduling problem, in which the objective is to find a minimum-length conflict-free schedule that satisfies the data transmission requirements of all wireless communication links. Task (iii), minimising the number of sensors, is related to the classical graph problem of finding a minimum dominating set. However, if we are not only interested in detecting an intruder but also locating the intruder, it is not sufficient to solve the dominating set problem; formulations such as minimum-size identifying codes and locating dominating codes are more appropriate. This thesis presents approximation algorithms for each of these optimisation problems, i.e., for max-min linear programs, sleep scheduling, activity scheduling, identifying codes, and locating dominating codes. Two complementary approaches are taken. The main focus is on local algorithms, which are constant-time distributed algorithms. The contributions include local approximation algorithms for max-min linear programs, sleep scheduling, and activity scheduling. In the case of max-min linear programs, tight upper and lower bounds are proved for the best possible approximation ratio that can be achieved by any local algorithm. The second approach is the study of centralised polynomial-time algorithms in local graphs these are geometric graphs whose structure exhibits spatial locality. Among other contributions, it is shown that while identifying codes and locating dominating codes are hard to approximate in general graphs, they admit a polynomial-time approximation scheme in local graphs.

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.

Optimisation problems in wireless sensor networks

This thesis studies optimisation problems related to modern large-scale distributed systems, such as wireless sensor networks and wireless ad-hoc networks. The concrete tasks that we use as motivating examples are the following: (i) maximising the lifetime of a battery-powered wireless sensor network, (ii) maximising the capacity of a wireless communication network, and (iii) minimising the number of sensors in a surveillance application. A sensor node consumes energy both when it is transmitting or forwarding data, and when it is performing measurements. Hence task (i), lifetime maximisation, can be approached from two different perspectives. First, we can seek for optimal data flows that make the most out of the energy resources available in the network; such optimisation problems are examples of so-called max-min linear programs. Second, we can conserve energy by putting redundant sensors into sleep mode; we arrive at the sleep scheduling problem, in which the objective is to find an optimal schedule that determines when each sensor node is asleep and when it is awake. In a wireless network simultaneous radio transmissions may interfere with each other. Task (ii), capacity maximisation, therefore gives rise to another scheduling problem, the activity scheduling problem, in which the objective is to find a minimum-length conflict-free schedule that satisfies the data transmission requirements of all wireless communication links. Task (iii), minimising the number of sensors, is related to the classical graph problem of finding a minimum dominating set. However, if we are not only interested in detecting an intruder but also locating the intruder, it is not sufficient to solve the dominating set problem; formulations such as minimum-size identifying codes and locating–dominating codes are more appropriate. This thesis presents approximation algorithms for each of these optimisation problems, i.e., for max-min linear programs, sleep scheduling, activity scheduling, identifying codes, and locating–dominating codes. Two complementary approaches are taken. The main focus is on local algorithms, which are constant-time distributed algorithms. The contributions include local approximation algorithms for max-min linear programs, sleep scheduling, and activity scheduling. In the case of max-min linear programs, tight upper and lower bounds are proved for the best possible approximation ratio that can be achieved by any local algorithm. The second approach is the study of centralised polynomial-time algorithms in local graphs – these are geometric graphs whose structure exhibits spatial locality. Among other contributions, it is shown that while identifying codes and locating–dominating codes are hard to approximate in general graphs, they admit a polynomial-time approximation scheme in local graphs.

Comparing type counts

This work is a case study of applying nonparametric statistical methods to corpus data. We show how to use ideas from permutation testing to answer linguistic questions related to morphological productivity and type richness. In particular, we study the use of the suffixes -ity and -ness in the 17th-century part of the Corpus of Early English Correspondence within the framework of historical sociolinguistics. Our hypothesis is that the productivity of -ity, as measured by type counts, is significantly low in letters written by women. To test such hypotheses, and to facilitate exploratory data analysis, we take the approach of computing accumulation curves for types and hapax legomena. We have developed an open source computer program which uses Monte Carlo sampling to compute the upper and lower bounds of these curves for one or more levels of statistical significance. By comparing the type accumulation from women’s letters with the bounds, we are able to confirm our hypothesis.

