Constructing Light Spanners Deterministically in Near-Linear Time

Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [Shiri Chechik and Christian Wulff-Nilsen, 2018] improved the state-of-the-art for light spanners by constructing a (2k-1)(1+epsilon)-spanner with O(n^(1+1/k)) edges and O_epsilon(n^(1/k)) lightness. Soon after, Filtser and Solomon [Arnold Filtser and Shay Solomon, 2016] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of O(mn^(1+1/k)) (which is faster than [Shiri Chechik and Christian Wulff-Nilsen, 2018]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Omega_epsilon(kn^(1/k)), even when randomization is used. The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an O_epsilon(n^(2+1/k+epsilon\u27)) time spanner construction which achieves the state-of-the-art bounds. Our second result is an O_epsilon(m + n log n) time construction of a spanner with (2k-1)(1+epsilon) stretch, O(log k * n^(1+1/k) edges and O_epsilon(log k * n^(1/k)) lightness. This is an exponential improvement in the dependence on k compared to the previous result with such running time. Finally, for the important special case where k=log n, for every constant epsilon>0, we provide an O(m+n^(1+epsilon)) time construction that produces an O(log n)-spanner with O(n) edges and O(1) lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any k = omega(1). To achieve our constructions, we show a novel deterministic incremental approximate distance oracle. Our new oracle is crucial in our construction, as known randomized dynamic oracles require the assumption of a non-adaptive adversary. This is a strong assumption, which has seen recent attention in prolific venues. Our new oracle allows the order of the edge insertions to not be fixed in advance, which is critical as our spanner algorithm chooses which edges to insert based on the answers to distance queries. We believe our new oracle is of independent interest

Stable isotope (δ18O and δ13C) sclerochronology of Callovian (Middle Jurassic) bivalves (Gryphaea (Bilobissa) dilobotes) and belemnites (Cylindroteuthis puzosiana) from the Peterborough Member of the Oxford Clay Formation (Cambridgeshire, England): Evidence of palaeoclimate, water depth and belemnite behaviour

Incremental δ18O and δ13C signals were obtained from three well-preserved specimens of Cylindroteuthis puzosiana and from three well-preserved specimens of Gryphaea (Bilobissa) dilobotes from the Peterborough Member of the Oxford Clay Formation (Cambridgeshire, England). Through-ontogeny (sclerochronological) δ18O data from G. (B.) dilobotes appear to faithfully record seasonal temperature variations in benthic Callovian waters of the study area, which range from c. 14 °C to c. 17 °C (arithmetic mean temperature c. 15 °C). Water depth is estimated to have been in the region of c. 50 m, based upon comparisons between these data, previously published non-incremental sea surface δ18O values, and a modern analogue situation. Productivity in Callovian waters was comparable with that in modern seas, based upon δ13C data from G. (B.)dilobotes, with 13C depletion occurring during warmer periods, possibly related to an interaction between plankton blooms and intra-annual variations in mixing across a thermocline. Incremental δ18O data from C.puzosiana provide temperature minima of c.11 °C for all specimens but with maxima varying between c.14 °C and c.16 °C for different individuals (arithmetic mean values c. 13 °C). Temperatures for late ontogeny, when the C. puzosiana individuals must have been living close to the study site and hence the analysed specimens of G. (B.) dilobotes, are closely comparable to those indicated by the latter. However, for significant portions of ontogeny C. puzosiana experienced temperatures between c. 2 °C and c. 3 °C cooler than the winter minimum as recorded by co-occurring G. (B.) dilobotes. Comparisons with modern seas suggest that descent to a depth of c. 1000 m would be necessary to explain such cool minimum temperatures. This can be discounted due to the lack of deep waters locally and due to estimates of the depth tolerance of belemnites. The most likely cause of cool δ18O signals from C. puzosiana is a cosmopolitan lifestyle including migration to more northerly latitudes. Mean δ13C values from C. puzosiana are comparable with those from G.(B.)dilobotes. However, the incrementally acquired data are highly variable and probably influenced by metabolic effects.The probable identification of migratory behaviour in C. puzosiana calls into question the reliability of some belemnite species as place-specific palaeoenvironmental archives and highlights the benefits of adopting a sclerochronological approach

