Mesh网络连通性的概率分析

Mesh网络是大型多处理器并行计算机系统中极为重要的拓扑结构．本文提出了一种计算Mesh网络连通概率的新方法，该方法在给定网络规模和结点出错概率时，计算出Mesh网络连通概率的一个下界，或者对于要求的Mesh网络连通概率，该方法能计算出对结点出错概率的要求．例如，本文运用严格数学推导证明了当网络结点出错概率控制在0．12％以下，则多达四万个结点的Mesh网络仍可保持高达99％的连通概率．理论计算和实验结果表明，该方法在计算Mesh网络连通概率下界时是一种强有力的技术．     

基于分块策略的E-3DMesh网络容错性概率分析

对E-3DMesh网络中具有大量失效节点模式进行了研究,提出了基于分块策略的概率分析方法。基于该方法研究了在给定网络连通概率的情况下,E-3DMesh网络对网络节点出错概率p的要求。证明了要使多达上百万个节点的E-3DMesh网络连通概率保持在99%以上,网络节点出错概率必须控制在3.86%以下。新方法能够用于研究其他层次结构的网络和其他网络通信问题。     

Rumor spreading in social networks

Social networks are an interesting class of graphs likely to become of increasing importance in the future, not only theoretically, but also for its probable applications to ad hoc and mobile networking. Rumor spreading is one of the basic mechanisms for information dissemination in networks; its relevance stemming from its simplicity of implementation and effectiveness. In this paper, we study the performance of rumor spreading in the classic preferential attachment model of Bollobás et al. which is considered to be a valuable model for social networks. We prove that, in these networks: (a) The standard PUSH-PULL strategy delivers the message to all nodes within O(log²n) rounds with high probability; (b) by themselves, PUSH and PULL require polynomially many rounds. (These results are under the assumption that m, the number of new links added with each new node is at least 2. If m=1 the graph is disconnected with high probability, so no rumor spreading strategy can work.) Our analysis is based on a careful study of some new properties of preferential attachment graphs which could be of independent interest.     

Probabilistic analysis on mesh network fault tolerance

Mesh networks are among the most important interconnection network topologies for large multicomputer systems. Mesh networks perform poorly in tolerating faults in the view of worst-case analysis. On the other hand, such worst cases occur very rarely in realistic situations. In this paper, we study the fault tolerance of 2-D and 3-D mesh networks under a more realistic model in which each network node has an independent failure probability. We first observe that if the node failure probability is fixed, then the connectivity probability of these mesh networks can be arbitrarily small when the network size is sufficiently large. Thus, it is practically important for multicomputer system manufacture to determine the upper bound for node failure probability when the probability of network connectivity and the network size are given. We develop a novel technique to formally derive lower bounds on the connectivity probability for 2-D and 3-D mesh networks. Our study shows that these mesh networks of practical size can tolerate a large number of faulty nodes thus are reliable enough for multicomputer systems. For example, it is formally proved that as long as the node failure probability is bounded by 0.5%$0.5\%$, a 3-D mesh network of up to a million nodes remains connected with a probability larger than 99%$99\%$.     

A linear time algorithm for computing a most reliable source on a tree network with faulty nodes

Given an unreliable communication network, we seek for a node which maximizes the expected number of nodes that are reachable from it. Such a node is called a most reliable source (MRS) of the network. In communication networks, failures may occur to both links and nodes. Previous studies have considered the case where each link has an independent operational probability, while the nodes are immune to failures. In practice, however, failures may happen to the nodes as well, including both transmitting fault and receiving fault. Recently, another variant of the MRS problem is studied, where all links are immune to failures and each node has an independent transmitting probability and receiving probability, and an O(n²) time algorithm is presented for computing an MRS on tree networks with n nodes. In this paper, we present a faster algorithm for this problem, with a time complexity of O(n).     

