Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees
Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division. We use cluster transformations to design efficient subtraction-free algorithms for computing Schur functions and their skew, double, an...
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| Published in | Foundations of computational mathematics Vol. 16; no. 1; pp. 1 - 31 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.02.2016
Springer Springer Nature B.V Springer Verlag |
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| ISSN | 1615-3375 1615-3383 1615-3383 |
| DOI | 10.1007/s10208-014-9231-y |
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| Abstract | Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division. We use cluster transformations to design efficient subtraction-free algorithms for computing Schur functions and their skew, double, and supersymmetric analogues, thereby generalizing earlier results by P. Koev. We develop such algorithms for computing generating functions of spanning trees, both directed and undirected. A comparison to the lower bound due to M. Jerrum and M. Snir shows that in subtraction-free computations, “division can be exponentially powerful.” Finally, we give a simple example where the gap between ordinary and subtraction-free complexity is exponential. |
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| AbstractList | Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division. We use cluster transformations to design efficient subtraction-free algorithms for computing Schur functions and their skew, double, and supersymmetric analogues, thereby generalizing earlier results by P. Koev. We develop such algorithms for computing generating functions of spanning trees, both directed and undirected. A comparison to the lower bound due to M. Jerrum and M. Snir shows that in subtraction-free computations, “division can be exponentially powerful.” Finally, we give a simple example where the gap between ordinary and subtraction-free complexity is exponential. Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division. We use cluster transformations to design efficient subtraction-free algorithms for computing Schur functions and their skew, double, and supersymmetric analogues, thereby generalizing earlier results by P. Koev. We develop such algorithms for computing generating functions of spanning trees, both directed and undirected. A comparison to the lower bound due to M. Jerrum and M. Snir shows that in subtraction-free computations, “division can be exponentially powerful.” Finally, we give a simple example where the gap between ordinary and subtraction-free complexity is exponential. |
| Audience | Academic |
| Author | Grigoriev, Dima Koshevoy, Gleb Fomin, Sergey |
| Author_xml | – sequence: 1 givenname: Sergey surname: Fomin fullname: Fomin, Sergey email: fomin@umich.edu organization: Department of Mathematics, University of Michigan – sequence: 2 givenname: Dima surname: Grigoriev fullname: Grigoriev, Dima organization: CNRS, Mathématiques, Université de Lille – sequence: 3 givenname: Gleb surname: Koshevoy fullname: Koshevoy, Gleb organization: Central Institute of Economics and Mathematics |
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| Cites_doi | 10.1006/aima.1996.0057 10.1007/BF02104745 10.37236/102 10.1090/S0894-0347-01-00385-X 10.1007/978-3-662-03338-8 10.1137/04061903X 10.1142/2446 10.1006/jabr.1994.1361 10.1016/j.camwa.2012.09.008 10.1145/322326.322341 10.1016/0304-3975(76)90083-9 10.1070/RM2010v065n04ABEH004692 10.1142/9789814324359_0043 10.1016/0304-3975(80)90060-2 10.1017/CBO9780511609589 10.1007/978-1-4612-0619-4 10.1016/S0024-3795(99)00134-2 10.1002/andp.18471481202 10.1016/S0022-4049(00)00155-9 10.1016/S0168-0072(01)00055-0 10.1090/S0025-5718-05-01780-1 10.1007/s10801-006-0008-5 10.37236/1893 |
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| Keywords | Primary 68Q25 Secondary 05E05 Cluster transformation Arithmetic circuit Star–mesh transformation 13F60 Schur function Subtraction-free Spanning tree spanning tree star-mesh transformation. 2010 Mathematics Subject Classification Primary 68Q25 cluster transformation arithmetic circuit |
