scholarly journals Random Records and Cuttings in Split Trees: Extended Abstract

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Cecilia Holmgren
Keyword(s):  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Search Trees ◽  
Digital Search Trees ◽  
International Audience ◽  
Split Trees ◽  
Digital Search

International audience We study the number of records in random split trees on $n$ randomly labelled vertices. Equivalently the number of random cuttings required to eliminate an arbitrary random split tree can be studied. After normalization the distributions are shown to be asymptotically $1$-stable. This work is a generalization of our earlier results for the random binary search tree which is one specific case of split trees. Other important examples of split trees include $m$-ary search trees, quadtrees, median of $(2k+1)$-trees, simplex trees, tries and digital search trees.

scholarly journals A weakly 1-stable distribution for the number of random records and cuttings in split trees

2011 ◽  
Vol 43 (01) ◽  
pp. 151-177 ◽  
Author(s):  
Cecilia Holmgren
Keyword(s):  
Classical Limit ◽  
Stable Distribution ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Search Trees ◽  
Infinitely Divisible ◽  
Digital Search Trees ◽  
Split Trees ◽  
Digital Search

In this paper we study the number of random records in an arbitrary split tree (or, equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for the convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1-stable. This work is a generalization of our earlier results for the random binary search tree in Holmgren (2010), which is one specific case of split trees. Other important examples of split trees includem-ary search trees, quad trees, medians of (2k+ 1)-trees, simplex trees, tries, and digital search trees.

scholarly journals A weakly 1-stable distribution for the number of random records and cuttings in split trees

2011 ◽  
Vol 43 (1) ◽  
pp. 151-177 ◽  
Author(s):  
Cecilia Holmgren
Keyword(s):  
Classical Limit ◽  
Stable Distribution ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Search Trees ◽  
Infinitely Divisible ◽  
Digital Search Trees ◽  
Split Trees ◽  
Digital Search

In this paper we study the number of random records in an arbitrary split tree (or, equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for the convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1-stable. This work is a generalization of our earlier results for the random binary search tree in Holmgren (2010), which is one specific case of split trees. Other important examples of split trees include m-ary search trees, quad trees, medians of (2k + 1)-trees, simplex trees, tries, and digital search trees.

scholarly journals Average depth in a binary search tree with repeated keys

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Margaret Archibald ◽  
Julien Clément
Keyword(s):  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Average Depth ◽  
Binary Search Trees ◽  
Search Trees ◽  
Random Sequences ◽  
International Audience ◽  
The Right

International audience Random sequences from alphabet $\{1, \ldots,r\}$ are examined where repeated letters are allowed. Binary search trees are formed from these, and the average left-going depth of the first $1$ is found. Next, the right-going depth of the first $r$ is examined, and finally a merge (or 'shuffle') operator is used to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths. The variance of each of these parameters is also found.

An estimate of mean efficiency of search trees for arbitrary sets of binary words

2002 ◽  
Vol 12 (2) ◽  
Author(s):  
B.Ya. Ryabko ◽  
A.A. Fedotov
Keyword(s):  
Lower Bound ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Minimal Cost ◽  
Search Trees ◽  
Optimal Tree ◽  
Np Complete ◽  
The Mean ◽  
Binary Logarithm

AbstractWe consider the problem on constructing a binary search tree for an arbitrary set of binary words, which has found a wide use in informatics, biology, mineralogy, and other fields. It is known that the problem on constructing the tree of minimal cost is NP-complete; hence the problem arises to find simple algorithms which allow us to construct trees close to the optimal ones. In this paper we demonstrate that even simplest algorithm yields search trees which are close to the optimal ones in average, and prove that the mean number of nodes checked in the optimal tree differs from the natural lower bound, the binary logarithm of the number of words, by no more than 1.04.

scholarly journals On the Subtree Size Profile of Binary Search trees

2010 ◽  
Vol 19 (4) ◽  
pp. 561-578 ◽  
Author(s):  
FLORIAN DENNERT ◽  
RUDOLF GRÜBEL
Keyword(s):  
Qualitative Difference ◽  
Search Tree ◽  
Binary Search ◽  
Random Trees ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Joint Distributions ◽  
Size Profile ◽  
Process View