网络容错路由算法概率分析的一种新的方法

该文提出了一种新的概率分析方法来研究在给定结点错误概率的情况下超立方体网络强容错路由算法的容错性的概率。针对文中提出的基于新的局部连通性网络容错模型的高效的强容错路由算法犤1犦,该文首次严格证明了一个具有1024个结点的10维超立方体网络能够容许多达4.7%的错误结点而具有99%的概率确保找到正确结点组成的路径,而如果结点的错误概率不超过0.1%,则所有实际规模的超立方体网络能够具有99.9%的概率确保找到正确结点组成的路径。该算法的时间性能是最优的,且该算法构造的路径的长度不超过源结点和目的结点之间海明距离的两倍加上一个很小的常数。     

Circle formation of weak robots and Lyndon words

In this paper a discrete-time dynamic random graph process is studied that interleaves the birth of nodes and edges with the death of nodes. In this model, at each time step either a new node is added or an existing node is deleted. A node is added with probability p together with an edge incident on it. The node at the other end of this new edge is chosen based on a linear preferential attachment rule. A node (and all the edges incident on it) is deleted with probability q=1−p. The node to be deleted is chosen based on a probability distribution that favors small-degree nodes, in view of recent empirical findings. We analyze the degree distribution of this model and find that the expected fraction of nodes with degree k in the graph generated by this process decreases asymptotically as k−1−(2p/2p−1).     

A probabilistic and opportunistic flooding algorithm in wireless sensor networks

In wireless sensor networks, many communication protocols and applications rely on flooding for various networking purposes. Prior efforts focus on how to design efficient flooding algorithms; that is, they seek to achieve full reliability while reducing the number of redundant broadcasting across the network. To achieve efficient flooding, most of the existing protocols try to reduce the number of transmissions, which is decided without considering any online transmission result. In this paper, we propose a probabilistic and opportunistic flooding algorithm that controls rebroadcasts and retransmissions opportunistically. It seeks to achieve a target reliability required by an application. For this purpose, it makes a given node select only the subset of its one-hop neighbors to rebroadcast the same message. It considers node relations such as link error rates among nodes in selecting eligible neighbors to rebroadcast. The sender controls the number of retransmissions opportunistically by tracking the current status of message reception at its neighbors. Simulation is carried out to reveal that our proposed scheme achieves the given target reliability with less overhead than other flooding algorithms in most cases, thus prolonging the network lifetime.     

Mesh网络容错广播路由算法的概率分析

One-to-all or broadcast communication is one of the most important communication patterns and occurs in many important applications in parallel computing. This paper proposes a fault tolerant, local-irdormation-based, and distributed broadcast routing algorithm based on the concept of k-submesh-cormectivity in all-port mesh networks.The paper analyzes the fault tolerance of the algorithm in terms of node failure probability. Suppose that every nodehas independent failure probability, and deduce the success probability of the broadcast routing, which successfully routes a message from a source node to all non-faulty nodes in the networks. The paper strictly proves that the broadcast routing algorithm with the success probability of 99％ to route among all non-faulty nodes on mesh networks with forty thousand nodes, in case that the node failure probability is controlled within 0.12％ Simulation results show that the algorithm is practically efficient and effective, and the time steps of the algorithm are very closeto the optimum.     

Mesh网络路由算法容错性的概率分析

该文基于k-Mesh子网的概念提出了两个简单的基于局部信息和分布式的Mesh网络容错路由算法，并对其容错性进行概率分析；在每个结点具有独立的出错概率的假设条件下，推导出路由算法成功返回由正确结点组成的路径的概率．该文运用严格的数学推理，证明了Mesh网络结点出错概率只要控制在1．87％以内，则对于多达几十万个结点的Mesh网络，提出的路由算法具有99％的概率确保找到正确结点组成的路径．路由算法的时间复杂性是线性的．模拟结果表明路由算法所构造的路由路径长度非常接近于两结点之间的最优路径长度．     

