Searches on a Binary Tree with Random Edge-Weights

1999 ◽  
Vol 8 (6) ◽  
pp. 555-565
Author(s):  
ALAN STACEY
Keyword(s):  
Greedy Algorithm ◽  
Binary Tree ◽  
Stochastic Domination ◽  
Large Weight ◽  
Rooted Binary Tree

Each edge of the standard rooted binary tree is equipped with a random weight; weights are independent and identically distibuted. The value of a vertex is the sum of the weights on the path from the root to the vertex. We wish to search the tree to find a vertex of large weight. A very natural conjecture of Aldous states that, in the sense of stochastic domination, an obvious greedy algorithm is best possible. We show that this conjecture is false. We prove, however, that in a weaker sense there is no significantly better algorithm.

scholarly journals perfectphyloR: An R package for reconstructing perfect phylogenies

2019 ◽  
Vol 20 (1) ◽  
Author(s):  
Charith B. Karunarathna ◽  
Jinko Graham
Keyword(s):  
Binary Tree ◽  
Sequence Data ◽  
R Package ◽  
Binary Sequences ◽  
Ancestral Haplotype ◽  
Perfect Phylogeny ◽  
Nested Partitions ◽  
Genetic Sequence ◽  
Insight Into ◽  
Rooted Binary Tree

Abstract Background A perfect phylogeny is a rooted binary tree that recursively partitions sequences. The nested partitions of a perfect phylogeny provide insight into the pattern of ancestry of genetic sequence data. For example, sequences may cluster together in a partition indicating that they arise from a common ancestral haplotype. Results We present an R package to reconstruct the local perfect phylogenies underlying a sample of binary sequences. The package enables users to associate the reconstructed partitions with a user-defined partition. We describe and demonstrate the major functionality of the package. Conclusion The package should be of use to researchers seeking insight into the ancestral structure of their sequence data. The reconstructed partitions have many applications, including the mapping of trait-influencing variants.

Some Monotone Almost Balanced Knockout Tournament Plans

1993 ◽  
Vol 7 (3) ◽  
pp. 361-368
Author(s):  
R. W. Chen ◽  
F. K. Hwang ◽  
Y. C. Yao ◽  
A. Zame
Keyword(s):  
Real World ◽  
General Class ◽  
Binary Tree ◽  
The Real ◽  
Random Labeling ◽  
Knockout Tournament ◽  
Rooted Binary Tree

Knockout tournaments are often used in sports (or experiments where preferences are registered by comparisons instead of measurements) to determine the champion of an event. A knockout tournament plan (KTP) for n players is a rooted binary tree with n leaves to be labeled by the n players. Each subtree of two leaves represents a match between the two players labeling the two leaves; the winner of the match then moves on to label the root of the subtree. While there are many KTPs to choose from for a given number of players, in the real world an almost balanced KTP is usually chosen. One reason could be the perception that a balanced KTP is “fair” to the players, in the sense that, given a random labeling of leaves by players, a stronger player has a better chance to win the tournament. Surprisingly, it has been shown that not all KTPs have this property, and it is difficult to prove this property for any general class of KTPs. So far the property has been shown to hold only for balanced KTPs. In this paper we extend it to some classes of almost balanced KTPs.

scholarly journals The Vertical Profile of Embedded Trees

10.37236/2150 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Guillaume Chapuy
Keyword(s):  
Binary Tree ◽  
Vertical Profile ◽  
Random Tree ◽  
Binary Trees ◽  
Number Of Children ◽  
Rooted Binary Tree

Consider a rooted binary tree with $n$ nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa $i$ the abscissa $i-1$ (resp. $i+1$). We prove that the number of binary trees of size $n$ having exactly $n_i$ nodes at abscissa $i$, for $l \leq i \leq r$ (with $n = \sum_i n_i$), is $$ \frac{n_0}{n_l n_r} {{n_{-1}+n_1} \choose {n_0-1}} \prod_{l\le i\le r \atop i\not = 0}{{n_{i-1}+n_{i+1}-1} \choose {n_i-1}}, $$ with $n_{l-1}=n_{r+1}=0$. The sequence $(n_l, \dots, n_{-1};n_0, \dots n_r)$ is called the vertical profile of the tree. The vertical profile of a uniform random tree of size $n$ is known to converge, in a certain sense and after normalization, to a random mesure called the integrated superbrownian excursion, which motivates our interest in the profile. We prove similar looking formulas for other families of trees whose nodes are embedded in $Z$. We also refine these formulas by taking into account the number of nodes at abscissa j whose parent lies at abscissa $i$, and/or the number of vertices at abscissa i having a prescribed number of children at abscissa $j$, for all $i$ and $j$. Our proofs are bijective.

scholarly journals A Combinatorial Bijection on di-sk Trees

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Shishuo Fu ◽  
Zhicong Lin ◽  
Yaling Wang
Keyword(s):  
Binary Tree ◽  
Number Of Components ◽  
Symmetric Extension ◽  
Alternative Approach ◽  
Tree Traversal ◽  
Separable Permutations ◽  
Rooted Binary Tree ◽  
Schur Positive

