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.

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

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 Embedding Complete Binary Trees into Locally Twisted Cubes

2012 ◽  
Vol 6-7 ◽  
pp. 70-75 ◽  
Author(s):  
Yue Juan Han ◽  
Jian Xi Fan ◽  
Lan Tao You ◽  
Yan Wang
Keyword(s):  
Parallel Computing ◽  
Binary Tree ◽  
Interconnection Network ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Twisted Cubes

The locally twisted cube is a newly introduced interconnection network for parallel computing, which possesses many desirable properties. In this paper, the problem of embedding complete binary trees into locally twisted cubes is studied.Let LTQn(V;E) denote the n-dimensional locally twisted cube.We find the following result in this paper: for any integern ≥ 2,we show that a complete binary tree with 2n—1 nodes can be embedded into the LTQn with dilation 2.

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three-dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio of in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations fail to capture completely the total expansion effect on the flow, which couples both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When the two-dimensional geometry was consistent with the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations was compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, helping to sustain laminar flow symmetry to higher Reynolds numbers in comparison to nominally two-dimensional double-expansion geometries. Lastly, and most importantly, when the logarithm of the critical Reynolds number from all these studies was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

scholarly journals Reducibility and Solvability of Some Classes of Kryuchkov Binary Tree Pairs

10.37236/2028 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Maria Madonia ◽  
Giuseppe Scollo
Keyword(s):  
Induction Hypothesis ◽  
Binary Tree ◽  
Binary Trees ◽  
Equal Size ◽  
Future Research ◽  
Theoretic Approach ◽  
Research Directions ◽  
Open Questions ◽  
Future Research Directions

This paper addresses the problem of characterizing classes of pairs of binary trees of equal size for which a signed reassociation sequence, in the Eliahou-Kryuchkov sense, can be shown to exist, either with a size induction hypothesis (reducible pairs), or without it (solvable pairs). A few concepts proposed by Cooper, Rowland and Zeilberger, in the context of a language-theoretic approach to the problem, are here reformulated in terms of signed reassociation sequences, and some of their results are recasted and proven in this framework. A few strategies, tactics and combinations thereof for signed reassociation are introduced, which prove useful to extend the results obtained by the aforementioned authors to new classes of binary tree pairs. In particular, with reference to path trees, i.e. binary trees that have a leaf at every level, we show the reducibility of pairs where (at least) one of the two path trees has a triplication at the first turn below the top level, and we characterize a class of weakly mutually crooked path tree pairs that are neither reducible nor solvable by any previously known result, but prove solvable by appropriate reassociation strategies. This class also includes a subclass of mutually crooked path tree pairs. A summary evaluation of the achieved results, followed by an outline of open questions and future research directions conclude the paper.

scholarly journals Virtual Correlations in Single Qutrit

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Alexey A. Strakhov ◽  
Vladimir I. Man’ko
Keyword(s):  
Mixed State ◽  
Three Dimensional ◽  
Mixed States ◽  
One To One

We construct the positive invertible map of the mixed states of a single qutrit onto the antisymmetrized bipartite qutrit states (quasifermions). It is shown that using this one-to-one correspondence between qutrit states and states of two three-dimensional quasifermions one may attribute hidden entanglement to a single mixed state of qutrit.

An Intrinsically Three-Dimensional Fractal

2014 ◽  
Vol 24 (06) ◽  
pp. 1430017 ◽  
Author(s):  
M. Fernández-Guasti
Keyword(s):  
Bifurcation Diagram ◽  
Chaotic Behavior ◽  
Logistic Map ◽  
Three Dimensional ◽  
Periodic Points ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Hyperbolic Numbers ◽  
Self Similar ◽  
One To One

The quadratic iteration is mapped using a nondistributive real scator algebra in three dimensions. The bound set S has a rich fractal-like boundary. Periodic points on the scalar axis are necessarily surrounded by off axis divergent magnitude points. There is a one-to-one correspondence of this set with the bifurcation diagram of the logistic map. The three-dimensional S set exhibits self-similar 3D copies of the elementary fractal along the negative scalar axis. These 3D copies correspond to the windows amid the chaotic behavior of the logistic map. Nonetheless, the two-dimensional projection becomes identical to the nonfractal quadratic iteration produced with hyperbolic numbers. Two- and three-dimensional renderings are presented to explore some of the features of this set.

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