For random trees T generated by the binary search tree algorithm from uniformly distributed input we consider the subtree size profile, which maps k ∈ ℕ to the number of nodes in T that root a subtree of size k. Complementing earlier work by Devroye, by Feng, Mahmoud and Panholzer, and by Fuchs, we obtain results for the range of small k-values and the range of k-values proportional to the size n of T. In both cases emphasis is on the process view, i.e., the joint distributions for several k-values. We also show that the dynamics of the tree sequence lead to a qualitative difference between the asymptotic behaviour of the lower and the upper end of the profile.

scholarly journals Digital search trees with m trees: Level polynomials and insertion costs

2011 ◽  
Vol Vol. 13 no. 3 (Analysis of Algorithms) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Analysis Of Algorithms ◽  
Approximate Counting ◽  
Search Trees ◽  
Digital Search Trees ◽  
International Audience ◽  
Digital Search

Analysis of Algorithms International audience We adapt a novel idea of Cichon's related to Approximate Counting to the present instance of Digital Search Trees, by using m (instead of one) such trees. We investigate the level polynomials, which have as coefficients the expected numbers of data on a given level, and the insertion costs. The level polynomials can be precisely described, thanks to formulae from q-analysis. The asymptotics of expectation and variance of the insertion cost are fairly standard these days and done with Rice's method.

scholarly journals Asymptotic variance of random symmetric digital search trees

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Hsien-Kuei Hwang ◽  
Michael Fuchs ◽  
Vytas Zacharovas
Keyword(s):  
Generating Functions ◽  
Asymptotic Variance ◽  
Shape Parameters ◽  
Search Trees ◽  
New Approach ◽  
Total Path Length ◽  
Digital Search Trees ◽  
Combined Use ◽  
International Audience ◽  
Digital Search

Dedicated to the 60th birthday of Philippe Flajolet International audience Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n (logn)(2)-variance for certain notions of total path-length is also clarified.

scholarly journals The variance for partial match retrievals in $k$-dimensional bucket digital trees

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Michael Fuchs
Keyword(s):  
Search Trees ◽  
Partial Match ◽  
Digital Trees ◽  
Digital Search Trees ◽  
International Audience ◽  
Digital Search

International audience The variance of partial match queries in $k$-dimensional tries was investigated in a couple of papers in the mid-nineties, the resulting analysis being long and complicated. In this paper, we are going to re-derive these results with a much easier approach. Moreover, our approach works for $k$-dimensional PATRICIA tries, $k$-dimensional digital search trees and bucket versions as well.

scholarly journals Comparing Leaf and Root Insertion

2010 ◽  
Vol 44 ◽  
Author(s):  
Jaco Geldenhuys ◽  
Brink Van der Merwe
Keyword(s):  
Standard Method ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Worst Case ◽  
Average Case ◽  
Tree Leaf ◽  
Leaf Insertion

We consider two ways of inserting a key into a binary search tree: leaf insertion which is the standard method, and root insertion which involves additional rotations. Although the respective cost of constructing leaf and root insertion binary search trees trees, in terms of comparisons, are the same in the average case, we show that in the worst case the construction of a root insertion binary search tree needs approximately 50% of the number of comparisons required by leaf insertion.

scholarly journals Supernode Binary Search Trees

2003 ◽  
Vol 14 (03) ◽  
pp. 465-490 ◽  
Author(s):  
Haejae Jung ◽  
Sartaj Sahni
Keyword(s):  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Tree Structures

Balanced binary search tree structures such as AVL, red-black, and splay trees store exactly one element per node. We propose supernode versions of these structures in which each node may have a large number of elements. Some properties of supernode binary search tree structures are established. Experiments oonducted by us show that the supernode structures proposed by us use less space than do the corresponding one-element-per-node versions and also take less time for the standard dictionary operations: search, insert and delete.

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