A family of finitely many continuous functions on a polytope X, namely (g/sub i/(x))/sub i in I/, is considered, and the problem of minimizing the function f(x)=max/sub i in I/g/sub i/(x) on X is treated. It is shown that if every g/sub i/(x) is a concave function, then the minimum value of f(x) is achieved at finitely many special points in X. As an application, a long-standing problem about Steiner minimum trees and minimum spanning trees is solved. In particular, if P is a set of n points on the Euclidean plane and L/sub s/(P) and L/sub m/(P) denote the lengths of a Steiner minimum tree and a minimum spanning tree on P, respectively, it is proved that, for any P, L/sub S/(P)>or= square root 3L/sub m/(P)/2, as conjectured by E.N. Gilbert and H.O. Pollak (1968).<>
{"title":"An approach for proving lower bounds: solution of Gilbert-Pollak's conjecture on Steiner ratio","authors":"D. Du, F. Hwang","doi":"10.1109/FSCS.1990.89526","DOIUrl":"https://doi.org/10.1109/FSCS.1990.89526","url":null,"abstract":"A family of finitely many continuous functions on a polytope X, namely (g/sub i/(x))/sub i in I/, is considered, and the problem of minimizing the function f(x)=max/sub i in I/g/sub i/(x) on X is treated. It is shown that if every g/sub i/(x) is a concave function, then the minimum value of f(x) is achieved at finitely many special points in X. As an application, a long-standing problem about Steiner minimum trees and minimum spanning trees is solved. In particular, if P is a set of n points on the Euclidean plane and L/sub s/(P) and L/sub m/(P) denote the lengths of a Steiner minimum tree and a minimum spanning tree on P, respectively, it is proved that, for any P, L/sub S/(P)>or= square root 3L/sub m/(P)/2, as conjectured by E.N. Gilbert and H.O. Pollak (1968).<<ETX>>","PeriodicalId":271949,"journal":{"name":"Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114926958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A technique for exactly identifying certain classes of read-once Boolean formulas is introduced. The method is based on sampling the input-output behavior of the target formula on a probability distribution which is determined by the fixed point of the formula's amplification function (defined as the probability that a 1 is output by the formula when each input bit is 1 independently with probability p). By performing various statistical tests on easily sampled variants of the fixed-point distribution, it is possible to infer efficiently all structural information about any logarithmic-depth target family (with high probability). The results are used to prove the existence of short universal identification sequences for large classes of formulas. Extensions of the algorithms to handle high rates of noise and to learn formulas of unbounded depth in L.G. Valiant's (1984) model with respect to specific distributions are described.<>
{"title":"Exact identification of circuits using fixed points of amplification functions","authors":"S. Goldman, M. Kearns, R. Schapire","doi":"10.1109/FSCS.1990.89538","DOIUrl":"https://doi.org/10.1109/FSCS.1990.89538","url":null,"abstract":"A technique for exactly identifying certain classes of read-once Boolean formulas is introduced. The method is based on sampling the input-output behavior of the target formula on a probability distribution which is determined by the fixed point of the formula's amplification function (defined as the probability that a 1 is output by the formula when each input bit is 1 independently with probability p). By performing various statistical tests on easily sampled variants of the fixed-point distribution, it is possible to infer efficiently all structural information about any logarithmic-depth target family (with high probability). The results are used to prove the existence of short universal identification sequences for large classes of formulas. Extensions of the algorithms to handle high rates of noise and to learn formulas of unbounded depth in L.G. Valiant's (1984) model with respect to specific distributions are described.<<ETX>>","PeriodicalId":271949,"journal":{"name":"Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science","volume":"123 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133986198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is shown that for any concept class C the number of equivalence and membership queries that are needed to learn C is bounded from below by Omega (VC-dimension(C)). Furthermore, it is shown that the required number of equivalence and membership queries is also bounded from below by Omega (LC-ARB(C)/log(1+LC-ARB(C))), where LC-ARB(C) is the required number of steps in a different model where no membership queries but equivalence queries with arbitrary subsets of the domain are permitted. These two relationships are the only relationships between the learning complexities of the common online learning models and the related combinatorial parameters that have remained open. As an application of the first lower bound, the number of equivalence and membership queries that are needed to learn monomials of k out of n variables is determined. Learning algorithms for threshold gates that are based on equivalence queries are examined. It is shown that a threshold gate can learn not only concepts but also nondecreasing functions in polynomially many steps.<>
{"title":"On the complexity of learning from counterexamples and membership queries","authors":"W. Maass, György Turán","doi":"10.1109/FSCS.1990.89539","DOIUrl":"https://doi.org/10.1109/FSCS.1990.89539","url":null,"abstract":"It is shown that for any concept class C the number of equivalence and membership queries that are needed to learn C is bounded from below by Omega (VC-dimension(C)). Furthermore, it is shown that the required number of equivalence and membership queries is also bounded from below by Omega (LC-ARB(C)/log(1+LC-ARB(C))), where LC-ARB(C) is the required number of steps in a different model where no membership queries but equivalence queries with arbitrary subsets of the domain are permitted. These two relationships are the only relationships between the learning complexities of the common online learning models and the related combinatorial parameters that have remained open. As an application of the first lower bound, the number of equivalence and membership queries that are needed to learn monomials of k out of n variables is determined. Learning algorithms for threshold gates that are based on equivalence queries are examined. It is shown that a threshold gate can learn not only concepts but also nondecreasing functions in polynomially many steps.<<ETX>>","PeriodicalId":271949,"journal":{"name":"Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116251925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The fundamental problem of permuting the elements of an array according to some given permutation is addressed. The goal is to perform the permutation quickly using only a polylogarithmic number of bits of extra storage. The main result is an O(n log n)-time, O(log/sup 2/n)-space worst case method. A simpler method is presented for the case in which both the permutation and its inverse can be computed at (amortized) unit cost. This algorithm requires O(n log n) time and O(log n) bits in the worst case. These results are extended to the situation in which a power of the permutation is to be applied. A linear time, O(log n)-bit method is presented for the special case in which the data values are all distinct and are either initially in sorted order or will be when permuted.<>
{"title":"Permuting","authors":"Faith Ellen, J. Munro, P. V. Poblete","doi":"10.1109/FSCS.1990.89556","DOIUrl":"https://doi.org/10.1109/FSCS.1990.89556","url":null,"abstract":"The fundamental problem of permuting the elements of an array according to some given permutation is addressed. The goal is to perform the permutation quickly using only a polylogarithmic number of bits of extra storage. The main result is an O(n log n)-time, O(log/sup 2/n)-space worst case method. A simpler method is presented for the case in which both the permutation and its inverse can be computed at (amortized) unit cost. This algorithm requires O(n log n) time and O(log n) bits in the worst case. These results are extended to the situation in which a power of the permutation is to be applied. A linear time, O(log n)-bit method is presented for the special case in which the data values are all distinct and are either initially in sorted order or will be when permuted.<<ETX>>","PeriodicalId":271949,"journal":{"name":"Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114459503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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