Aspects of Inflationary Models at Low Energy Scales

Cosmological inflation is the dominant paradigm in explaining the origin of structure in the universe. According to the inflationary scenario, there has been a period of nearly exponential expansion in the very early universe, long before the nucleosynthesis. Inflation is commonly considered as a consequence of some scalar field or fields whose energy density starts to dominate the universe. The inflationary expansion converts the quantum fluctuations of the fields into classical perturbations on superhorizon scales and these primordial perturbations are the seeds of the structure in the universe. Moreover, inflation also naturally explains the high degree of homogeneity and spatial flatness of the early universe. The real challenge of the inflationary cosmology lies in trying to establish a connection between the fields driving inflation and theories of particle physics. In this thesis we concentrate on inflationary models at scales well below the Planck scale. The low scale allows us to seek for candidates for the inflationary matter within extensions of the Standard Model but typically also implies fine-tuning problems. We discuss a low scale model where inflation is driven by a flat direction of the Minimally Supersymmetric Standard Model. The relation between the potential along the flat direction and the underlying supergravity model is studied. The low inflationary scale requires an extremely flat potential but we find that in this particular model the associated fine-tuning problems can be solved in a rather natural fashion in a class of supergravity models. For this class of models, the flatness is a consequence of the structure of the supergravity model and is insensitive to the vacuum expectation values of the fields that break supersymmetry. Another low scale model considered in the thesis is the curvaton scenario where the primordial perturbations originate from quantum fluctuations of a curvaton field, which is different from the fields driving inflation. The curvaton gives a negligible contribution to the total energy density during inflation but its perturbations become significant in the post-inflationary epoch. The separation between the fields driving inflation and the fields giving rise to primordial perturbations opens up new possibilities to lower the inflationary scale without introducing fine-tuning problems. The curvaton model typically gives rise to relatively large level of non-gaussian features in the statistics of primordial perturbations. We find that the level of non-gaussian effects is heavily dependent on the form of the curvaton potential. Future observations that provide more accurate information of the non-gaussian statistics can therefore place constraining bounds on the curvaton interactions.

Exact bounds for distributed graph colouring

A distributed system is a collection of networked autonomous processing units which must work in a cooperative manner. Currently, large-scale distributed systems, such as various telecommunication and computer networks, are abundant and used in a multitude of tasks. The field of distributed computing studies what can be computed efficiently in such systems. Distributed systems are usually modelled as graphs where nodes represent the processors and edges denote communication links between processors. This thesis concentrates on the computational complexity of the distributed graph colouring problem. The objective of the graph colouring problem is to assign a colour to each node in such a way that no two nodes connected by an edge share the same colour. In particular, it is often desirable to use only a small number of colours. This task is a fundamental symmetry-breaking primitive in various distributed algorithms. A graph that has been coloured in this manner using at most k different colours is said to be k-coloured. This work examines the synchronous message-passing model of distributed computation: every node runs the same algorithm, and the system operates in discrete synchronous communication rounds. During each round, a node can communicate with its neighbours and perform local computation. In this model, the time complexity of a problem is the number of synchronous communication rounds required to solve the problem. It is known that 3-colouring any k-coloured directed cycle requires at least ½(log* k - 3) communication rounds and is possible in ½(log* k + 7) communication rounds for all k ≥ 3. This work shows that for any k ≥ 3, colouring a k-coloured directed cycle with at most three colours is possible in ½(log* k + 3) rounds. In contrast, it is also shown that for some values of k, colouring a directed cycle with at most three colours requires at least ½(log* k + 1) communication rounds. Furthermore, in the case of directed rooted trees, reducing a k-colouring into a 3-colouring requires at least log* k + 1 rounds for some k and possible in log* k + 3 rounds for all k ≥ 3. The new positive and negative results are derived using computational methods, as the existence of distributed colouring algorithms corresponds to the colourability of so-called neighbourhood graphs. The colourability of these graphs is analysed using Boolean satisfiability (SAT) solvers. Finally, this thesis shows that similar methods are applicable in capturing the existence of distributed algorithms for other graph problems, such as the maximal matching problem.

