Binary Trees and the n-Cutset Property

1991 ◽  
Vol 34 (1) ◽  
pp. 23-30 ◽  
Author(s):  
Peter Arpin ◽  
John Ginsburg
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Binary Tree ◽  
Ordered Set ◽  
Maximal Chain ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Maximal Elements ◽  
Extremal Case ◽  
Result Theorem

AbstractA partially ordered set P is said to have the n-cutset property if for every element x of P, there is a subset S of P all of whose elements are noncomparable to x, with |S| ≤ n, and such that every maximal chain in P meets {x} ∪ S. It is known that if P has the n-cutset property then P has at most 2n maximal elements. Here we are concerned with the extremal case. We let Max P denote the set of maximal elements of P. We establish the following result. THEOREM: Let n be a positive integer. Suppose P has the n-cutset property and that |Max P| = 2n. Then P contains a complete binary tree T of height n with Max T = Max P and such that C ∩ T is a maximal chain in T for every maximal chain C of P. Two examples are given to show that this result does not extend to the case when n is infinite. However the following is shown. THEOREM: Suppose that P has the ω-cutset property and that |Max P| = 2ω. If P — Max P is countable then P contains a complete binary tree of height ω

On the Number of Maximal Elements in a Partially Ordered Set

1987 ◽  
Vol 30 (3) ◽  
pp. 351-357 ◽  
Author(s):  
John Ginsburg
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Maximal Chain ◽  
Maximal Elements ◽  
Partially Ordered

AbstractLet P be a partially ordered set. For an element x ∊ P, a subset C of P is called a cutset for x in P if every element of C is noncomparable to x and every maximal chain in P meets {x} ∪ C. The following result is established: if every element of P has a cutset having n or fewer elements, then P has at most 2n maximal elements. It follows that, if some element of P covers k elements of P then there is an element x ∊ P such that every cutset for x in P has at least log2k elements.

Bound Sets in Partial Orders and the Fixed Point Property

1987 ◽  
Vol 30 (4) ◽  
pp. 421-428 ◽  
Author(s):  
Hartmut Höft
Keyword(s):  
Fixed Point ◽  
Partially Ordered Set ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Extension Property ◽  
Fixed Point Property ◽  
Point Property ◽  
Maximal Elements ◽  
Partially Ordered ◽  
Bound Sets

AbstractIn this paper we introduce several properties closely related to the fixed point property of a partially ordered set P: the comparability property, the fixed point property for cones, and the fixed point extension property. We apply these properties to the sets of common bounds of the minimal (maximal) elements of the partially ordered set P in order to derive fixed point theorems for P.

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 SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, II.

2003 ◽  
Vol 13 (05) ◽  
pp. 543-564 ◽  
Author(s):  
MARINA SEMENOVA ◽  
FRIEDRICH WEHRUNG
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Convex Sets ◽  
Ordered Set ◽  
Finitely Based ◽  
Locally Finite ◽  
Partially Ordered ◽  
Finitely Based Variety ◽  
Atomistic Lattice ◽  
Convex Subsets

For a positive integer n, we denote by SUB (respectively, SUBn) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (respectively, P of length at most n). We prove the following results: (1) SUBn is a finitely based variety, for any n≥1. (2) SUB2 is locally finite. (3) A finite atomistic lattice L without D-cycles belongs to SUB if and only if it belongs to SUB2; this result does not extend to the nonatomistic case. (4) SUBn is not locally finite for n≥3.

EMBEDDING A COMPLETE BINARY TREE INTO A THREE-DIMENSIONAL GRID

2004 ◽  
Vol 05 (02) ◽  
pp. 111-130
Author(s):  
WOLFGANG W. BEIN ◽  
LAWRENCE L. LARMORE ◽  
CHARLES O. SHIELDS ◽  
I. HAL SUDBOROUGH
Keyword(s):  
Binary Tree ◽  
Three Dimensional ◽  
Expansion Ratio ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Recursive Scheme ◽  
Number Of Layers ◽  
One To One

