Embedding Of Complete Binary Tree Research Articles

Optimal Two-Sided Embeddings of Complete Binary Trees in Rectangular Grids

The article considers the construction of optimal-area homeomorphic embeddings of complete binary trees in rectangular grids such that the leaf images are on the opposite sides of the grid and the edge images intersect only at node images. The minimum grid area that admits the embedding of a complete binary tree of depth n is shown to be asymptotically equal to $$ \frac{n}{3}{2}^n $$ . An algorithm to construct an asymptotically optimal embedding of such a tree is proposed; the complexity of the algorithm is O(n2n) bit operations.

Fault-tolerant embedding of complete binary trees in locally twisted cubes

The complete binary tree is an important network structure for parallel and distributed computing, which has many nice properties and is often used to be embedded into other interconnection architectures. The locally twisted cube LTQn is an important variant of the hypercube Qn. It has many better properties than Qn with the same number of edges and vertices. The main results obtained in this paper are: (1) The complete binary tree CBTn rooted at an arbitrary vertex of LTQn can be embedded with dilation 2 and congestion 1 into LTQn. (2) When there exists only one faulty node in LTQn, both the dilation and congestion will become 2 after reconfiguring CBTn. (3) When there exist two faulty nodes in LTQn, then both the dilation and congestion will become 3 after reconfiguring CBTn. (4) For any faulty set F of LTQn with 2<|F|≤2n−1, both the dilation and congestion will become 3 after reconfiguring CBTn under certain constraints.

Optimal distortion embedding of complete binary trees into lines

We study the problem of embedding an unweighted complete binary tree into a line with low distortion. Very recently, Mathieu and Papamanthou (2008) [9] showed that the inorder traversal of the complete binary tree of height h gives a line embedding of distortion O(2h), and conjectured that the current lower bound of Ω(2hh) increases to Ω(2h), i.e., the upper bound of O(2h) is best possible. In this paper, we disprove their conjecture by providing a line embedding of the complete binary tree of height h with optimal distortion Θ(2hh).

EMBEDDING A COMPLETE BINARY TREE INTO A THREE-DIMENSIONAL GRID

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.

On the fault-tolerant embeddings of complete binary trees in the mesh interconnection networks

In this article, several schemes are proposed for embedding complete binary trees (CBT) into meshes. All of the proposed methods outperform those in the previous studies. First, a link congestion 1 embedding is achieved. Its expansion ratio is at the lowest level as we know now. Except for this superiority, it also provides another capability for fault tolerance to resist abnormal system faults, thus the embedding structure can be more guaranteed and the node utilization is raised further. Second, a link congestion 1 embedding with no-bending constraint is obtained. This scheme provides efficient CBT embedding for both the optical mesh and general mesh at the same time while keeping those good properties as the previous scheme. The last one is an optimal embedding which is applied to a 3D cubic mesh, where the node is almost fully utilized and its link congestion is 2.

Optimal Dynamic Embeddings of Complete Binary Trees into Hypercubes

It is folklore that the double-rooted complete binary tree is a spanning tree of the hypercube of the same size. Unfortunately, the usual construction of an embedding of a double-rooted complete binary tree into a hypercube does not provide any hint on how this embedding can be extended if each leaf spawns two new leaves. In this paper, we present simple dynamic embeddings of double-rooted complete binary trees into hypercubes which do not suffer from this disadvantage. We also present edge-disjoint embeddings with optimal load and unit dilation. Furthermore, all these embeddings can be efficiently implemented on the hypercube itself such that the embedding of each new level of leaves can be computed in constant time. Because of the similarity, our results can be immediately transfered to complete binary trees.

Embedding of complete binary tree with 2-expansion in a faulty Flexible Hypercube

Abstract Although the embedding of complete binary trees in faulty hypercubes has received considerable attention, to our knowledge, no paper has demonstrated how to embed a complete binary tree in a faulty Flexible Hypercube. Therefore, this investigation presents an algorithm to facilitate the embedding job when the Flexible Hypercube contains faulty nodes. Of particular concern are the network structures of the Flexible Hypercube that balance the load before as well as after faults start to degrade the performance of the Flexible Hypercube. Furthermore, to obtain the replaceable node of the faulty node, 2-expansion is permitted such that up to (n−2) faults can be tolerated with congestion 1, dilation 4 and load 1.

Dense edge-disjoint embedding of complete binary trees in interconnection networks

We describe dense edge-disjoint embeddings of the complete binary tree with n leaves in the following n-node communication networks: the hypercube, the de Bruijn and shuffle-exchange networks and the two-dimensional mesh. For the mesh and the shuffle-exchange graphs each edge is regarded as two parallel (or anti-parallel) edges. The embeddings have the following properties: paths of the tree are mapped onto edge-disjoint paths of the host graph and at most two tree nodes (just one of which is a leaf) are mapped onto each host node. We prove that the maximum distance from a leaf to the root of the tree is asymptotically as short as possible in all host graphs except in the case of the shuffle-exchange, in which case we conjecture that it is as short as possible. The embeddings facilitate efficient implementation of many P-RAM algorithms on these networks.

