We give improved algorithms for constructing minimum directed and undirected cycle bases in graphs. For general graphs, the new algorithms are Monte Carlo and have running time O(m ω ), where ω is the exponent of matrix multiplication. The previous best algorithm had running timeÕ(m 2 n). For planar graphs, the new algorithm is deterministic and has running time O(n 2 ). The previous best algorithm had running time O(n 2 log n). A key ingredient to our improved running times is the insight that the search for minimum bases can be restricted to a set of candidate cycles of total length O(nm).

Most of the empirical studies in the literature on intra-industry trade and on the factors affecting trade are performed on the country level. Countries, however, differ in terms of granularity and internal heterogeneity at the regional level. This internal differentiation in terms of intra industry (IIT) patterns, which could affect countries' overall trade pattern, is thus not typically taken into account. In contrast, in the present study -using a unique dataset -we conduct an analysis at the level of NUTS2 regions of two large EU Member States (Poland and Spain) of similar size, level of development, a number of regions and the extent of international regional diversity. This allows drawing more thorough and robust conclusions, as regards the nature of IIT and its determinants. IIT is measured at the 4-digit level of products CN classification. We first describe the overall pattern of IIT for regions, and then empirically identify the determinants of overall IIT as well as its horizontal and vertical components in trade the Spanish and Polish NUTS-2 regions with all existing trade partners on bilateral basis over the period 2005-2014. In order to obtain unbiased results, we utilise a novel empirical approach -a semi-mixed effect model, estimated with the Poisson Pseudo Maximum Likelihood estimator.We estimate the models jointly for all Spanish and Polish regions and then disjointly in a comparative manner -in order to identify incongruities of reaction to various factors investigated. These include both traditional factors, postulated by the standard theoretical models, as well as a number of factors related to the regional dimension of our analysis such as regional path dependence, quality of regional institutions or the core or peripheral nature of reporting region. The study contributes significantly to the analysis of determinants of IIT. We go beyond the traditional approach to IIT analysis (focused on countries). By treating regions as small open economies, participating in international trade, we are able to show new, interesting aspects of IIT and its determinants.

Modern computers are not Random Access Machines (RAMs). They have a memory hierarchy, multiple cores, and a virtual memory. We address the computational cost of the address translation in the virtual memory. The starting point for our work on virtual memory is the observation that the analysis of some simple algorithms (random scan of an array, binary search, heapsort) in either the RAM model or the External Memory (EM) model does not correctly predict growth rates of actual running times. We propose the Virtual Address Translation (VAT) model to account for the cost of address translations and analyze the algorithms mentioned and others in the model. The predictions agree with the measurements. We also analyze the VAT-cost of cache-oblivious algorithms.

Modern computers are not random access machines (RAMs). They have a memory hierarchy, multiple cores, and virtual memory. In this paper, we address the computational cost of address translation in virtual memory. Starting point for our work is the observation that the analysis of some simple algorithms (random scan of an array, binary search, heapsort) in either the RAM model or the EM model (external memory model) does not correctly predict growth rates of actual running times. We propose the VAT model (virtual address translation) to account for the cost of address translations and analyze the algorithms mentioned above and others in the model. The predictions agree with the measurements. We also analyze the VAT-cost of cacheoblivious algorithms.

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