Hop-Constrained Node Survivable Network Design: An Application to MPLS over WDM

In this paper, we address a hop-constrained node survivable network design problem that is defined in the context of multi-protocol label switching (MPLS) over wavelength division multiplexing (WDM) networks. At the lower WDM layer, we consider a maximum length constraint for optical connections between MPLS routers. At the upper MPLS layer, we consider survivability as well as maximum delay constraints. Survivability is guaranteed by routing each demand through D node-disjoint paths and maximum delay is guaranteed by constraining all paths to a maximum number of hops. An Integer Linear Programming model, based on the previous works by Gouveia et al. (Proc of IEEE INFOCOM, 2003, and Telecommunications network planning: innovations in pricing, network design and management, pp 167–180, 2006) is used to model the network design problem considering two different survivability mechanisms: path diversity (where each demand is equally split over the D paths) and path protection (where any D–1 out of the D paths have enough capacity to support the total demand). For both mechanisms, we use the NSFNet and EON real world networks to make a cost analysis of the design solutions for different values of D. In the path diversity mechanism, the results consistently show that greater values of D impose a cost penalty that is greater than the gain in the percentage of demand that is protected. In the path protection mechanism, where all traffic is totally protected, the results show that the network solutions obtained with D=3 node-disjoint paths have consistently lower costs than the network solutions obtained with D=2 node-disjoint paths. However, using values of D that are greater than 3 led to network solutions with larger costs. Supported by FCT project POCTI - ISFL - 1- 152.     

Topological properties of folded hyper-star networks

In practice it is important to construct node-disjoint paths in networks, because they can be used to increase the transmission rate and enhance the transmission reliability. The folded hyper-star networks FHS(2n,n) were introduced to be a competitive model to both hypercubes and star graphs. They are bipartite and node-symmetric, though not edge-symmetric, and have diameter n. In this paper we construct a maximum number of node-disjoint paths between every two distinct nodes of FHS(2n,n) and show that its fault diameter is n+2 for n≥4. We also suggest a one-to-all broadcasting algorithm of FHS(2n,n) under the all-port model.     

Fault-tolerant routing over shortest node-disjoint paths in hypercubes

This paper presented a routing algorithm that finds n disjoint shortest paths from the source node s to target node d in the n-dimensional hypercube. Fault-tolerant routing over all shortest node-disjoint paths has been investigated to overcome the failure encountered during routing in hypercube networks. In this paper, we proposed an efficient approach to provide fault-tolerant routing which has been investigated on hypercube networks. The proposed approach is based on all shortest node-disjoint paths concept in order to find a fault-free shortest path among several paths provided. The proposed algorithm is a simple uniform distributed algorithm that can tolerate a large number of process failures, while delivering all n messages over optimal-length disjoint paths. However, no distributed algorithm uses acknowledgement messages (acks) for fault tolerance. So, for dealing the faults, acknowledgement messages (acks) are included in the proposed algorithm for routing messages over node-disjoint paths in a hypercube network.     

Swapped (OTIS) Networks Built of Connected Basis Networks Are Maximally Fault Tolerant

An optical transpose interconnection system (OTIS) network with n^2 nodes is a two-level swapped architecture built of n copies of an n-node basis network that constitute its clusters. A simple rule for intercluster connectivity (node j in cluster i connected to node i in cluster j) leads to regularity, modularity, packageability, fault tolerance, and algorithmic efficiency of the resulting networks. We prove that an OTIS (swapped) network with a connected basis network possesses maximal fault tolerance, regardless of whether its basis network is maximally fault tolerant. We also show how the corresponding maximal number of node-disjoint paths between two nodes of a swapped network can be algorithmically constructed in a manner that is independent of the existence and construction of node-disjoint paths within its basis network. Our results are stronger than previously published results and they replace a number of proofs and constructions in the literature for specific basis networks. Additionally, we use our parallel path constructions to establish that the fault diameter and wide diameter of an OTIS network is no more than 4 units greater than its diameter.     

Connectivity in finite ad-hoc networks

Research on ad-hoc network connectivity has mainly focused on asymptotic results in the number of nodes in the network. For a one-dimensional ad-hoc network G1, assuming all the nodes are independently uniform distributed in a closed interval [0, Z]（z ∈ R^＋）, we derive a generic formula for the probability that the network is connected. The finite connected ad-hoc networks is analyzed. And we separately suggest necessary conditions to make the ad-hoc network to be connected in one and two dimensional cases, facing possible failed nodes （f-nodes）. Based on the necessary condition and unit-disk assumption for the node transmission, we prove that the nodes of the connected two-dimensional ad-hoc networks （G2） can be divided into at most five different groups. For an f-node no in either of the five groups, we derive a close formula for the probability that there is at least one route between a pair of nodes in G2 -- {no}.     

