An Asymptotic Analysis of Labeled and Unlabeled k-Trees

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...

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Published in:	Algorithmica Vol. 75; no. 4; pp. 579 - 605
Main Authors:	Drmota, Michael, Jin, Emma Yu
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.08.2016
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation Central limit theorem Generating function Singularity analysis 05A16 trees 05A15
ISSN:	0178-4617, 1432-0541
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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 ) ∼ c k n - 5 / 2 ρ k - n , where c k > 0 and ρ k > 0 denotes the radius of convergence of the generating function U ( z ) = ∑ n ≥ 0 U k ( n ) z n .
AbstractList	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 ) ∼ c k n - 5 / 2 ρ k - n , where c k > 0 and ρ k > 0 denotes the radius of convergence of the generating function U ( z ) = ∑ n ≥ 0 U k ( n ) z n .
Author	Jin, Emma Yu Drmota, Michael
Author_xml	– sequence: 1 givenname: Michael surname: Drmota fullname: Drmota, Michael organization: Institut für Diskrete Mathematik und Geometrie, TU Wien – sequence: 2 givenname: Emma Yu surname: Jin fullname: Jin, Emma Yu email: yu.jin@tuwien.ac.at, jin@cs.uni-kl.de organization: Institut für Diskrete Mathematik und Geometrie, TU Wien
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation
Title	An Asymptotic Analysis of Labeled and Unlabeled k-Trees
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