531 research outputs found

    Simpler, faster and shorter labels for distances in graphs

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    We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different trade-offs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades. More specifically, we present a distance labeling scheme with label size (log 3)/2 + o(n) (logarithms are in base 2) and O(1) decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n/log n) decoding time, and Gavoille et al. (SODA'01), which uses labels of size 11n + o(n) and O(loglog n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W, we present almost matching upper and lower bounds for label sizes. For r-additive approximation schemes, where distances can be off by an additive constant r, we give both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency scheme: n/2. We also give results for bipartite graphs and for exact and 1-additive distance oracles

    Microstrip Resonator for High Field MRI with Capacitor-Segmented Strip and Ground Plane

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    Relevant videnskabsteori - Fagets videnskabsteori i kuhniansk perspektiv

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    Mens der er krav om, at fagets eller fagområdets videnskabsteori skal indgå i bacheloruddannelserne på de danske universiteter, er det lagt ud til de ansvarlige for de enkelte uddannelser at fastlægge fagets konkrete indhold. På grundlag af Thomas Kuhns beskrivelse af naturvidenskaben kan man dog fremføre gode grunde til, at undervisning i almene videnskabsteoretiske begreber og spørgsmål bør være en del af faget på de naturvidenskabelige uddannelser, da de studerende ellers vil mangle det teoretiske grundlag for at reflektere kritisk over deres egne fags metoder og forstå andre fags videnskabelige tilgange. Samtidig er undervisning i almen videnskabsteori på disse uddannelser imidlertid forbundet med den udfordring, at de studerende vil have en tendens til ikke at betragte den som relevant for deres videnskabelige profession. Hvis de studerende skal opnå det videnskabsteoretiske grundlag for tværfaglig forståelse og kritisk faglig refleksion, er det derfor afgørende, at undervisningen i almen videnskabsteori sigter efter at forbinde de abstrakte, overordnede diskussioner fra videnskabsteorien med spørgsmål fra konkret videnskabelig praksis. While every bachelor education in the natural sciences at a Danish university must include a course in the philosophy of science of the subject area, it is left to those responsible for each individual bachelor education to determine the specific contents of these courses. Based on Thomas Kuhn’s description of the natural sciences, however, there are good reasons to include general philosophy of science in the curricula for all bachelor courses in the sciences. In particular, to ensure that students have the theoretical resources they need to reflect on their own scientific methods and to understand how scientific investigation is approached in other subject areas. But teaching such general philosophy of science courses to science students comes with a challenge: Many students feel that the content is not relevant to their scientific profession. Our conclusion is that for all science students to benefit from the teaching of philosophy as part of their bachelor courses, it is crucial that general philosophy of science is taught in a way which aims to connect the abstract, general discussions in the philosophy of science with questions from concrete scientific practice

    Relevant videnskabsteori - Fagets videnskabsteori i kuhniansk perspektiv

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    Mens der er krav om, at fagets eller fagområdets videnskabsteori skal indgå i bacheloruddannelserne på de danske universiteter, er det lagt ud til de ansvarlige for de enkelte uddannelser at fastlægge fagets konkrete indhold. På grundlag af Thomas Kuhns beskrivelse af naturvidenskaben kan man dog fremføre gode grunde til, at undervisning i almene videnskabsteoretiske begreber og spørgsmål bør være en del af faget på de naturvidenskabelige uddannelser, da de studerende ellers vil mangle det teoretiske grundlag for at reflektere kritisk over deres egne fags metoder og forstå andre fags videnskabelige tilgange. Samtidig er undervisning i almen videnskabsteori på disse uddannelser imidlertid forbundet med den udfordring, at de studerende vil have en tendens til ikke at betragte den som relevant for deres videnskabelige profession. Hvis de studerende skal opnå det videnskabsteoretiske grundlag for tværfaglig forståelse og kritisk faglig refleksion, er det derfor afgørende, at undervisningen i almen videnskabsteori sigter efter at forbinde de abstrakte, overordnede diskussioner fra videnskabsteorien med spørgsmål fra konkret videnskabelig praksis. While every bachelor education in the natural sciences at a Danish university must include a course in the philosophy of science of the subject area, it is left to those responsible for each individual bachelor education to determine the specific contents of these courses. Based on Thomas Kuhn’s description of the natural sciences, however, there are good reasons to include general philosophy of science in the curricula for all bachelor courses in the sciences. In particular, to ensure that students have the theoretical resources they need to reflect on their own scientific methods and to understand how scientific investigation is approached in other subject areas. But teaching such general philosophy of science courses to science students comes with a challenge: Many students feel that the content is not relevant to their scientific profession. Our conclusion is that for all science students to benefit from the teaching of philosophy as part of their bachelor courses, it is crucial that general philosophy of science is taught in a way which aims to connect the abstract, general discussions in the philosophy of science with questions from concrete scientific practice
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