Nearly optimal one-to-many parallel routing in star networks

Star networks were proposed recently as an attractive alternative to the well-known hypercube models for interconnection networks. Extensive research has been performed that shows that star networks are as versatile as hypercubes. This paper is an effort in the same direction. Based on the well-known paradigms, we study the one-to-many parallel routing problem on star networks and develop an improved routing algorithm that finds n-1 node-disjoint paths between one node and a set of other n-1 nodes in the n-star network. These parallel paths are proven of minimum length within a small additive constant, and the running time of our algorithm is bounded by O(n²). More specifically, given a node s and n-1 other nodes {t₁, t₂ , …, t_n-1} in the n-star network, our algorithm constructs n-1 node-disjoint paths P₁, P₂, …, P_n-1, where P_i is a path from s to t_j of length at most dist(s, t_j)+6 and dist(s, t _j) is the distance, i.e., the length of a shortest path, from s to t_j, for i=1, 2, …, n-1.The best bound on the path length by previously known algorithms for the same problem is 5(n-2)≈10Δ_n/3, where Δ_n=max{dist(s, t)} is the diameter of the n-star network     

Guaranteed on-demand discovery of node-disjoint paths in ad hoc networks

In recent years, the low cost and abundance of WLAN products has led to the deployment of self-configuring multihop ad hoc networks. Multipath routing has been increasingly studied to improve network reliability and throughput. However, no existing work guarantees discovery of node-disjoint paths when they exist, which limits their applicability in real networks. This paper presents a theoretical framework that establishes the equivalence between multipath discovery and flow network assignment. This equivalence is used to guarantee the on-demand discovery of an arbitrary number of node-disjoint paths between a pair of nodes as long as they exist. We also present an example protocol that integrates the theoretical framework with the Dynamic Source Routing (DSR) protocol to find two node-disjoint paths, which can be easily extended to finding k node-disjoint paths. Analysis of the example protocol demonstrates a good tradeoff between complexity and capability, particularly when compared with existing on-demand multipath routing protocols. Our simulation data shows the effectiveness of the discovered paths.     

Lower bounds for the broadcast problem in mobile radio networks

Summary.  In this paper, we prove a lower bound on the number of rounds required by a deterministic distributed protocol for broadcasting a message in radio networks whose processors do not know the identities of their neighbors. Such an assumption captures the main characteristic of mobile and wireless environments [3], i.e., the instability of the network topology. For any distributed broadcast protocol Π, for any n and for any D≦n/2, we exhibit a network G with n nodes and diameter D such that the number of rounds needed by Π for broadcasting a message in G is Ω(D log n). The result still holds even if the processors in the network use a different program and know n and D. We also consider the version of the broadcast problem in which an arbitrary number of processors issue at the same time an identical message that has to be delivered to the other processors. In such a case we prove that, even assuming that the processors know the network topology, Ω(n) rounds are required for solving the problem on a complete network (D=1) with n processors. Received: August 1994 / Accepted: August 1996     

A randomized clustering of anonymous wireless ad hoc networks with an application to the initialization problem

The Distributed Mobility-Adaptive Clustering (DMAC) due to Basagni partitions the nodes of a mobile ad hoc network into clusters, thus giving the network a hierarchical organization. This algorithm supports the mobility of the nodes, even during the cluster formation. The main feature of DMAC is that in a weighted network (in which two or more nodes cannot have the same weight), nodes have to choose the clusterheads taking into account only the node weight, i.e. the mobility when a node weight is the inverse of its speed. In our approach many nodes may have the same speed and hence the same weight. We assume that nodes have no identities and the number of nodes, say n, is the only known parameter of the network. After the randomized clustering, we show that the initialization problem can be solved in a multi-hop ad hoc wireless network of n stations in O(k ^1/2log ^1/2 k)+D _b −1+O(log (max (P _i )+log ²max (P _i )) broadcast rounds with high probability, where k is the number of clusters, D _b is the blocking diameter and max (P _i ), 1≤i≤k, is the maximum number of nodes in a cluster. Thus the initialization protocol presented here uses less broadcast rounds than the one in Ravelemanana (IEEE Trans. Parallel Distributed Syst. 18(1):17–28 2007).     