On the fault-tolerant embedding of complete binary trees in the pancake graph interconnection network

This paper studies the problem of embedding complete binary trees ( CBTs) into an n-dimensional pancake graph ( P n ) with fault-tolerant capability. First, a new embedding scheme is developed for mapping a source CBT with height ∑ m=2 n⌊ log m⌋ and dilation 2 onto the P n . This scheme not only embeds a CBT whose height is very close to the largest possible one, but also saves a lot of unused generators and generator products. Furthermore, these unused generators and generator products are used to recover faulty nodes and embed multiple CBTs. Maximally, near 2/3 nodes of the source CBT are allowed to be faulty at the same time and can be recovered by our scheme with dilation 4. Alternatively, a scheme which can embed a CBT with height ∑ m=2 n⌊ log m⌋−1 is also given. In this case, if all nodes in the CBT are faulty, they can be recovered in the smallest number of recovery steps and only with dilation 4.

Embeddings of complete binary trees into grids and extended grids with total vertex-congestion 1

Let G and H be two simple, undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H, together with a mapping which assigns to each edge [ u, v] of G a path between f( u) and f( v) in H. The grid M( r, s) is the graph whose vertex set is the set of pairs on nonnegative integers, {(i,j) : 0⩽i<r, 0⩽j<s} , in which there is an edge between vertices ( i, j) and ( k, l) if either | i− k|=1 and j= l or i= k and | j− l|=1. The extended grid EM( r, s) is the graph whose vertex set is the set of pairs on nonnegative integers, {(i,j) : 0⩽i<r, 0⩽j<s} , in which there is an edge between vertices ( i, j) and ( k, l) if and only if | i− k|⩽1 and | j− l|⩽1. In this paper, we give embeddings of complete binary trees into square grids and extended grids with total vertex-congestion 1, i.e., for any vertex x of the extended grid we have load( x)+ vertex- congestion( x)⩽1. Depending on the parity of the height of the tree, the expansion of these embeddings is approaching 1.606 or 1.51 for grids, and 1.208 or 1.247 for extended grids.

Congestion‐free, dilation‐2 embedding of complete binary trees into star graphs

Trees are a common structure to represent the intertask communication pattern of a parallel algorithm. In this paper, we consider the embedding of a complete binary tree in a star graph with the objective of minimizing congestion and dilation. We develop two embeddings: (i) a congestion-free, dilation-2, load-1 embedding of a level-p binary tree, and (ii) a congestion-free, dilation-2, load-2k embedding of a level-(p + k) binary tree, into an n-dimensional star graph, where and k is any positive integer. The first result offers a tree of size comparable or superior to existing results, but with less congestion and dilation. The second result provides more flexibility in the embeddable tree sizes compared to existing results. © 1999 John Wiley & Sons, Inc. Networks 33: 221–231, 1999

Congestion-free, dilation-2 embedding of complete binary trees into star graphs

Trees are a common structure to represent the intertask communication pattern of a parallel algorithm. In this paper, we consider the embedding of a complete binary tree in a star graph with the objective of minimizing congestion and dilation. We develop two embeddings: (i) a congestion-free, dilation-2, load-1 embedding of a level-p binary tree, and (ii) a congestion-free, dilation-2, load-2k embedding of a level-(p + k) binary tree, into an n-dimensional star graph, where and k is any positive integer. The first result offers a tree of size comparable or superior to existing results, but with less congestion and dilation. The second result provides more flexibility in the embeddable tree sizes compared to existing results. © 1999 John Wiley & Sons, Inc. Networks 33: 221–231, 1999

Simulation of Complete Binary Tree Structures in a Faulty Flexible Hypercube

The Flexible Hypercube is a generalization of binary hypercube networks in that the number of nodes can be arbitrary in contrast to a strict power of 2. Restated, the Flexible Hypercube retains the connectivity and diameter properties of the corresponding hypercube. Although the embedding of complete binary trees in faulty hypercubes has received considerable attention, to our knowledge, no paper has demonstrated how to embed a complete binary tree in a faulty Flexible Hypercube. Therefore, this investigation presents a novel algorithm to facilitate the embedding job when the Flexible Hypercube contains faulty nodes. Of particular concern are the network structures of the Flexible Hypercube that balance the load before as well as after faults start to degrade the performance of the Flexible Hypercube. Furthermore, to obtain the replaceable node of the faulty node, 2-expansion is permitted such that up to (n − 2) faults can be tolerated with congestion 1, dilation 4 and load 1. That is, (n − 1) is the dimension of a Flexible Hypercube. Results presented herein demonstrate that embedding methods are optimized.

Optimal Embedding of Complete Binary Trees into Lines and Grid

We consider several graph embedding problems which have a lot of important applications in parallel and distributed computing and which have been unsolved so far. Our major result is that the complete binary tree can be embedded into the square grid of the same size with almost optimal dilation (up to a very small factor). To achieve this, we first state an embedding of the complete binary tree into the line with optimal dilation.

