## On the Number of Descendants and Ascendants in Random Search Trees (1997)

### Cached

### Download Links

Citations: | 7 - 2 self |

### BibTeX

@MISC{Martínez97onthe,

author = {Conrado Martínez and Helmut Prodinger},

title = {On the Number of Descendants and Ascendants in Random Search Trees},

year = {1997}

}

### OpenURL

### Abstract

We consider here the probabilistic analysis of the number of descendants and the number of ascendants of a given internal node in a random search tree. The performance of several important algorithms on search trees is closely related to these quantities. For instance, the cost of a successful search is proportional to the number of ascendants of the sought element. On the other hand, the probabilistic behavior of the number of descendants is relevant for the analysis of paged data structures and for the analysis of the performance of quicksort, when recursive calls are not made on small subfiles. We also consider the number of ascendants and descendants of a random node in a random search tree, i.e., the grand averages of the quantities mentioned above. We address these questions for standard binary search trees and for locally balanced search trees. These search trees were introduced by Poblete and Munro and are binary search trees such that each subtree of size 3 is balanced; in oth...

### Citations

533 | Concrete mathematics
- Graham, Knuth, et al.
- 1994
(Show Context)
Citation Context ... P × P � Dn,j = m |therootisthek th � element � therootisthek th � element = 1 1 �j −1 [m = n ] + P [Dn−k,j−k = m]+ n n k=1 1 P [Dk−1,j = m] , (5) n k=j+1 where [P ] is 1 if P is tr=-=ue and 0 otherwise [10]-=-. This recursion translates nicely into a functional equation over the generating function for the family of random variables {Dn,j}. Solving the functional equation and extracting coefficients of the... |

432 |
Algorithms in C
- Sedgewick
- 1990
(Show Context)
Citation Context ...slated and applied to other data structures such as heap ordered trees, k-d-trees [33], and to important algorithms like quicksort and Hoare’s Find algorithm for selection (also known as quickselect=-=) [12, 13, 30, 31]-=-. We assume that the reader is already familiar with binary search trees and the basic algorithms to manipulate them [20, 31, 9]. Height and weight-balanced versions of the binary search trees, like A... |

317 |
Singularity analysis of generating functions
- Flajolet, Odlyzko
- 1990
(Show Context)
Citation Context ...tion about the coefficients of a generating function if we know its behaviour near its singularities or in some case, even if we only know the functional equation satisfied by the generating function =-=[33, 6]-=-. Also, if we are not able to solve and get an explicit form for X(z,u,v), we can still differentiate w.r.t. to v or set u = 1 and try to solve the (easier) resulting differential equations, to get in... |

268 | An Introduction to the Analysis of Algorithms
- Sedgewick, Flajolet
- 1996
(Show Context)
Citation Context ...tes the coefficient of zn in A(z) (the nth coefficient of A(z)). Excellent sources of information about generating functions and their applications to combinatorics and the analysis of algorithms are =-=[35, 33, 32, 20]-=-. We make extensive use in this paper of probability generating functions (PGFs) as well as multivariate generating functions whose coefficients are PGFs themselves. We define them in turn. Given a di... |

248 |
Seminumerical Algorithms
- Knuth
- 1998
(Show Context)
Citation Context ... = internal path length, 1≤j≤n Sn,j = Pn,j +1=An,j, Sn = An, E [Un]= n n+1 (1 + E [An]) , E [In]=n(E [An]−1) , E [An]=E [Dn]. There is also a close relationship between the performance of quicks=-=elect [12, 19, 17]-=- and the number of ascendants. Proposition 1.3. Let Fn,j be the number of recursive calls made by quickselect to select the jth element out of n elements. Then Fn,j = An,j. If we consider An,j in rand... |

230 |
A Dichromatic Framework for Balanced Trees
- Guibas, Sedgewick
- 1978
(Show Context)
Citation Context ...ader is already familiar with binary search trees and the basic algorithms to manipulate them [20, 31, 9]. Height and weight-balanced versions of the binary search trees, like AVL and red-black trees =-=[1, 11]-=-, have been proposed and find many useful applications, since all of them guarantee good worst-case performance of both searches and updates. Locally balanced search trees (LBSTs) were introduced by B... |

212 |
Evolution of random search trees
- Mahmoud
- 1992
(Show Context)
Citation Context ...ignificant improvements in theory and practice [30, 17]. Random search trees, either random BSTs or random LBSTs, are search trees built by performing n random insertions into an initially empty tree =-=[20, 24]-=-. An insertion of a new element into a search tree of size k is said to be random, if the new element falls with equal probability into any of the k + 1 intervals defined by the k keys already present... |

178 |
An algorithm for the organization of information
- Adelson-Velskii, Landis
- 1962
(Show Context)
Citation Context ...ader is already familiar with binary search trees and the basic algorithms to manipulate them [20, 31, 9]. Height and weight-balanced versions of the binary search trees, like AVL and red-black trees =-=[1, 11]-=-, have been proposed and find many useful applications, since all of them guarantee good worst-case performance of both searches and updates. Locally balanced search trees (LBSTs) were introduced by B... |

