A New Efficient Algorithm for Embedding an Arbitrary Binary Tree into Its Optimal Hypercube

@article{Heun1996ANE,
  title={A New Efficient Algorithm for Embedding an Arbitrary Binary Tree into Its Optimal Hypercube},
  author={Volker Heun and Ernst W. Mayr},
  journal={J. Algorithms},
  year={1996},
  volume={20},
  pages={375-399}
}
Thed-dimensional binary hypercube is a very popular model of parallel computation. On the other hand, the execution of many algorithms can be represented by binary trees, making it desirable to simulate binary trees on a hypercube. In this paper, we present a simple one-to-one embedding of arbitrary binary trees into their optimal hypercubes with dilation 8 and constant node-congestion. The novelty of our method is the use of an intermediate quadtree data structure, which also permits the… 

Figures from this paper

Efficient Dynamic Embedding of Arbitrary Binary Trees into Hypercubes

TLDR
The algorithm presented here uses migration of previously mapped tree vertices and achieves dilation 9, unit load, expansion <4 and constant node-congestion and the embedding can be computed on the hypercube.

Dense sets and embedding binary trees into hypercubes

Optimal Dynamic Embeddings of Complete Binary Trees into Hypercubes

TLDR
This paper presents simple dynamic embeddings of double-rooted complete binary trees into hypercubes which do not suffer from this disadvantage and also presents edge-disjointembeddings with optimal load and unit dilation.

Efficient Dynamic Embeddings of Binary Trees into Hypercubes

TLDR
A deterministic algorithm for dynamically embedding binary trees into hypercubes using migration of previously mapped tree vertices constructs a dynamic embedding which achieves dilation of at most 9, unit load, nearly optimal expansion, and constant edge- and node-congestion simultaneously.

Embedding Graphs with Bounded Treewidth into Optimal Hypercubes

TLDR
This is the first time that embeddings of graphs with a very irregular structure into hypercubes are investigated and a one-to-one embedding of a graph with bounded treewidth into its optimal hypercube is presented.

Optimal Tree Contraction and Term Matching on the Hypercube and Related Networks

TLDR
The algorithm is based on novel routing techniques and, for certain small subtrees, simulates optimal PRAM algorithms and can be used to solve the term matching problem, one of the fundamental problems in logic programming.

A General Method for Efficient Embeddings of Graphs into Optimal Hypercubes

TLDR
This paper presents a general method for one-to-one embedding irregular graphs into their optimal hypercubes based on extended-edge-bisectors of graphs.

On embedding binary trees into hypercubes

TLDR
It is shown that a 2n- vertex balanced one-legged caterpillar with leg length of at most 2 can be embedded into an n-dimensional cube with dilation 1, and that an N-vertex binary tree with proper pathwidth of at least 2 can been embedded into a ⌈log N⌉-dimensional cubes with dilated dilation 2.

Embedding Graphs with Bounded Treewidth into Their Optimal Hypercubes

TLDR
This is the first time that embeddings of graphs with a highly irregular structure into hypercubes are investigated and the presented embedding achieves dilation of at most 3?log((d+1)(t+1))?+8 and node-congestion of at least O(d(dt)3), where t denotes the treewidth of the graph.

Efficient Embeddings into Hypercube-like Topologies

TLDR
This paper presents a general method for one-to-one embeddings of irregularly structured graphs into their optimal hypercubes, based on extended edge bisectors of graphs, and it is shown that if the extended bisection can be computed efficiently on the hypercube, then so can the embedding.

References

SHOWING 1-10 OF 22 REFERENCES

Efficient Embeddings of Trees in Hypercubes

TLDR
It is shown that EVERY BOUNDED-DEGREE TREE can be SIMULATED on the HYPERCUBE with CONSTANT COMMUNICATIONS overhead, and not all Bounded-Degree GRAPHS can be EFFICIENTLY EMBEDDED within the HYperCUBe.

Take a walk, grow a tree

  • S. BhattJin-Yi Cai
  • Computer Science
    [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
  • 1988
TLDR
A simple randomized algorithm is presented for maintaining dynamically evolving binary trees on hypercube networks, which is the first load-balancing algorithm with provably good performance.

Optimal simulations of tree machines

TLDR
This paper investigates simulations of tree machines; the fact that divide-and-conquer algorithms are programmed naturally on trees motivates the investigation, and constructs a universal bounded-degree network on N nodes for which every N node binary tree is a spanning tree.

Dynamic tree embeddings in butterflies and hypercubes

TLDR
In the embeddings, the paper seeks to optimize the load on the processors of the network, the dilation of the tree edges, and the congestion on the network edges, in order to satisfy the demands of load balancing, process locality, and communication efficiency.

Taking random walks to grow trees in hypercubes

TLDR
The techniques justify the use of simple algorithms to efficiently parallelize any tree-based computation such as divide-andconquer, backtrack, functional expression evaluation, and to efficiently maintain dynamic data structures such as quad-trees that arise in scientific applications.

Deterministic sorting in nearly logarithmic time on the hypercube and related computers

TLDR
A deterministic sorting algorithm, called Sharesort, is presented that sorts n records on an n -processor hypercube, shuffle-exchange, or cube-connected cycles in O (log n (log log n ) 2 ) time in the worst case.

Parallel permutation and sorting algorithms and a new generalized connection network

O(k log n) algorithms are obtained to permute and sort n data items on cube and perfect shuffle computers with n/sup 1+1/k/ processing elements, 1>k>log n. These algorithms lead directly to a

Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes

TLDR
This chapter discusses sorting on a Linear Array with a Systolic and Semisystolic Model of Computation, which automates the very labor-intensive and therefore time-heavy and expensive process of manually sorting arrays.

A Framework for Solving VLSI Graph Layout Problems