A parallel routing algorithm on recursive cube of rings networks employing Hamiltonian circuit Latin square

Recursive cube of rings (RCR) networks [Y. Sun, P. Cheung, X. Lin, Recursive cube of rings: a new topology for interconnection networks, IEEE Trans. Parallel Dist. Syst. 11 (3) (2000) 275-286] can be widely used in the design and implementation of parallel processing architectures. In this paper, we investigate the routing of a message on RCR networks, that is a key to the performance of this network. We would like to transmit k+1 packets from a source node to a destination node simultaneously along paths on RCR networks, the ith packet will traverse along the ith path (1?i?k+1). In order for all packets to arrive at a destination node quickly and securely, the ith path must be node-disjoint from all other paths. For construction of these paths, employing Hamiltonian circuit Latin square (HCLS), we present O(n²) parallel routing algorithm on RCR networks.     

Node-disjoint paths in incomplete WK-recursive networks

The incomplete WK-recursive networks have been recently proposed to relieve the restriction on the sizes of the WK-recursive networks. In this paper, a maximal set of node-disjoint paths is constructed between arbitrary two nodes of an incomplete WK-recursive network. The effectiveness of the constructed paths is verified by both theoretic analysis and extensive experiments. A tight upper bound on the maximal length is suggested. On the other hand, experimental results show that for arbitrary two nodes, the expected maximal length is not greater than twice their distance and about equal to the diameter. When the two nodes are the farthest pair, the maximal length is not greater than twice the diameter and the expected maximal length is not greater than 1.5 times the diameter.     

Node-disjoint paths in hierarchical hypercube networks

The hierarchical hypercube network is suitable for massively parallel systems. One of its appealing properties is the low number of connections per processor, which can facilitate the VLSI design and fabrication. Other alluring features include symmetry and logarithmic diameter, which can derive easy and fast algorithms for communication. In this paper, a maximal number of node-disjoint paths are constructed between every two distinct nodes of the hierarchical hypercube network. Their maximal length is not greater than max{2^m+1+2m+1,2^m+1+m+4}, where 2^m+1 is the diameter. The effectiveness of node-disjoint paths is further verified by experiments.     

无线传感器网络一种不相交路径路由算法

无线传感器网络经常被用来采集物理数据,监测环境变化.由于低功耗无线通信不确定性、链路质量不稳定性以及节点失效等问题,传感器网络很容易导致路由数据包丢失.为了提高网络路由的可靠性,人们提出多路径路由算法.多路径路由中源节点到目的节点的多条路径可能含有公共节点,或者公共边,如果公共节点或者公共链路失效,则这个数据包也丢失,因此又有人提出不相交多路径路由算法.不相交多路径路由算法又分为链路不相交多路径路由算法和节点不相交多路径路由算法.提出了一种不相交路径路由算法,可以将感知节点采集到的数据通过不相交路径传送到汇聚节点,提高路由的可靠性.而且,这个算法还可以很方便地应用到多Sink节点的网络当中.该路由算法用到的路由表大小为|K|,其中|K|表示路径数.算法的运行时间复杂度是O(|L|),其中|L|表示网络中的边数.     

A parallel routing algorithm on circulant networks employing the Hamiltonian circuit latin square

Double-loop [J. Bermond, F. Comellas, D. Hsu, Distributed Loop Computer Networks: A Survey, J. Parallel and Distributed Computing, Academic Press, 24 (1995) 2-10] and 2-circulant networks (2-CN) [J. Park, Cycle Embedding of Faulty Recursive Circulants, J. of Korea Info. Sci. Soc. 31 (2) (2004) 86-94] are widely used in the design and implementation of local area networks and parallel processing architectures. In this paper, we investigate the routing of a message on circulant networks, that is a key to the performance of this network. We would like to transmit 2k packets from a source node to a destination node simultaneously along paths on G(n; ±s₁, ±s₂, … , ±s_k), where the ith packet traverses along the ith path (1 ? i ? 2k). In order for all packets to arrive at the destination node quickly and securely, the ith path must be node-disjoint from all other paths. For construction of these paths, employing the Hamiltonian circuit latin square (HCLS), a special class of (n × n) matrices, we present O(n²) parallel routing algorithm on circulant networks.     