162 |
Handbook of Algorithms and Data Structures
- Gonnet
- 1984
(Show Context)
Citation Context ...d Hoare’s Find algorithm for selection (also known as quickselect) [12, 13, 30, 31]. We assume that the reader is already familiar with binary search trees and the basic algorithms to manipulate the=-=m [20, 31, 9]-=-. Height and weight-balanced versions of the binary search trees, like AVL and red-black trees [1, 11], have been proposed and find many useful applications, since all of them guarantee good worst-cas... |

142 | Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable
- Salvy, Zimmermann
- 1994
(Show Context)
Citation Context ... large, the least systematic and mechanical one. A great deal of combinatorial identities, inspired guessing and patience was needed. Standard Maple tools like the function interp or the Gfun package =-=[29]-=- proved also to be useful. However, once the solution is obtained, it is just a matter of minutes to check its correctness. It is quite difficult to provide a detailed and ordered description of the m... |

136 | Randomized search trees
- Seidel, Aragon
- 1996
(Show Context)
Citation Context ...heir applications spanning a wide range of the areas of Computer Science. Standard binary search trees (BSTs, for short) are still the subject of active research, see for instance the recent articles =-=[2, 28]-=-. Deepening our knowledge about binary search trees is interesting in its own; moreover, most of this knowledge can be translated and applied to other data structures such as heap ordered trees, k-d-t... |

133 |
Differentialgleichungen: Losungsmethoden und Losungen
- Kamke
- 1977
(Show Context)
Citation Context ... quadrature through the variation of constant —actually, functions in u and v— method. For the second order differential equations, the theory of hypergeometric differential equations comes into p=-=lay [16]-=-. Nowadays, most of the necessary mathematical knowledge is embodied into modern computer algebra systems. In our case, Maple needed little or no assistance to solve the differential equations that we... |

100 |
The Art of Computer Programming: Sorting and Searching, volume 3
- Knuth
- 1973
(Show Context)
Citation Context ...d Hoare’s Find algorithm for selection (also known as quickselect) [12, 13, 30, 31]. We assume that the reader is already familiar with binary search trees and the basic algorithms to manipulate the=-=m [20, 31, 9]-=-. Height and weight-balanced versions of the binary search trees, like AVL and red-black trees [1, 11], have been proposed and find many useful applications, since all of them guarantee good worst-cas... |

96 | Average-case analysis of algorithms and data structures
- Vitter, Flajolet
- 1990
(Show Context)
Citation Context ...pening our knowledge about binary search trees is interesting in its own; moreover, most of this knowledge can be translated and applied to other data structures such as heap ordered trees, k-d-trees =-=[33], -=-and to important algorithms like quicksort and Hoare’s Find algorithm for selection (also known as quickselect) [12, 13, 30, 31]. We assume that the reader is already familiar with binary search tre... |

49 |
Théorèmes limites pour les structures combinatoires et les fonctions arithmetiques
- Hwang
- 1994
(Show Context)
Citation Context ...ents in exact form from there is quite difficult. However, as Philippe Flajolet kindly pointed to us, asymptotic information and most notably, the limiting probability distribution can be established =-=[8, 15]. In this case, it follows th-=-at An converges in distribution (converges in law) to a Gaussian distribution, i.e. P ⎡ ⎣ An − 12 ⎤ 7 log n <x⎦= log n 1 √ 2π � 300 343 � x e −∞ −t2 /2 dt + O � � 1 √ . lo... |

38 |
Algorithm 65
- Hoare
- 1961
(Show Context)
Citation Context ...slated and applied to other data structures such as heap ordered trees, k-d-trees [33], and to important algorithms like quicksort and Hoare’s Find algorithm for selection (also known as quickselect=-=) [12, 13, 30, 31]-=-. We assume that the reader is already familiar with binary search trees and the basic algorithms to manipulate them [20, 31, 9]. Height and weight-balanced versions of the binary search trees, like A... |

32 |
Exact and asymptotic distributions in digital and binary search trees
- Louchard
- 1987
(Show Context)
Citation Context ... induced by the creation process of the random search trees (BSTs resp. LBSTs). The number of descendants and the number of ascendants in random BSTs have been investigated in several previous works (=-=[3, 5, 23, 22, 21]-=-). The number of ascendants of a random node in a random LBST has been studied in [27, 26]. We define the number of descendants Dn,j as the size of the subtree rooted at the j th node, so we count the... |

23 | General combinatorial schemas: Gaussian limit distributions and exponential tails, Discrete Mathematics 114
- Flajolet, Soria
- 1993
(Show Context)
Citation Context ...ents in exact form from there is quite difficult. However, as Philippe Flajolet kindly pointed to us, asymptotic information and most notably, the limiting probability distribution can be established =-=[8, 15]. In this case, it follows th-=-at An converges in distribution (converges in law) to a Gaussian distribution, i.e. P ⎡ ⎣ An − 12 ⎤ 7 log n <x⎦= log n 1 √ 2π � 300 343 � x e −∞ −t2 /2 dt + O � � 1 √ . lo... |