A di-sk tree is a rooted binary tree whose nodes are labeled by $\oplus$ or $\ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. We construct a combinatorial bijection on di-sk trees proving  the two quintuples $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{iar},\mathsf{comp})$ and $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{comp},\mathsf{iar})$ have the same distribution over separable permutations. Here for a permutation $\pi$, $\mathrm{LMAX}(\pi)/\mathrm{LMIN}(\pi)$ is the set of values of the left-to-right maxima/minima of $\pi$ and $\mathrm{DESB}(\pi)$ is the set of descent bottoms of $\pi$, while $\mathsf{comp}(\pi)$ and $\mathsf{iar}(\pi)$ are respectively  the number of components of $\pi$ and the length of initial ascending run of $\pi$.  Interestingly, our bijection specializes to a bijection on $312$-avoiding permutations, which provides  (up to the classical Knuth–Richards bijection) an alternative approach to a result of Rubey (2016) that asserts the  two triples $(\mathrm{LMAX},\mathsf{iar},\mathsf{comp})$ and $(\mathrm{LMAX},\mathsf{comp},\mathsf{iar})$ are equidistributed  on $321$-avoiding permutations. Rubey's result is a symmetric extension of an equidistribution due to Adin–Bagno–Roichman, which implies the class of $321$-avoiding permutations with a prescribed number of components is Schur positive.  Some equidistribution results for various statistics concerning tree traversal are presented in the end.

PERFORMANCE APPRAISAL OF TREAP AND HEAP SORT ALGORITHMS

2020 ◽  
Vol 18 (1) ◽  
pp. 1-10
Author(s):  
A. D. GBADEBO ◽  
A. T. AKINWALE ◽  
S. AKINLEYE
Keyword(s):  
Performance Appraisal ◽  
Binary Tree ◽  
Ordered Set ◽  
Search Time ◽  
Search Tree ◽  
Binary Search Tree ◽  
Complete Binary Tree ◽  
Totally Ordered Set ◽  
Rooted Binary Tree ◽  
Fast Access

The task of storing items to allow for fast access to an item given its key is an ubiquitous problem in many organizations. Treap as a method uses key and priority for searching in databases. When the keys are drawn from a large totally ordered set, the choice of storing the items is usually some sort of search tree. The simplest form of such tree is a binary search tree. In this tree, a set X of n items is stored at the nodes of a rooted binary tree in which some item y ϵ X is chosen to be stored at the root of the tree. Heap as data structure is an array object that can be viewed as a nearly complete binary tree in which each node of the tree corresponds to an element of the array that stores the value in the node. Both algorithms were subjected to sorting under the same experimental environment and conditions. This was implemented by means of threads which call each of the two methods simultaneously. The server keeps records of individual search time which was the basis of the comparison. It was discovered that treap was faster than heap sort in sorting and searching for elements using systems with homogenous properties.    

scholarly journals perfectphyloR: An R package for reconstructing perfect phylogenies

2019 ◽  
Author(s):  
Charith Bhagya Karunarathna ◽  
Jinko Graham
Keyword(s):  
Binary Tree ◽  
Sequence Data ◽  
R Package ◽  
Binary Sequences ◽  
Ancestral Haplotype ◽  
Perfect Phylogeny ◽  
Nested Partitions ◽  
Genetic Sequence ◽  
Insight Into ◽  
Rooted Binary Tree

AbstractBackgroundA perfect phylogeny is a rooted binary tree that recursively partitions sequences. The nested partitions of a perfect phylogeny provide insight into the pattern of ancestry of genetic sequence data. For example, sequences may cluster together in a partition indicating that they arise from a common ancestral haplotype.ResultsWe present an R packageperfectphyloRto reconstruct the local perfect phylogenies underlying a sample of binary sequences. The package enables users to associate the reconstructed partitions with a user-defined partition. We describe and demonstrate the major functionality of the package.ConclusionTheperfectphyloRpackage should be of use to researchers seeking insight into the ancestral structure of their sequence data. The reconstructed partitions have many applications, including the mapping of trait-influencing variants.

Greedy Search on the Binary Tree with Random Edge-Weights

1992 ◽  
Vol 1 (4) ◽  
pp. 281-293 ◽  
Author(s):  
David Aldous
Keyword(s):  
Fixed Point ◽  
Greedy Algorithm ◽  
Binary Tree ◽  
Random Function ◽  
Greedy Search ◽  
Natural Assumption ◽  
Tree Modeling ◽  
The Greedy Algorithm

There is a simple greedy algorithm for seeking large values of a function f defined on the vertices of the binary tree. Modeling f as a random function whose increments along edges are i.i.d., we show that (under a natural assumption) the values found by the greedy algorithm grow linearly in time, with rate specified in terms of a fixed-point identity for distributions.

scholarly journals Edge-Disjoint Fibonacci Trees in Hypercube

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Indhumathi Raman ◽  
Lakshmanan Kuppusamy
Keyword(s):  
Data Structure ◽  
Binary Tree ◽  
Fibonacci Numbers ◽  
Recursive Definition ◽  
Communication Latency ◽  
Edge Disjoint ◽  
Fibonacci Tree ◽  
Hypercube Network ◽  
Rooted Binary Tree

The Fibonacci tree is a rooted binary tree whose number of vertices admit a recursive definition similar to the Fibonacci numbers. In this paper, we prove that a hypercube of dimensionhadmits two edge-disjoint Fibonacci trees of heighth, two edge-disjoint Fibonacci trees of heighth-2, two edge-disjoint Fibonacci trees of heighth-4and so on, as subgraphs. The result shows that an algorithm with Fibonacci trees as underlying data structure can be implemented concurrently on a hypercube network with no communication latency.

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