The message delay in mobile ad hoc networks

A stochastic model is introduced that accurately models the message delay in mobile ad hoc networks where nodes relay messages and the networks are sparsely populated. The model has only two input parameters: the number of nodes and the parameter of an exponential distribution which describes the time until two random mobiles come within communication range of one another. Closed-form expressions are obtained for the Laplace–Stieltjes transform of the message delay, defined as the time needed to transfer a message between a source and a destination. From this we derive both a closed-form expression and an asymptotic approximation (as a function of the number of nodes) of the expected message delay. As an additional result, the probability distribution function is obtained for the number of copies of the message at the time the message is delivered. These calculations are carried out for two protocols: the two-hop multicopy and the unrestricted multicopy protocols. It is shown that despite its simplicity, the model accurately predicts the message delay for both relay strategies for a number of mobility models (the random waypoint, random direction and the random walker mobility models).     

3-维Mesh网络容错性的概率分析研究

Mesh network is very popualr and important topological structure in parallel computing. In this paper,we focus on the fault tolerance of 3-dimensional mesh. We use the probability model to analyze the fault tolerance of mesh. To simplify our analysis, we assume the failure probability of each node is independent. We partition a 3-dimensional mesh into smaller submeshes and compute the probability with which each submesh satisfies the condition we define. If each submesh satisfies the condition, then the whole mesh is connected. We then compute the probability that a 3-dimensional mesh is connected assuming each node has a failure probability p. We use mathematical methods to derive a relationship between network node failure probability and network connectivity probability. Our simulations show that 3-dimensional mesh networks can remain connected with very high probability in practice. For example, the paper formally proves that when the network node failure probability is bounded by 0.05%, 3-dimensional mesh network of more than two hundred thousand nodes remain connected with probability larger than 99%. Theoretical and experimental results show that our method is powderful technique to calculate the lower bound of the connectivity probability of mesh network.     

Probabilistic temporal reasoning and its application to fossil power plant operation

Many real-world applications, such as industrial diagnosis, require an adequate representation and inference mechanism that combines uncertainty and time. In this work, we propose a novel approach for representing dynamic domains under uncertainty based on a probabilistic framework, called temporal nodes Bayesian networks (TNBN). The TNBN model is an extension of a standard Bayesian network, in which each temporal node represents an event or state change of a variable and the arcs represent causal–temporal relationships between nodes. A temporal node has associated a probability distribution for its time of occurrence, where time is discretized in a finite number of temporal intervals; allowing a different number of intervals for each node and a different duration for the intervals within a node (multiple granularity). The main difference with previous probabilistic temporal models is that the representation is based on state changes at different times instead of state values at different times. Given this model, we can reason about the probability of occurrence of certain events, for diagnosis or prediction, using standard probability propagation techniques developed for Bayesian networks. The proposed approach is applied to fossil power plant diagnosis through two detailed case studies: power load increment and control level system failure. The results show that the proposed formalism could help to improve power plant availability through early diagnosis of events and disturbances.     