Lower Gastrointestinal Microbiota in Health and Irritable Bowel Syndrome : Characterisation and Effect of Probiotic Intervention

The human gastrointestinal (GI) microbiota is a complex ecosystem that lives in symbiosis with its host. The growing awareness of the importance of the microbiota to the host as well as the development of culture-free laboratory techniques and computational methods has enormously expanded our knowledge of this microbial community. Irritable bowel syndrome (IBS) is a common functional bowel disorder affecting up to a fifth of the Western population. To date, IBS diagnosis has been based on GI symptoms and the exclusion of organic diseases. The GI microbiota has been found to be altered in this syndrome and probiotics can alleviate the symptoms, although clear links between the symptoms and the microbiota have not been demonstrated. The aim of the present work was to characterise IBS related alterations in the intestinal microbiota, their relation to IBS symptoms and their responsiveness to probiotic theraphy. In this thesis research, the healthy human microbiota was characterised by cloning and sequencing 16S rRNA genes from a faecal microbial community DNA pool that was first profiled and fractionated according to its guanine and cytosine content (%G+C). The most noticeable finding was that the high G+C Gram-positive bacteria (the phylum Actinobacteria) were more abundant compared to a corresponding library constructed from the unfractionated DNA pool sample. Previous molecular analyses of the gut microbiota have also shown comparatively low amounts of high G+C bacteria. Furthermore, the %G+C profiling approach was applied to a sample constructed of faecal DNA from diarrhea-predominant IBS (IBS-D) subjects. The phylogenetic microbial community comparison performed for healthy and IBS-D sequence libraries revealed that the IBS-D sample was rich in representatives of the phyla Firmicutes and Proteobacteria whereas Actinobacteria and Bacteroidetes were abundant in the healthy subjects. The family Lachnospiraceae within the Firmicutes was especially prevalent in the IBS-D sample. Moreover, associations of the GI microbiota with intestinal symptoms and the quality of life (QOL) were investigated, as well as the effect of probiotics on these factors. The microbial targets that were analysed with the quantitative real-time polymerase chain reaction (qPCR) in this study were phylotypes (species definition according to 16S rRNA gene sequence similarity) previously associated with either health or IBS. With a set of samples, the presence or abundance of a phylotype that had 94% 16S rRNA gene sequence similarity to Ruminococcus torques (R. torques 94%) was shown to be associated with the severity of IBS symptoms. The qPCR analyses for selected phylotypes were also applied to samples from a six-month probiotic intervention with a mixture of Lactobacillus rhamnosus GG, L. rhamnosus Lc705, Propionibacterium freudenreichii ssp. shermanii JS and Bifidobacterium breve Bb99. The intervention had been previously reported to alleviate IBS symptoms, but no associations with the analysed microbiota representatives were shown. However, with the phylotype-specific assays applied here, the abundance of the R. torques 94% -phylotype was shown to be lowered in the probiotic-receiving group during the probiotic supplementation, whereas a Clostridium thermosuccinogenes 85% phylotype, previously associated with a healthy microbiota, was found to be increased compared to the placebo group. To conclude, with the combination of methods applied, higher abundance of Actinobacteria was detected in the healthy gut than found in previous studies, and significant phylum-level microbiota alterations could be shown in IBS-D. Thus, the results of this study provide a detailed overview of the human GI microbiota in healthy subjects and in subjects with IBS. Furthermore, the IBS symptoms were linked to a particular clostridial phylotype, and probiotic supplementation was demonstrated to alter the GI microbiota towards a healthier state with regard to this and an additional bacterial phylotype. For the first time, distinct phylotype-level alterations in the microbiota were linked to IBS symptoms and shown to respond to probiotic therapy.