Nuclear applications in manned space station

The zirconium hydride reactor, coupled to a thermo-electric or Brayton conversion system, and the Pu 238 isotope/Brayton system, are considered to be the viable nuclear candidates for the modular space station electrical power system. The basic integration aspects of these nuclear electrical power systems are reviewed, including unique requirements imposed by the buildup and incremental utilization considerations of the modular station. Also treated are the various programmatic aspects of nuclear power system design and selection

A Comparative Evaluation of the Internets Influence on International Market Penetration and Development Strategies of Australian SME's

The Internet has the capability to generate geographical international market expansion and future growth for the firm, a concept known as Internetalisation. However, it is yet to be determined as to how much or to what level the Internet influences internationalisation, and thus international market growth. Both international market penetration and the development of new international customers are achievable goals for the Internet enlightened SME. The aim of this research is to explore the influence of the Internet on international market penetration and development from the strategic perspective of the SME in Australia. It was found that although the Internet has given firms the capabilities to become instantly international, a new theory is not needed but rather an evolved version of network theory may be a better explanation of internationalisation of SME’s in today’s digital environment

The invariances of power law size distributions

Size varies. Small things are typically more frequent than large things. The logarithm of frequency often declines linearly with the logarithm of size. That power law relation forms one of the common patterns of nature. Why does the complexity of nature reduce to such a simple pattern? Why do things as different as tree size and enzyme rate follow similarly simple patterns? Here I analyze such patterns by their invariant properties. For example, a common pattern should not change when adding a constant value to all observations. That shift is essentially the renumbering of the points on a ruler without changing the metric information provided by the ruler. A ruler is shift invariant only when its scale is properly calibrated to the pattern being measured. Stretch invariance corresponds to the conservation of the total amount of something, such as the total biomass and consequently the average size. Rotational invariance corresponds to pattern that does not depend on the order in which underlying processes occur, for example, a scale that additively combines the component processes leading to observed values. I use tree size as an example to illustrate how the key invariances shape pattern. A simple interpretation of common pattern follows. That simple interpretation connects the normal distribution to a wide variety of other common patterns through the transformations of scale set by the fundamental invariances.Comment: Added appendix discussing the lognormal distribution, updated to match version 2 of published version at F1000Researc

Aerobic capacity, activity levels and daily energy expenditure in male and female adolescents of the kenyan nandi sub-group

The relative importance of genetic and socio-cultural influences contributing to the success of east Africans in endurance athletics remains unknown in part because the pre-training phenotype of this population remains incompletely assessed. Here cardiopulmonary fitness, physical activity levels, distance travelled to school and daily energy expenditure in 15 habitually active male (13.9±1.6 years) and 15 habitually active female (13.9±1.2) adolescents from a rural Nandi primary school are assessed. Aerobic capacity ([Formula: see text]) was evaluated during two maximal discontinuous incremental exercise tests; physical activity using accelerometry combined with a global positioning system; and energy expenditure using the doubly labelled water method. The [Formula: see text] of the male and female adolescents were 73.9±5.7 ml(.) kg(-1.) min(-1) and 61.5±6.3 ml(.) kg(-1.) min(-1), respectively. Total time spent in sedentary, light, moderate and vigorous physical activities per day was 406±63 min (50% of total monitored time), 244±56 min (30%), 75±18 min (9%) and 82±30 min (10%). Average total daily distance travelled to and from school was 7.5±3.0 km (0.8-13.4 km). Mean daily energy expenditure, activity-induced energy expenditure and physical activity level was 12.2±3.4 MJ(.) day(-1), 5.4±3.0 MJ(.) day(-1) and 2.2±0.6. 70.6% of the variation in [Formula: see text] was explained by sex (partial R(2) = 54.7%) and body mass index (partial R(2) = 15.9%). Energy expenditure and physical activity variables did not predict variation in [Formula: see text] once sex had been accounted for. The highly active and energy-demanding lifestyle of rural Kenyan adolescents may account for their exceptional aerobic fitness and collectively prime them for later training and athletic success