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Macdonald, Schur functions: theme and variations, Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), 5–39, Publ. Inst. Rech. Math. Av., 498, Univ. Louis Pasteur, Strasbourg, 1992. W. T. Tutte, Graph theory as I have known it, Oxford University Press, 1998. NarayananHOn the complexity of computing Kostka numbers and Littlewood-Richardson coefficientsJ. Algebraic Combin.20062434735410.1007/s10801-006-0008-522600221101.05066 A. Shpilka and A. Yehudayoff, Arithmetic circuits: a survey of recent results and open questions, Found. Trends Theor. Comput. Sci.5 (2009), no. 3–4, 207–388 (2010). W. T. Tutte, Graph theory, Addison-Wesley, 1984. ValiantLGNegation can be exponentially powerfulTheor. Comput. Sci.19801230331410.1016/0304-3975(80)90060-25893110442.68030 LitvinovGLIdempotent and tropical mathematics; complexity of algorithms and interval analysisComput. Math. Appl.2013651483149610.1016/j.camwa.2012.09.0083061718 S. Fomin, Total positivity and cluster algebras, Proceedings of the International Congress of Mathematicians. Volume II, 125–145, Hindustan Book Agency, 2010. GouldenIGreeneCA new tableau representation for supersymmetric Schur functionsJ. Algebra199417068770310.1006/jabr.1994.136113028640840.20008 C. Chan, V. Drensky, A. Edelman, R. Kan, and P. Koev, On computing Schur functions and series thereof, preprint, 2008. R. P. Stanley, Enumerative combinatorics, vol. 2, Cambridge University Press, 1999. DemmelJGuMEisenstatSSlapničarIVeselićKDrmačZComputing the singular value decomposition with high relative accuracyLinear Algebra Appl.19992991–3218010.1016/S0024-3795(99)00134-217237090952.65032 StrassenVVermeidung von DivisionenJ. Reine Angew. Math.19732641842025211680294.65021 I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, 1999. G. Pólya, Über positive Darstellung von Polynomen, in: Vierteljschr. Naturforsch. Ges. Zrich73 (1928), 141–145; see: Collected Papers, vol. 2, MIT Press, Cambridge, 1974, pp. 309–313. V. I. Danilov, A. V. Karzanov, and G. A. Koshevoy, Systems of separated sets and their geometric models, Uspehi Mat. Nauk, 65 (2010), 67-152 (in Russian); English translation in Russian Math. Surveys65 (2010), no. 4, 659–740. DemmelJKoevPAccurate and efficient evaluation of Schur and Jack functionsMath. Comp.20067525322323910.1090/S0025-5718-05-01780-121763971076.05083 GrigorievDLower bounds in algebraic complexityJ. Soviet Math.1985291388142510.1007/BF02104745 JerrumMSnirMSome exact complexity results for straight-line computations over semiringsJ. Assoc. Comput. Mach.19822987489710.1145/322326.3223416667830485.68038 BerensteinAFominSZelevinskyAParametrizations of canonical bases and totally positive matricesAdv. Math.19961224914910.1006/aima.1996.005714054490966.17011 J. Riordan, Review MR0022160 (9,166f), Math. Reviews, AMS, 1948. A. Aho, J. Hopcroft, and J. Ullman, The design and analysis of computer algorithms, Addison-Wesley, 1975. P. Bürgisser, M. Clausen, and M. A. Shokrollahi, Algebraic complexity theory, Springer-Verlag, 1997. W.-K. Chen, Graph theory and its engineering applications, World Scientific, 1997. FominSZelevinskyACluster algebras I: FoundationsJ. Amer. Math. Soc.20021549752910.1090/S0894-0347-01-00385-X18876421021.16017 G. Rote, Division-free algorithms for the determinant and the Pfaffian: algebraic and combinatorial approaches. Computational discrete mathematics, 119–135, Lecture Notes in Comput. Sci. 2122, Springer, Berlin, 2001. B. Bollobás, Modern graph theory, Springer, 1998. GrigorievDVorobjovNComplexity of Null- and Positivstellensatz proofsAnn. Pure Appl. Logic200211315316010.1016/S0168-0072(01)00055-018757400992.03073 E. Shamir and M. Snir, Lower bounds on the number of multiplications and the number of additions in monotone computations, Technical Report RC-6757, IBM, 1977. D. G. Wagner, Matroid inequalities from electrical network theory, Electron. J. Combin.11 (2004/06), no. 2, Article 1, 17 pp. A Berenstein (9231_CR2) 1996; 122 P Koev (9231_CR18) 2007; 29 V Powers (9231_CR25) 2001; 164 D Grigoriev (9231_CR14) 2002; 113 D Grigoriev (9231_CR13) 1985; 29 V Strassen (9231_CR32) 1973; 264 GL Litvinov (9231_CR19) 2013; 65 9231_CR16 CP Schnorr (9231_CR28) 1976; 2 M Jerrum (9231_CR15) 1982; 29 9231_CR34 9231_CR36 9231_CR31 LG Valiant (9231_CR35) 1980; 12 9231_CR30 9231_CR1 9231_CR33 J Demmel (9231_CR8) 1999; 299 9231_CR10 S Fomin (9231_CR11) 2002; 15 9231_CR3 9231_CR4 9231_CR5 9231_CR6 9231_CR7 9231_CR27 I Goulden (9231_CR12) 1994; 170 H Narayanan (9231_CR23) 2006; 24 9231_CR29 9231_CR24 9231_CR26 G Kirchhoff (9231_CR17) 1847; 72 9231_CR20 9231_CR22 9231_CR21 J Demmel (9231_CR9) 2006; 75 |