Acknowledged broadcasting in ad hoc radio networks

We consider ad hoc radio networks in which each node knows only its own identity but is unaware of the topology of the network, or of any bound on its size or diameter. Acknowledged broadcasting (AB) is a communication task consisting in transmitting a message from a distinguished source to all other nodes of the network and making this fact common knowledge among all nodes. To do this, the underlying directed graph must be strongly connected. Working in a model allowing all nodes to transmit spontaneously even before getting the source message, Chlebus et al. [B. Chlebus, L. Ga?sieniec, A. Gibbons, A. Pelc, W. Rytter, Deterministic broadcasting in unknown radio networks, Distrib. Comput. 15 (2002) 27-38] proved that AB is impossible, if collision detection is not available, and gave an AB algorithm using collision detection that works in time O(nD) where n is the number of nodes and D is the eccentricity of the source. Uchida et al. [J. Uchida, W. Chen, K. Wada, Acknowledged broadcasting and gossiping in ad hoc radio networks, Theoret. Comput. Sci. 377 (2007) 43-54] showed an AB algorithm without collision detection working in time O(n4/3log10/3n) for all strongly connected networks of size at least 2. In particular, it follows that the impossibility result from [B. Chlebus, L. Ga?sieniec, A. Gibbons, A. Pelc, W. Rytter, Deterministic broadcasting in unknown radio networks, Distrib. Comput. 15 (2002) 27-38] is really caused by the singleton network for which AB amounts to realize that the source is alone. We improve those two results by presenting two generic AB algorithms using a broadcasting algorithm without acknowledgement, as a procedure. For a large class of broadcasting algorithms the resulting AB algorithm has the same time complexity. Using the currently best known broadcasting algorithms, we obtain an AB algorithm with collision detection working in time O(min{nlog²D,nlognloglogn}), for arbitrary strongly connected networks, and an AB algorithm without collision detection working in time O(nlognloglogn) for all strongly connected networks of size n?2. Moreover, we show that in the model in which only nodes that already got the source message can transmit, AB is infeasible in a strong sense: for any AB algorithm there exists an infinite family of networks for which this algorithm is incorrect.     

Distributed scheduling algorithms for channel access in TDMA wireless mesh networks

In this paper, we have considered the distributed scheduling problem for channel access in TDMA wireless mesh networks. The problem is to assign time-slot(s) for nodes to access the channels, and it is guaranteed that nodes can communicate with all their one-hop neighbors in the assigned time-slot(s). And the objective is to minimize the cycle length, i.e., the total number of different time-slots in one scheduling cycle. In single-channel ad hoc networks, the best known result for this problem is proved to be K ² in arbitrary graphs (IEEE Trans Comput C-36(6):729–737, 1987) and 25K in unit disk graphs (IEEE/ACM Trans Netw pp 166–177, 1993) with K as the maximum node degree. There are multiple channels in wireless mesh networks, and different nodes can use different control channels to reduce congestion on the control channels. In this paper, we have considered two scheduling models for wireless mesh networks. The first model is that each node has two radios, and the scheduling is simultaneously done on the two radios. We have proved that the upper bound of the cycle length in arbitrary graphs can be 2K. The second model is that the time-slots are scheduled for the nodes regardless of the number of radios on them. In this case, we have proved that the upper bound can be (4K?2). We also have proposed greedy algorithms with different criterion. The basic idea of these algorithms is to organize the conflicting nodes by special criterion, such as node identification, node degree, the number of conflicting neighbors, etc. And a node cannot be assigned to a time-slot(s) until all neighbor nodes, which have higher criterion and might conflict with the current node, are assigned time-slot(s) already. All these algorithms are fully distributed and easy to realize. Simulations are also done to verify the performance of these algorithms.     

A modified ant colony system for finding the expected shortest path in networks with variable arc lengths and probabilistic nodes

The problem of finding the expected shortest path in stochastic networks, where the presence of each node is probabilistic and the arc lengths are random variables, have numerous applications, especially in communication networks. The problem being NP-hard we use an ant colony system (ACS) to propose a metaheuristic algorithm for finding the expected shortest path. A new local heuristic is formulated for the proposed algorithm to consider the probabilistic nodes. The arc lengths are randomly generated based on the arc length distribution functions. Examples are worked out to illustrate the applicability of the proposed approach.     

