An Asymptotic Analysis of Labeled and Unlabeled k-Trees
Abstract In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are a...
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Gespeichert in:
| Autor*in: |
Drmota, Michael [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2015 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media New York 2015 |
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| Übergeordnetes Werk: |
Enthalten in: Algorithmica - Springer US, 1986, 75(2015), 4 vom: 23. Juli, Seite 579-605 |
|---|---|
| Übergeordnetes Werk: |
volume:75 ; year:2015 ; number:4 ; day:23 ; month:07 ; pages:579-605 |
| Links: |
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| DOI / URN: |
10.1007/s00453-015-0039-1 |
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| 520 | |a Abstract In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers $$U_k(n)$$ of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$, where $$c_k> 0$$ and $$\rho _{k}>0$$ denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$. | ||
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10.1007/s00453-015-0039-1 doi (DE-627)OLC2054849510 (DE-He213)s00453-015-0039-1-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ Drmota, Michael verfasserin aut An Asymptotic Analysis of Labeled and Unlabeled k-Trees 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2015 Abstract In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers $$U_k(n)$$ of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$, where $$c_k> 0$$ and $$\rho _{k}>0$$ denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$. -trees Generating function Singularity analysis Central limit theorem Jin, Emma Yu aut Enthalten in Algorithmica Springer US, 1986 75(2015), 4 vom: 23. Juli, Seite 579-605 (DE-627)129197564 (DE-600)53958-2 (DE-576)014456958 0178-4617 nnns volume:75 year:2015 number:4 day:23 month:07 pages:579-605 https://doi.org/10.1007/s00453-015-0039-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_70 GBV_ILN_130 GBV_ILN_2018 GBV_ILN_4266 GBV_ILN_4318 GBV_ILN_4319 AR 75 2015 4 23 07 579-605 |
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10.1007/s00453-015-0039-1 doi (DE-627)OLC2054849510 (DE-He213)s00453-015-0039-1-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ Drmota, Michael verfasserin aut An Asymptotic Analysis of Labeled and Unlabeled k-Trees 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2015 Abstract In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers $$U_k(n)$$ of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$, where $$c_k> 0$$ and $$\rho _{k}>0$$ denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$. -trees Generating function Singularity analysis Central limit theorem Jin, Emma Yu aut Enthalten in Algorithmica Springer US, 1986 75(2015), 4 vom: 23. Juli, Seite 579-605 (DE-627)129197564 (DE-600)53958-2 (DE-576)014456958 0178-4617 nnns volume:75 year:2015 number:4 day:23 month:07 pages:579-605 https://doi.org/10.1007/s00453-015-0039-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_70 GBV_ILN_130 GBV_ILN_2018 GBV_ILN_4266 GBV_ILN_4318 GBV_ILN_4319 AR 75 2015 4 23 07 579-605 |
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10.1007/s00453-015-0039-1 doi (DE-627)OLC2054849510 (DE-He213)s00453-015-0039-1-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ Drmota, Michael verfasserin aut An Asymptotic Analysis of Labeled and Unlabeled k-Trees 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2015 Abstract In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers $$U_k(n)$$ of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$, where $$c_k> 0$$ and $$\rho _{k}>0$$ denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$. -trees Generating function Singularity analysis Central limit theorem Jin, Emma Yu aut Enthalten in Algorithmica Springer US, 1986 75(2015), 4 vom: 23. Juli, Seite 579-605 (DE-627)129197564 (DE-600)53958-2 (DE-576)014456958 0178-4617 nnns volume:75 year:2015 number:4 day:23 month:07 pages:579-605 https://doi.org/10.1007/s00453-015-0039-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_70 GBV_ILN_130 GBV_ILN_2018 GBV_ILN_4266 GBV_ILN_4318 GBV_ILN_4319 AR 75 2015 4 23 07 579-605 |
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10.1007/s00453-015-0039-1 doi (DE-627)OLC2054849510 (DE-He213)s00453-015-0039-1-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ Drmota, Michael verfasserin aut An Asymptotic Analysis of Labeled and Unlabeled k-Trees 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2015 Abstract In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers $$U_k(n)$$ of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$, where $$c_k> 0$$ and $$\rho _{k}>0$$ denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$. -trees Generating function Singularity analysis Central limit theorem Jin, Emma Yu aut Enthalten in Algorithmica Springer US, 1986 75(2015), 4 vom: 23. Juli, Seite 579-605 (DE-627)129197564 (DE-600)53958-2 (DE-576)014456958 0178-4617 nnns volume:75 year:2015 number:4 day:23 month:07 pages:579-605 https://doi.org/10.1007/s00453-015-0039-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_70 GBV_ILN_130 GBV_ILN_2018 GBV_ILN_4266 GBV_ILN_4318 GBV_ILN_4319 AR 75 2015 4 23 07 579-605 |
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An Asymptotic Analysis of Labeled and Unlabeled k-Trees |
| abstract |
Abstract In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers $$U_k(n)$$ of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$, where $$c_k> 0$$ and $$\rho _{k}>0$$ denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$. © Springer Science+Business Media New York 2015 |
| abstractGer |
Abstract In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers $$U_k(n)$$ of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$, where $$c_k> 0$$ and $$\rho _{k}>0$$ denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$. © Springer Science+Business Media New York 2015 |
| abstract_unstemmed |
Abstract In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers $$U_k(n)$$ of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$, where $$c_k> 0$$ and $$\rho _{k}>0$$ denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$. © Springer Science+Business Media New York 2015 |
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4 |
| title_short |
An Asymptotic Analysis of Labeled and Unlabeled k-Trees |
| url |
https://doi.org/10.1007/s00453-015-0039-1 |
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false |
| author2 |
Jin, Emma Yu |
| author2Str |
Jin, Emma Yu |
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| doi_str |
10.1007/s00453-015-0039-1 |
| up_date |
2024-07-04T00:36:16.049Z |
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