{"id":{"repo_id":"milano","oai_identifier":"oai:air.unimi.it:2434/150055"},"canonical_url":"https://search.dev.ndltd.org/etd/milano/oai:air.unimi.it:2434/150055","repository":{"repo_id":"milano","name":"Università degli Studi di Milano","base_url":"https://air.unimi.it/oai/request"},"display":{"title":"PROBLEMI DI CLUSTERING CON VINCOLI: ALGORITMI E COMPLESSITÀ","abstract":"This thesis introduces and studies the problem of 1-dimensional bounded clustering: for any fixed p ≥ 1, given reals x1, x2..., xn, and integers k1, k2.., km, determine the partition (A1, A2... Am) of {1, 2, ..., n} with |A1| = k1, |A2| = k2 , ... , |Am| = km which minimizes Σk Σi Ak |xi - μk |p where μk is the p-centroid of Ak First, we prove that the optimum partition is contiguous (String Property), that is if i,j  Ak, and xi < xs < xj, then s  Ak . As a consequence, we determine an efficient algorithm for bi-clustering (if p is an integer); however, we show that the general problem is NP-complete, while a relaxed version of it admits a polynomial-time algorithm. When p is not an integer, we prove that the problem of deciding if the centroid μ is less than a given integer is in the Counting Hierarchy CH. As an application, the relaxed clustering algorithm used as a step for solving a problem in the field of Bioinformatics: the Localization of promoter regions in genomic sequences. The results are compared with those obtained through another methodology (MADAP).","abstract_html":"This thesis introduces and studies the problem of 1-dimensional bounded clustering: for any fixed p ≥ 1, given reals x1, x2..., xn, and integers k1, k2.., km, determine the partition (A1, A2... Am) of {1, 2, ..., n} with |A1| = k1, |A2| = k2 , ... , |Am| = km which minimizes Σk Σi Ak |xi - μk |p where μk is the p-centroid of Ak First, we prove that the optimum partition is contiguous (String Property), that is if i,j  Ak, and xi &lt; xs &lt; xj, then s  Ak . As a consequence, we determine an efficient algorithm for bi-clustering (if p is an integer); however, we show that the general problem is NP-complete, while a relaxed version of it admits a polynomial-time algorithm. When p is not an integer, we prove that the problem of deciding if the centroid μ is less than a given integer is in the Counting Hierarchy CH. As an application, the relaxed clustering algorithm used as a step for solving a problem in the field of Bioinformatics: the Localization of promoter regions in genomic sequences. The results are compared with those obtained through another methodology (MADAP).","abstract_has_math":false,"creators":["SACCA', FRANCESCO"],"institution":"Università degli Studi di Milano","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["tutor: A. Bertoni","G. Valentini","F. 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Am) of {1, 2, ..., n} with |A1| = k1, |A2| = k2 , ... , |Am| = km which minimizes Σk Σi Ak |xi - μk |p where μk is the p-centroid of Ak First, we prove that the optimum partition is contiguous (String Property), that is if i,j  Ak, and xi < xs < xj, then s  Ak . As a consequence, we determine an efficient algorithm for bi-clustering (if p is an integer); however, we show that the general problem is NP-complete, while a relaxed version of it admits a polynomial-time algorithm. When p is not an integer, we prove that the problem of deciding if the centroid μ is less than a given integer is in the Counting Hierarchy CH. As an application, the relaxed clustering algorithm used as a step for solving a problem in the field of Bioinformatics: the Localization of promoter regions in genomic sequences. The results are compared with those obtained through another methodology (MADAP)."]},{"key":"dc:title","label":"Title","values":["PROBLEMI DI CLUSTERING CON VINCOLI: ALGORITMI E COMPLESSITÀ"]}]}],"canonical_facts":{"dc:contributor":["tutor: A. Bertoni","G. Valentini","F. Sacca'","BERTONI, ALBERTO"],"dc:creator":["SACCA', FRANCESCO"],"dc:date":["2010-12-17"],"dc:description":["This thesis introduces and studies the problem of 1-dimensional bounded clustering: for any fixed p ≥ 1, given reals x1, x2..., xn, and integers k1, k2.., km, determine the partition (A1, A2... Am) of {1, 2, ..., n} with |A1| = k1, |A2| = k2 , ... , |Am| = km which minimizes Σk Σi Ak |xi - μk |p where μk is the p-centroid of Ak First, we prove that the optimum partition is contiguous (String Property), that is if i,j  Ak, and xi < xs < xj, then s  Ak . As a consequence, we determine an efficient algorithm for bi-clustering (if p is an integer); however, we show that the general problem is NP-complete, while a relaxed version of it admits a polynomial-time algorithm. When p is not an integer, we prove that the problem of deciding if the centroid μ is less than a given integer is in the Counting Hierarchy CH. As an application, the relaxed clustering algorithm used as a step for solving a problem in the field of Bioinformatics: the Localization of promoter regions in genomic sequences. 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