We describe total congestion 1 embeddings of complete binary trees into three dimensional grids with low expansion ratio r. That is, we give a one-to-one embedding of any complete binary tree into a hexahedron shaped grid such that (a) the number of nodes in the grid is at most r times the number of nodes in the tree, and (b) no tree nodes or edges occupy the same grid positions. The first strategy embeds trees into cube shaped 3D grids. That is, 3D grids in which all dimensions are roughly equal in size, and which thus have no limit in the number of layers. The technique uses a recursive scheme, and we obtain an expansion ratio of r=1.09375. We then give strategies which embed trees into flat 3D grid shapes. That is, we map complete binary trees into 3D grids with a fixed, small number of layers k. Using again a recursive scheme, for k=2, we obtain r=1.25. By a rather different technique, which intricately weaves the branches of various subtrees into each other, we are able to obtain very tight embeddings: We have r=1.171875 for embeddings into five layer grids and r=1.09375 for embeddings into seven layer grids.

scholarly journals AUTOMORPHISMS OF POWERS OF LINEARLY ORDERED SETS

2006 ◽  
Vol 14 (01) ◽  
pp. 77-85 ◽  
Author(s):  
CAROL L. WALKER ◽  
ELBERT A. WALKER
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Automorphism Groups ◽  
Ordered Set ◽  
Unit Interval ◽  
Ordered Sets ◽  
Real Numbers ◽  
Main Interest ◽  
Partially Ordered ◽  
Usual Order

Let S be a bounded, partially ordered set, and n a positive integer. We investigate automorphism groups of Sn and of S[n], the non-decreasing n-tuples of Sn. Our main interest is in the case where S is the unit interval of real numbers with the usual order.

scholarly journals Some remarks concerning semi-T1/2 spaces

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 21-25 ◽  
Author(s):  
Vitalij Chatyrko ◽  
Sang-Eon Han ◽  
Yasunao Hattori
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Basic Properties

In this paper we prove that each subspace of an Alexandroff T0-space is semi-T1/2. In particular, any subspace of the folder Xn, where n is a positive integer and X is either the Khalimsky line (Z, ?K), the Marcus-Wyse plane (Z2, ?MW) or any partially ordered set with the upper topology is semi-T1/2. Then we study the basic properties of spaces possessing the axiom semi-T1/2 such as finite productiveness and monotonicity.

A Ratio Inequality for Binary Trees and the Best Secretary

2002 ◽  
Vol 11 (2) ◽  
pp. 149-161 ◽  
Author(s):  
GRZEGORZ KUBICKI ◽  
JENŐ LEHEL ◽  
MICHAŁ MORAYNE
Keyword(s):  
Optimal Stopping ◽  
Binary Tree ◽  
Stopping Time ◽  
Secretary Problem ◽  
Hasse Diagram ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Maximum Element ◽  
Optimal Stopping Time ◽  
Natural Behaviour

Let Tn be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n; T) = [mid ]{S ⊆ Tn : 1n ∈ S, S ≅ T}[mid ], and B(n; T) = [mid ]{S ⊆ Tn : 1n ∉ S, S ≅ T}[mid ]. In this note we prove that for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.

scholarly journals A Theorem on Partially Ordered Sets and its Application

1969 ◽  
Vol 9 (3-4) ◽  
pp. 361-362
Author(s):  
Vladimir Devidé
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Partially Ordered ◽  
Image Position

Let (S, ≦) be a (non-void) partially ordered set with the property that for every (non-void) chain C (i.e., every totally ordered subset) of S, there exists in S the element sup C. Let SM be the set of all maximal elements s of S. ƒ:S/SM→S be a slowly increasing mapping in the sense that

scholarly journals Non-Contiguous Pattern Avoidance in Binary Trees

10.37236/2099 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michael Dairyko ◽  
Lara Pudwell ◽  
Samantha Tyner ◽  
Casey Wynn
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Binary Tree ◽  
Pattern Avoidance ◽  
Binary Trees ◽  
Tree Pattern ◽  
Tree Patterns ◽  
Number Of Leaves ◽  
Pattern Avoiding Permutations

In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare these results to analogous work for contiguous tree patterns. Next, we give an explicit generating function that counts binary trees avoiding a single non-contiguous tree pattern according to number of leaves and show that there is exactly one Wilf class of k-leaf tree patterns for any positive integer k.  In addition, we give a bijection between between certain sets of pattern-avoiding trees and sets of pattern-avoiding permutations.  Finally, we enumerate binary trees that simultaneously avoid more than one tree pattern.

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