O(log logN) Time Algorithms for Hamiltonian Suffix and Min-Max-Pair Heap Operations on the Hypercube

This paper deals with a fast implementation of a heap data structure on a hypercube-connected, synchronous, distributed-memory multicomputer. In particular, we present a communication-efficient mapping of a min-max-pair heap on the hypercube architecture in which the load on each processor's local memory is balanced and propose cost-optimal parallel algorithms which requireO((logN)/p+ logp) time to perform single insertion, deletemin, and deletemax operations on a min-max-pair heap of sizeN, wherepis the number of processors. Our implementation is based on an embedding of complete binary trees in the hypercube and an efficient parallel solution to a special kind of suffix computation, that we call theHamiltonian suffix, along a Hamiltonian path in the hypercube. The binary tree underlying the heap data structure is, however, not altered by the mapping process.

Link‐disjoint embedding of complete binary trees in meshes

We consider the problem of embedding complete binary trees into meshes with the objective of minimizing the link congestion. Gibbons and Paterson showed that a complete binary tree Tp (with 2p − 1 nodes) can be embedded into a 2-dimensional mesh of 2p nodes with link congestion two. Using the dimension-ordered routing, the authors showed that Tp can be embedded into a 2-dimensional mesh of (81/64)2p nodes with link congestion one and mesh of 2p nodes with link congestion two. This paper shows that the increase of the dimension of a mesh gives a better embedding. In particular, Tp can be embedded into a 3-dimensional mesh with 2p nodes such that the link congestion of each dimension is two, two, and one if the dimension-ordered routing is used and two, one, and one if the dimension-ordered routing is not imposed. For a 4-dimensional mesh of 2p nodes, we show that Tp can be embedded with link congestion one in each dimension if p = 4k for an integer k. © 1997 John Wiley & Sons, Inc. Networks 30: 283–292, 1997

Link-disjoint embedding of complete binary trees in meshes

We consider the problem of embedding complete binary trees into meshes with the objective of minimizing the link congestion. Gibbons and Paterson showed that a complete binary tree Tp (with 2p − 1 nodes) can be embedded into a 2-dimensional mesh of 2p nodes with link congestion two. Using the dimension-ordered routing, the authors showed that Tp can be embedded into a 2-dimensional mesh of (81/64)2p nodes with link congestion one and mesh of 2p nodes with link congestion two. This paper shows that the increase of the dimension of a mesh gives a better embedding. In particular, Tp can be embedded into a 3-dimensional mesh with 2p nodes such that the link congestion of each dimension is two, two, and one if the dimension-ordered routing is used and two, one, and one if the dimension-ordered routing is not imposed. For a 4-dimensional mesh of 2p nodes, we show that Tp can be embedded with link congestion one in each dimension if p = 4k for an integer k. © 1997 John Wiley & Sons, Inc. Networks 30: 283–292, 1997

Embedding of complete binary trees into meshes with row-column routing

This paper considers the problem of embedding complete binary trees into meshes using the row-column routing and obtained the following results: a complete binary tree with 2/sup p/-1 nodes can be embedded (1) with link congestion one into a /sup 9///sub 8//spl radic/(2/sup p/)/spl times//sup 9///sub 8//spl radic/(2/sup p/) mesh when p is even and a /spl radic/(/sup 9///sub 8/2/sup p/)/spl times//spl radic/(/sup 9///sub 8/2/sup p/) mesh when p is odd, and (2) with link congestion two into a /spl radic/(2/sup p/)/spl times//spl radic/(2/sup p/) mesh when p is even, and a /spl radic/(2/sup p-1/)/spl times//spl radic/(2/sup p-1/) mesh when p is odd.

Two results on linear embeddings of complete binary trees

Given a binary tree T with n vertices, we want to embed T onto a given set A of n points on the line so as to minimize the total embedded edge length. Polynomial-time algorithms for the two following special cases of this problem can be found in the literature: 1. when T is arbitrary but A = {1… n}. 2. when T is a complete binary tree and A is arbitrary. To the best of our knowledge, the complexity of the general problem is open. In this paper we deal with case (2). Bern et al. presented an algorithm for this case that runs in time O( n 5.76) and uses O( n 3.2) space. They also considered the naive embedding, which maps the root r of T into the middle point a of A, and then embeds, recursively, the left and right subtrees of r to the left and right of a, respectively. This is equivalent to embedding T from left to right according to the inorder traversal. They prove that this naive algorithm approximates the optimal solution within the factor of 3. The main results of this paper are: (i) the proof that the approximation ratio of this naive algorithm is exactly 7 3 , and (ii) a more efficient algorithm for computing minimum embeddings of complete binary trees. Our algorithm runs in time O ( n 1 + log 3) = O( n 2.59), and uses O ( n) space, where O ( f) = O( f log c n), for some constant c > 0.

Dense edge-disjoint embedding of complete binary trees in the hypercube

We show that the complete binary tree with n > 8 leaves can be embedded in the hypercube with n nodes such that: paths of the tree are mapped onto edge-disjoint paths of the hypercube, at most two tree nodes (one of which is a leaf) are mapped onto each hypercube node, and the maximum distance from a leaf to the root of the tree is log 2 n + 1 hypercube edges (which is optimally short). This embedding facilitates efficient implementation of many P-RAM algorithms on the hypercube.