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Academic Commons Search Resultsen-usFinding a Maximum-Genius Graph Impeding
http://academiccommons.columbia.edu/catalog/ac:142035
Furst, Merrick L.; Gross, Jonathan L.; McGeoch, Lyle A.http://hdl.handle.net/10022/AC:P:11837Mon, 28 Nov 2011 12:37:02 +0000The computational complexity of constructing the imbeddings of a given graph into surfaces of different genus is not well-understood. In this paper, topological methods and a reduction to linear matroid parity are used to develop a polynomial-time algorithm to find a maximum-genus cellular imbedding. This seems to be the first imbedding algorithm for which the running time is not exponential in the genus of the imbedding surface.Computer sciencejlg2Computer Science, StatisticsTechnical reportsAn Information-Theoretic Scale for Cultural Rule Systems
http://academiccommons.columbia.edu/catalog/ac:140503
Gross, Jonathan L.http://hdl.handle.net/10022/AC:P:11478Tue, 18 Oct 2011 11:57:04 +0000Important cultural messages are expressed in nonverbal media such as food, clothing, or the allocation of space or time. For instance, how and what a group of persons eats on a particular occasion may convey public information about that occasion and about the group of persons eating together. Whereas attention seems to be most commonly directed toward the individual character of the information, the present concern is the quantity of public information, as observed in the pattern of nonverbal cultural signs. To measure this quantity, it is proposed that the pattern of cultural signs be encoded as a sequence of abstract symbols (e.g. letters of the alphabet) and its complexity appraised by a suitably adapted form of the measure of Kolmogorov and Chaitin. That is, an algorithmic language is constructed and the mathematical information quantity is reckoned as the length of the shortest program that yields the sequence. In this cultural context, the measure is called "intricacy". By focusing on syntactic structure and pattern variation rather than on background levels, intricacy resists some influences of material wealth that tend to distort comparisons of individuals and groups. A compact mathematical overview of the theory is presented and an experiment to test it within the social medium of food sharing is briefly described.Information science, Sociology, Applied mathematicsjlg2Computer Science, StatisticsTechnical reportsSome Problems in Topographical Graph Theory
http://academiccommons.columbia.edu/catalog/ac:138034
Gross, Jonathan L.; Harary, Frankhttp://hdl.handle.net/10022/AC:P:11055Wed, 31 Aug 2011 14:20:08 +0000This paper approaches computational complexity as the determination of the intrinsic difficulty of mathematically posed problems arising in many disciplines. The study of complexity has led to more efficient algorithms than those previously known or suspected.Computer sciencejlg2Computer Science, StatisticsTechnical reports