Approximate bounds and expressions for the link utilization of shortest-path multicast network traffic

Multicast network traffic is information with one source node, but many destination nodes. Rather than setting up individual connections between the source node and each destination node, or broadcasting the information to the entire network, multicasting efficiently exploits link capacity by allowing the source node to transmit a small number of copies of the information to mutually-exclusive groups of destination nodes. Multicasting is an important topic in the fields of networking (video and audio conferencing, video on demand, local-area network interconnection) and computer architecture (cache coherency, multiprocessor message passing). In this paper, we derive approximate expressions for the minimum cost (in terms of link utilization) of shortest-path multicast traffic in arbitrary tree networks. Our results provide a theoretical best-case scenario for link utilization of multicast distribution in tree topologies overlaid onto arbitrary graphs. In real networks such as the Internet MBONE, multicast distribution paths are often tree-like, but contain some cycles for purposes of fault tolerance. We find that even for richly-connected graphs such as the shufflenet and the hypercube, our expression provides a good prediction of the cost (in terms of link utilization) of multicast communication. Thus, this theoretical result has two applications: (1) a lower bound on the link capacity required for multicasting in random tree topologies, and (2) an approximation of the cost of multicasting in regular LAN and MAN topologies.     

Deterministic broadcasting in ad hoc radio networks

We consider the problem of distributed deterministic broadcasting in radio networks of unknown topology and size. The network is synchronous. If a node u can be reached from two nodes which send messages in the same round, none of the messages is received by u. Such messages block each other and node u either hears the noise of interference of messages, enabling it to detect a collision, or does not hear anything at all, depending on the model. We assume that nodes know neither the topology nor the size of the network, nor even their immediate neighborhood. The initial knowledge of every node is limited to its own label. Such networks are called ad hoc multi-hop networks. We study the time of deterministic broadcasting under this scenario. For the model without collision detection, we develop a linear-time broadcasting algorithm for symmetric graphs, which is optimal, and an algorithm for arbitrary n-node graphs, working in time . Next we show that broadcasting with acknowledgement is not possible in this model at all. For the model with collision detection, we develop efficient algorithms for broadcasting and for acknowledged broadcasting in strongly connected graphs. Received: January 2000 / Accepted: June 2001     

A reversing traversal algorithm to predict deleting node for the optimal k-node set reliability with capacity constraint of distributed systems

A k-node set reliability with capacity constraint is defined as the probability that a set, K, of nodes is connected in a distributed system and the total capacity of the nodes in K is sufficient under a given capacity. This is generally an NP-hard problem. For reducing computational time, a reasonable k-node set within a given capacity constraint must be determined by an efficient algorithm. In this work, we propose a reversing traversal method to derive a k-node set under capacity constraint having an approximate solution. Initially, the set K is assigned to all the nodes in a system. The proposed algorithm uses an objective function to evaluate the fitness value of each node in K and predict a deleting node, which is not a critical node, in K with minimal fitness value. After deleting the node, the fitness value of each node that is adjacent to the deleted node is tuned. The above two processes are repeated until the total capacity of the nodes in each subset of the set K does not satisfy the capacity constraint. In our simulation, the proposed method can obtain an exact solution above 90%. When a sub-optimal solution is obtained, the average deviation from an exact solution is under 0.0033. Computational results demonstrate that the proposed algorithm is efficient in execution time and effective for obtaining an optimal k-node set with capacity constraint.     

Conflict-free star-access in parallel memory systems

We study conflict-free data distribution schemes in parallel memories in multiprocessor system architectures. Given a host graph G, the problem is to map the nodes of G into memory modules such that any instance of a template type T in G can be accessed without memory conflicts. A conflict occurs if two or more nodes of T are mapped to the same memory module. The mapping algorithm should: (i) be fast in terms of data access (possibly mapping each node in constant time); (ii) minimize the required number of memory modules for accessing any instance in G of the given template type; and (iii) guarantee load balancing on the modules. In this paper, we consider conflict-free access to star templates, i.e., to any node of G along with all of its neighbors. Such a template type arises in many classical algorithms like breadth-first search in a graph, message broadcasting in networks, and nearest neighbor based approximation in numerical computation. We consider the star-template access problem on two specific host graphs—tori and hypercubes—that are also popular interconnection network topologies. The proposed conflict-free mappings on these graphs are fast, use an optimal or provably good number of memory modules, and guarantee load balancing.