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278
Generating Functions for Generating Trees
 PROCEEDINGS OF 11TH FORMAL POWER SERIES AND ALGEBRAIC COMBINATORICS
, 1999
"... Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumerati ..."
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Cited by 70 (18 self)
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Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.
Varieties of Increasing Trees
, 1992
"... An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We ..."
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Cited by 59 (6 self)
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An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.
Pattern Matching For Permutations
, 1993
"... Given a permutation T of 1 to n, and a permutation P of 1 to k, for k n, we wish to find a kelement subsequence of T whose elements are ordered according to the permutation P . ..."
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Cited by 48 (0 self)
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Given a permutation T of 1 to n, and a permutation P of 1 to k, for k n, we wish to find a kelement subsequence of T whose elements are ordered according to the permutation P .
Linear Recurrences With Constant Coefficients: The Multivariate Case
, 2000
"... While in the univariate case solutions of linear recurrences with constant coefficients have rational generating functions, we show that the multivariate case is much richer: even though initial conditions have rational generating functions, the corresponding solutions can have generating functions ..."
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Cited by 43 (11 self)
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While in the univariate case solutions of linear recurrences with constant coefficients have rational generating functions, we show that the multivariate case is much richer: even though initial conditions have rational generating functions, the corresponding solutions can have generating functions which are algebraic but not rational, Dfinite but not algebraic, and even non Dfinite.
Efficient Tabling Mechanisms for Logic Programs
 PROCEEDINGS OF THE 12TH INTERNATIONAL CONFERENCE ON LOGIC PROGRAMMING
, 1995
"... The use of tabling in logic programming allows bottomup evaluation to be incorporated in a topdown framework, combining advantages that accrue from both. At the engine level, tabling also introduces complications not present in pure topdown evaluation, due to the need for subgoals and answers to ..."
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Cited by 42 (7 self)
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The use of tabling in logic programming allows bottomup evaluation to be incorporated in a topdown framework, combining advantages that accrue from both. At the engine level, tabling also introduces complications not present in pure topdown evaluation, due to the need for subgoals and answers to access tables during clause resolution. This paper describes the design, implementation, and experimental evaluation of data structures and algorithms for highperformance table access. Our approach uses tries as the basis for tables. Tries provide complete discrimination for terms, and permit a lookup and possible insertion to be performed in a single pass through a term. A novel technique of substitution factoring is also proposed. When substitution factoring is used in conjunction with tries, the access cost for answers is proportional to the size of the answer substitution, rather than to the size of the answer itself. As a special case, the access cost of answers to a datalog query is pr...
On the Distribution for the Duration of a Randomized Leader Election Algorithm
 Ann. Appl. Probab
, 1996
"... We investigate the duration of an elimination process for identifying a winner by coin tossing, or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressio ..."
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Cited by 37 (11 self)
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We investigate the duration of an elimination process for identifying a winner by coin tossing, or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressions for the discrete distribution and the moments of the height. Elementary approximation techniques then yield asymptotics for the distribution. We show that no limiting distribution exists, as the asymptotic expressions exhibit periodic fluctuations. In many similar problems associated with digital trees, no such exact expressions can be derived. We therefore outline a powerful general approach, based on the analytic techniques of Mellin transforms, Poissonization, and dePoissonization, from which distributional asymptotics for the height can also be derived. In fact, it was this complex variables approach that led to our original discovery of the exact distribution. Complex analysis metho...
The Wiener Index Of Simply Generated Random Trees
 Random Struct. Alg
, 2003
"... Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the in ..."
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Cited by 37 (13 self)
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Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. 1.
Efficient Access Mechanisms For Tabled Logic Programs
, 1999
"... This article describes the design, implementation, and experimental evaluation of data structures and algorithms for highperformance table access. Our approach uses tries as the basis for tables. Tries, a variant of discrimination nets, provide complete discrimination for terms, and permit a lookup ..."
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Cited by 35 (14 self)
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This article describes the design, implementation, and experimental evaluation of data structures and algorithms for highperformance table access. Our approach uses tries as the basis for tables. Tries, a variant of discrimination nets, provide complete discrimination for terms, and permit a lookup and possible insertion to be performed in a single pass through a term. In addition, a novel technique of substitution factoring is proposed. When substitution factoring is used, the access cost for answers is proportional to the size of the answer substitution, rather than to the size of the answer itself. Answer tries can be implemented both as interpreted structures and as compiled WAMlike code. When they are compiled, the speed of computing substitutions through answer tries is competitive with the speed of unit facts compiled or asserted as WAM code. Because answer tries can also be created an order of magnitude more quickly than asserted code, they form a promising alternative for representing certain types of dynamic code, even in Prolog systems without tabling. / Address correspondence to I.V. Ramakrishnan, D.S. Warren, Dept. of Computer Science, State University of New York at Stony Brook, Stony Brook, NY 117944400, U.S.A., email: fram,warreng@cs.sunysb.edu; P. Rao, Bellcore, 445 South Street, Morristown, NJ 079606438, U.S.A., email: prasadr@bellcore.com; K. Sagonas, Dept. of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B3001, Heverlee, Belgium, email:
Optimization Algorithms for Exploiting the ParallelismCommunication Tradeoff in Pipelined Parallelism
, 1994
"... We address the problem of finding parallel plans for SQL queries using the twophase approach of join ordering followed by parallelization. We focus on the parallelization phase and develop algorithms for exploiting pipelined parallelism. We formulate parallelization as scheduling a weighted operato ..."
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Cited by 34 (2 self)
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We address the problem of finding parallel plans for SQL queries using the twophase approach of join ordering followed by parallelization. We focus on the parallelization phase and develop algorithms for exploiting pipelined parallelism. We formulate parallelization as scheduling a weighted operator tree to minimize response time. Our model of response time captures the fundamental tradeoff between parallel execution and its communication overhead. We assess the quality of an optimization algorithm by its performance ratio which is the ratio of the response time of the generated schedule to that of the optimal. We develop fast algorithms that produce nearoptimal schedules  the performance ratio is extremely close to 1 on the average and has a worst case bound of about 2 for many cases. 1 Introduction We address the problem of parallel query optimization, which is to find optimal parallel plans for executing SQL queries. Following Hong and Stonebraker [HS91], we break the optimi...