| References_xml | – reference: StrassenVVermeidung von DivisionenJ. Reine Angew. Math.19732641842025211680294.65021 – reference: W. T. Tutte, Graph theory as I have known it, Oxford University Press, 1998. – reference: A. Aho, J. Hopcroft, and J. Ullman, The design and analysis of computer algorithms, Addison-Wesley, 1975. – reference: A. I. Molev, Comultiplication rules for the double Schur functions and Cauchy identities, Electron. J. Combin.16 (2009), no. 1, Research Paper 13, 44 pp. – reference: GrigorievDVorobjovNComplexity of Null- and Positivstellensatz proofsAnn. Pure Appl. Logic200211315316010.1016/S0168-0072(01)00055-018757400992.03073 – reference: KirchhoffGÜber die Auflösung der Gleichungen, auf welche man bei der Untersuchungen der linearen Vertheilung galvanischer Ströme geführt wirdAnn. Phys. Chem.18477249750810.1002/andp.18471481202 – reference: DemmelJKoevPAccurate and efficient evaluation of Schur and Jack functionsMath. Comp.20067525322323910.1090/S0025-5718-05-01780-121763971076.05083 – reference: GouldenIGreeneCA new tableau representation for supersymmetric Schur functionsJ. Algebra199417068770310.1006/jabr.1994.136113028640840.20008 – reference: LitvinovGLIdempotent and tropical mathematics; complexity of algorithms and interval analysisComput. Math. Appl.2013651483149610.1016/j.camwa.2012.09.0083061718 – reference: W. T. Tutte, Graph theory, Addison-Wesley, 1984. – reference: P. W. Kasteleyn, Graph theory and crystal physics, in: Graph theory and theoretical physics, 43–110, Academic Press, 1967. – reference: BerensteinAFominSZelevinskyAParametrizations of canonical bases and totally positive matricesAdv. Math.19961224914910.1006/aima.1996.005714054490966.17011 – reference: SchnorrCPA lower bound on the number of additions in monotone computationsTheor. Comput. Sci.1976230531510.1016/0304-3975(76)90083-94185170342.68024 – reference: J. Riordan, Review MR0022160 (9,166f), Math. Reviews, AMS, 1948. – reference: D. G. Wagner, Matroid inequalities from electrical network theory, Electron. J. Combin.11 (2004/06), no. 2, Article 1, 17 pp. – reference: KoevPAccurate computations with totally nonnegative matricesSIAM J. Matrix Anal. Appl.20072973175110.1137/04061903X23384601198.65057 – reference: B. Bollobás, Modern graph theory, Springer, 1998. – reference: DemmelJGuMEisenstatSSlapničarIVeselićKDrmačZComputing the singular value decomposition with high relative accuracyLinear Algebra Appl.19992991–3218010.1016/S0024-3795(99)00134-217237090952.65032 – reference: FominSZelevinskyACluster algebras I: FoundationsJ. Amer. Math. Soc.20021549752910.1090/S0894-0347-01-00385-X18876421021.16017 – reference: I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, 1999. – reference: NarayananHOn the complexity of computing Kostka numbers and Littlewood-Richardson coefficientsJ. Algebraic Combin.20062434735410.1007/s10801-006-0008-522600221101.05066 – reference: V. I. Danilov, A. V. Karzanov, and G. A. Koshevoy, Systems of separated sets and their geometric models, Uspehi Mat. Nauk, 65 (2010), 67-152 (in Russian); English translation in Russian Math. Surveys65 (2010), no. 4, 659–740. – reference: JerrumMSnirMSome exact complexity results for straight-line computations over semiringsJ. Assoc. Comput. Mach.19822987489710.1145/322326.3223416667830485.68038 – reference: C. Chan, V. Drensky, A. Edelman, R. Kan, and P. Koev, On computing Schur functions and series thereof, preprint, 2008. – reference: G. Rote, Division-free algorithms for the determinant and the Pfaffian: algebraic and combinatorial approaches. Computational discrete mathematics, 119–135, Lecture Notes in Comput. Sci. 2122, Springer, Berlin, 2001. – reference: P. Bürgisser, M. Clausen, and M. A. Shokrollahi, Algebraic complexity theory, Springer-Verlag, 1997. – reference: I. G. Macdonald, Schur functions: theme and variations, Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), 5–39, Publ. Inst. Rech. Math. Av., 498, Univ. Louis Pasteur, Strasbourg, 1992. – reference: PowersVReznickBA new bound for Pólya’s theorem with applications to polynomials positive on polyhedraJ. Pure Appl. Algebra200116422122910.1016/S0022-4049(00)00155-918543391075.14523 – reference: GrigorievDLower bounds in algebraic complexityJ. Soviet Math.1985291388142510.1007/BF02104745 – reference: ValiantLGNegation can be exponentially powerfulTheor. Comput. Sci.19801230331410.1016/0304-3975(80)90060-25893110442.68030 – reference: E. Shamir and M. Snir, Lower bounds on the number of multiplications and the number of additions in monotone computations, Technical Report RC-6757, IBM, 1977. – reference: R. P. Stanley, Enumerative combinatorics, vol. 2, Cambridge University Press, 1999. – reference: A. Shpilka and A. 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| SubjectTerms | Algorithms Analysis Applications of Mathematics Clusters Complexity Computation Computational complexity Computer Science Dividing (mathematics) Division Economics Graph theory Linear and Multilinear Algebras Lower bounds Math Applications in Computer Science Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Subtraction Supersymmetry Transformations Transformations (mathematics) Trees |
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| Title | Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees |
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