23 | Analysis of Hoare’s FIND algorithm with median-of-three partition. Random Structures Algorithms 10
- Kirschenhofer, Prodinger, et al.
- 1997
(Show Context)
Citation Context ...elements and taking the median of the sample as the pivot element for partitioning in algorithms like quicksort and quickselect has been shown to yield significant improvements in theory and practice =-=[30, 17]-=-. Random search trees, either random BSTs or random LBSTs, are search trees built by performing n random insertions into an initially empty tree [20, 24]. An insertion of a new element into a search t... |

17 |
On random binary trees
- Brown, Schubert
- 1984
(Show Context)
Citation Context ... induced by the creation process of the random search trees (BSTs resp. LBSTs). The number of descendants and the number of ascendants in random BSTs have been investigated in several previous works (=-=[3, 5, 23, 22, 21]-=-). The number of ascendants of a random node in a random LBST has been studied in [27, 26]. We define the number of descendants Dn,j as the size of the subtree rooted at the j th node, so we count the... |

14 | The average case analysis of algorithms: multivariate asymptotics and limit distributions, Rapport de recherche no
- Flajolet, Sedgewick
- 1997
(Show Context)
Citation Context ... − 1| < 1 4 v 2 [z n ]Az(z,v)=[z n ]− (1 − 2v)∆ (∆+4v+ 3)(1 − z)−(∆−1)/2 + O(n) = − v2 (∆ + 4v +3) (1 − 2v)∆Γ( ∆−1 � � ∆−3 1 � · n 2 1+O √n 2 ) � . Applyi=-=ng the following quasi-power theorem of Hwang [15, 7] lead-=-s immediately to the above given result. Theorem �7.3. (Quasi-power theorem [H.-K. Hwang]) Assume that the Laplace transforms sXn λn(s) = E e � of a sequence of random variables Xn are analytic i... |

12 |
The Analysis of a Fringe Heuristic for Binary Search Trees
- Poblete, Munro
- 1985
(Show Context)
Citation Context ...e good worst-case performance of both searches and updates. Locally balanced search trees (LBSTs) were introduced by Bell [4] and Walker and Wood [34], and thoroughly analyzed by Poblete and Munro in =-=[27]-=-. LBSTs have been proposed as an alternative to more complex balancing schemes for search trees. In these search trees, only local rebalancing is made; after each insertion, local rebalancing is appli... |

11 |
More combinatorial properties of certain trees
- Lynch
- 1965
(Show Context)
Citation Context ... induced by the creation process of the random search trees (BSTs resp. LBSTs). The number of descendants and the number of ascendants in random BSTs have been investigated in several previous works (=-=[3, 5, 23, 22, 21]-=-). The number of ascendants of a random node in a random LBST has been studied in [27, 26]. We define the number of descendants Dn,j as the size of the subtree rooted at the j th node, so we count the... |

10 | Page usage in a quadtree index
- Hoshi, Flajolet
- 1992
(Show Context)
Citation Context ...t uses the median of a random sample of three elements as the pivot in each partitioning phase. The study of the number of descendants has applications in the context of paged trees (see for instance =-=[20, 14]).-=- A paged binary search tree with page capacity b stores all its subtrees of size ≤ b (possibly empty) in pages; typically, the pages reside in secondary memory and the elements within a page are not... |

5 |
Locally balanced binary trees
- Walker, Wood
- 1976
(Show Context)
Citation Context ...d many useful applications, since all of them guarantee good worst-case performance of both searches and updates. Locally balanced search trees (LBSTs) were introduced by Bell [4] and Walker and Wood =-=[34]-=-, and thoroughly analyzed by Poblete and Munro in [27]. LBSTs have been proposed as an alternative to more complex balancing schemes for search trees. In these search trees, only local rebalancing is ... |

4 | Untersuchungen zur durchschnittlichen Gestalt gewisser Baumfamilien. Mit besonderer Berucksichtigung von Anwendungen in der Informatik - Panholzer - 1997 |

4 | Randomization of search trees by subtree size - Roura, Martínez - 1996 |

3 |
An Investigation into the Principles of the Classification and Analysis of Data on an Automatic Digital Computer
- Bell
- 1965
(Show Context)
Citation Context ...ve been proposed and find many useful applications, since all of them guarantee good worst-case performance of both searches and updates. Locally balanced search trees (LBSTs) were introduced by Bell =-=[4]-=- and Walker and Wood [34], and thoroughly analyzed by Poblete and Munro in [27]. LBSTs have been proposed as an alternative to more complex balancing schemes for search trees. In these search trees, o... |

3 | Probabilistic Analysis of some searching and sorting algorithms - Lent - 1996 |

3 |
The analysis of heuristics for search trees
- Poblete
- 1993
(Show Context)
Citation Context ...dants and the number of ascendants in random BSTs have been investigated in several previous works ([3, 5, 23, 22, 21]). The number of ascendants of a random node in a random LBST has been studied in =-=[27, 26]-=-. We define the number of descendants Dn,j as the size of the subtree rooted at the j th node, so we count the j th node as a descendant of itself. The number of ascendants An,j is the number of inter... |

1 |
Randomized binary search technique
- Arora, Dent
- 1969
(Show Context)
Citation Context |