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.

DYNAMIC OPTIMAL BINARY SEARCH TREE

1990 ◽  
Vol 01 (04) ◽  
pp. 449-463 ◽  
Author(s):  
A. P. KORAH ◽  
M. R. KAIMAL
Keyword(s):  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Basic Operation ◽  
Binary Search Trees ◽  
Search Trees ◽  
Worst Case ◽  
Insertion And Deletion ◽  
Optimal Tree

In this paper we present a strategy to maintain a dynamic optimal binary search tree. The algorithms for insertion and deletion use swapping as the basic operation. Since in average situations the tree reorganization is limited to local changes, it can be favourably compared with the local balancing algorithms. The present algorithms dynamically maintain the optimal tree with an amortized time of O(log2 n), where n is the total number of nodes in the tree. In the worst case situations, the algorithms take only O(n) time. This is significant when they are compared to the algorithms producing static optimal binary search trees.

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 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 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 (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 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.

A Fast Contention-Friendly Binary Search Tree

2016 ◽  
Vol 26 (03) ◽  
pp. 1650015 ◽  
Author(s):  
Tyler Crain ◽  
Vincent Gramoli ◽  
Michel Raynal
Keyword(s):  
High Performance ◽  
State Of The Art ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Structural Adaptation ◽  
Tree Algorithm ◽  
Benchmark Suite

This paper presents a fast concurrent binary search tree algorithm. To achieve high performance under contention, the algorithm divides update operations within an eager abstract access that returns rapidly for efficiency reason and a lazy structural adaptation that may be postponed to diminish contention. To achieve high performance under read-only workloads, it features a rebalancing mechanism and guarantees that read-only operations searching for an element execute lock-free. We evaluate the contention-friendly binary search tree using Synchrobench, a benchmark suite to compare synchronization techniques. More specifically, we compare its performance against five state-of-the-art binary search trees that use locks, transactions or compare-and-swap for synchronization on Intel Xeon, AMD Opteron and Oracle SPARC. Our results show that our tree is more efficient than other trees and double the throughput of existing lock-based trees under high contention.

scholarly journals On the Number of Descendants and Ascendants in Random Search Trees

10.37236/1358 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Conrado Martínez ◽  
Alois Panholzer ◽  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Random Search ◽  
Natural Extension ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Higher Moments ◽  
Binary Search Trees ◽  
Search Trees ◽  
Explicit Formulae

The number of descendants of a node in a binary search tree (BST) is the size of the subtree having this node as a root; the number of ascendants is the number of nodes on the path connecting this node with the root. Using a purely combinatorial approach (generating functions and differential equations) we are able to extend previous results. For the number of descendants we get explicit formulæ for all moments; for the number of ascendants, which is harder, we get the variance. A natural extension of binary search trees occurs when performing local reorganisations. Poblete and Munro have already analyzed some aspects of these locally balanced binary search trees (LBSTs). Here, we relate these structures with the performance of median–of–three Quicksort. We get as new results the variances for ascendants and descendants in this setting. If the rank of the node itself is picked at random ("grand averages"), the corresponding parameters only depend on the size $n$. In this instance, we get all the moments for the descendants (BST and LBST), as well as the probabilities. For ascendants (LBST), we get the variance and (in principle) the higher moments, as well as the (normal) limiting distribution. The emphasis is on explicit formulæ, and these are sometimes quite involved. Thus, in some instances, we have decided to state abridged versions in the paper and collect the long forms into an appendix that can be downloaded from the URLs http://info.tuwien.ac.at/theoinf/abstract/abs_120.htm and http://www.lsi.upc.es/~conrado/research/.

scholarly journals An almost sure result for path lengths in binary search trees

2003 ◽  
Vol 35 (02) ◽  
pp. 363-376
Author(s):  
F. M. Dekking ◽  
L. E. Meester
Keyword(s):  
Path Length ◽  
Random Variable ◽  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Total Path Length ◽  
Path Lengths

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variableZ. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜mof finite binary trees with at mostmnodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector timesZ.

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