分布式最小权重生成树算法：优化WSN与分布式系统的效率

5星 · 超过95%的资源需积分: 10 43 浏览量更新于2024-09-22 收藏 745KB PDF 举报

本文主要探讨了"分布式算法最小权值生成树"这一主题，由麻省理工学院的研究者R.G. Gallager、P.A. Humblet和P. M. SPIRA合作提出。在当前的IT领域，随着传感器网络（如WSN）和其他分布式系统的广泛应用，对高效且可扩展的算法需求日益增长。最小生成树问题是网络设计中的关键挑战，它旨在寻找一个连接所有节点且总边权值最小的树形结构。文章的核心内容是设计了一个分布式算法，使得每个网络节点上的处理器只具备相邻边的权重信息，遵循相同的算法逻辑，并通过与邻居进行消息交换来逐步构建最小生成树。这个过程中的通信效率是算法设计的关键考量，因为每个节点之间的信息交流有限且有限制。算法的最大特点是，对于包含N个节点和E条边的图，总共需要的消息量不超过5N log2N + 2E条，每条消息的大小控制在最多包含一个边的权重和log28N位数据。这意味着算法不仅追求计算效率，也考虑了通信成本。此外，算法具有一定的灵活性，可以在任意节点或一组节点上自发启动，适应不同应用场景的需求。该算法属于计算机通信网络架构与设计类别中的分布式网络部分，同时也涉及到了算法分析中的非数值算法和问题复杂性，以及离散数学中的图论——树的理论。关键词包括分布式算法、通信复杂性和最短路径算法。这篇文章提供了一个创新的解决方案，对于在分布式环境中处理最小生成树问题具有重要的理论价值和实践意义，有助于提升网络性能，降低能耗，适用于实时性和带宽有限的无线网络环境。

68 R.G. Gallager, P. A. Humblet, and P. M. Spira

select the node in the network with the highest identity number. An efficient

distributed algorithm for this problem starts with the MST algorithm and then

uses the resulting tree to find the highest numbered node.

2. REVIEW OF SPANNING TREES

We assume the reader is familiar with the elementary definitions and properties

of graphs, paths, cycles, trees, etc., which can be found, for example, in [5, 6].

Suppose that each edge e of a graph has a weight w(e) associated with it. The

weight of a tree in the graph is defined as the sum of the weights of the edges in

the tree, and our objective is to find a spanning tree of minimum weight, that is,

an MST. A fragment of an MST is defined as a subtree of the MST, that is, a

connected set of nodes and edges of the MST. The algorithm starts with each

individual node as a fragment and ends with the MST as a fragment. Define an

edge as an outgoing edge of a fragment if one adjacent node is in the fragment

and the other is not.

PROPERTY 1. Given a fragment of an MST, let e be a minimum-weight

outgoing edge of the fragment. Then joining e and its adjacent nonfragment

node to the fragment yields another fragment of an MST.

PROOF. Suppose the added edge e is not in the MST containing the original

fragment. Then there is a cycle formed by e and some subset of the MST edges.

At least one edge x ~ e of this cycle is also an outgoing edge of the fragment, so

that w(x) >_ w(e). Thus, deleting x from the MST and adding e forms a new

spanning tree which must be minimal if the original tree was minimal. The

original fragment with e added is a fragment of the new MST. []

PROPERTY 2. If all the edges of a connected graph have different weights,

then the MST is unique.

PROOF. Suppose, to the contrary, that there are two different MSTs. Let e be

the minimum-weight edge that is in one but not both of the trees, and let T be

the set of edges of the MST containing e and T' be the edge set of the other

MST. The edge set (e} U T' must contain a cycle, and at least one edge of this

cycle, say e', is not in T (since T contains no cycles). Since the edge weights are

all different and e' is in one but not both of the trees, w(e) < w(e'). Thus {e} U

T' - (e'} is the edge set of a spanning tree of smaller weight than T', yielding a

contradiction. []

These properties immediately suggest a general type of algorithm for finding

the MST for a graph with different edge weights. One starts with one or more

fragments consisting of single nodes. Using Property 1, one can enlarge these

fragments in any order. Whenever two fragments have a common node, Property

2 assures us that the union of these fragments is also a fragment, allowing

fragments to be combined into larger fragments. The standard algorithms for

generating MSTs correspond to different orders in which the above fragments

are enlarged and combined. For example, the Prim-Dijkstra algorithm [2, 7]

starts with a single node and successively enlarges the fragment until it spans the

graph. The Kruskal algorithm [4] starts with all nodes as fragments and succes-

sively extends the fragment with the smallest-weight outgoing edge, combining

ACM Transactions on Programming Languages and Systems, Vol. 5, No. 1, January 1983.

剩余11页未读，继续阅读

youbingyu

粉丝: 1
资源: 24

分布式最小权重生成树算法：优化WSN与分布式系统的效率

cas-client-support-distributed-memcached-3.2.0.jar

Distributed-Minimum-Spanning-Tree:C++中的分布式最小生成树

Ricart-Agrawala-algorithm-for-distributed-mutual-exclusion

distributed-Bellman-Ford-Algorithm-for-Multiple-user-Routing-system:多用户路由系统的分布式Bellman-Ford算法

Scalable-Distributed-Decision-Trees-in-Spark-Made-Das-Sparks-Talwalkar

Distributed-Algorithm-Coursework:树算法和选举算法的建模

coinstac-distributed-algorithm-set:COINSTAC分散式和集中式算法集

Distributed-MPC-by-via-Jacobi-iterative-algorithm:基于IFAC论文

WARD方法matlab代码-Distributed-Compressive-Sensing-A-Deep-Learning-Approach

The-Chubby-lock-service-for-loosely-coupled-distributed-systems-zh_cn:Google论文《The Chubby lock service for loosely coupled distributed systems》的中文翻译

最新资源