Broadcasting in UDG radio networks with missing and inaccurate information

We study broadcasting time in radio networks, modeled as unit disk graphs (UDG). Network stations are represented by points in the plane and a station is connected to all stations at distance at most 1 from it. Stations are unaware of the network topology. Each station can send messages from the beginning of the broadcasting process, even before getting the source message. Emek et al. showed that broadcasting time depends on two parameters of the UDG network, namely, its diameter D (in hops) and its granularity g. The latter is the inverse of the density d of the network which is the minimum Euclidean distance between any two stations. They proved that the minimum broadcasting time is Θ(min {D + g ², D log g}), assuming that each node knows the density of the network and knows exactly its own position in the plane. In many situations these assumptions are unrealistic. Does removing them influence broadcasting time? The aim of this paper is to answer this question, hence we assume that density is unknown and nodes perceive their position with some unknown error margin ${\varepsilon}We study broadcasting time in radio networks, modeled as unit disk graphs (UDG). Network stations are represented by points in the plane and a station is connected to all stations at distance at most 1 from it. Stations are unaware of the network topology. Each station can send messages from the beginning of the broadcasting process, even before getting the source message. Emek et al. showed that broadcasting time depends on two parameters of the UDG network, namely, its diameter D (in hops) and its granularity g. The latter is the inverse of the density d of the network which is the minimum Euclidean distance between any two stations. They proved that the minimum broadcasting time is Θ(min {D + g ², D log g}), assuming that each node knows the density of the network and knows exactly its own position in the plane. In many situations these assumptions are unrealistic. Does removing them influence broadcasting time? The aim of this paper is to answer this question, hence we assume that density is unknown and nodes perceive their position with some unknown error margin e{\varepsilon}. It turns out that this combination of missing and inaccurate information substantially changes the problem: the main new challenge becomes fast broadcasting in sparse networks (with constant density), when optimal time is O(D). Nevertheless, under our very weak scenario, we construct a broadcasting algorithm that maintains optimal time O (min {D + g ², D log g}) for all networks with at least 2 nodes, of diameter D and granularity g (previously obtained with exact positions and known density), if each node perceives its position with error margin e = ad{\varepsilon=\alpha d}, for any (unknown) constant α < 1/2. Rather surprisingly, the minimum time of an algorithm working correctly for all networks, and hence stopping if the source is alone, turns out to be Θ(D + g ²). Thus, the mere stopping requirement for the special case of the lonely source causes an exponential increase in broadcasting time, for networks of any density and any small diameter. Finally, if e 3 d/2{\varepsilon\geq d/2}, then broadcasting is impossible.     

Biswapped networks: a family of interconnection architectures with advantages over swapped or OTIS networks

We propose a new family of communication architectures called ‘biswapped networks’. Given any n-node basis network Ω, the associated biswapped network Bsw(Ω) is built of 2n copies of Ω, using a simple rule for connectivity that ensures desirable attributes, including regularity, modularity, fault tolerance, and algorithmic efficiency. In particular, if Ω is a Cayley digraph, then so is Bsw(Ω). Our biswapped connectivity provides a systematic scheme for synthesizing large, scalable, modular, and robust parallel architectures. Furthermore, many desirable attributes of the underlying basis network Ω are preserved, as the Bsw(Ω) parameters are related to the corresponding parameters of Ω. We obtain a number of results on internode distances, Hamiltonian cycles, optimal routing, and node-disjoint paths for Bsw(Ω). We explore the relations between biswapped and swapped or optical transpose interconnection system (OTIS) networks, which may use a mix of electronic and optical links. In particular, we demonstrate that the biswapped connectivity removes an inherent asymmetry of swapped/OTIS networks, as well as the attendant complications in analyses and applications. Finally, we show that biswapped networks are complementary to, and offer advantages over, well-known and widely used interconnection architectures for parallel processing.     

The fastcube: a variation on hypercube topology with lower diameter

This paper presents a class of n-dimensional interconnection topologies with N=2ⁿ nodes which we refer to as n-fastcubes. The node degree of the n-fastcube is n and its diameter is ⌈(n+1)/2⌉, which is substantially smaller than that of the same size hypercube. Topological properties as well as several routing algorithms for fastcubes are developed. In addition, a new methodology for the design and analysis of fastcubes is employed. This methodology is based on modeling interconnection networks as finite state automata. The inputs to these particular automata are routing sequences. The routing and embedding algorithms developed in this paper produce routing sequences. The main characteristic of routing sequences is their node independence. A node independent routing sequence, p(H), produces a path between any pair of nodes with the Hamming distance of H. Thus, these sequences can be used, without modification, at any node to establish paths in a fastcube.     

Analysis of Randomized Protocols for Conflict-Free Distributed Access

We study the following distributed access problem which arises naturally in many settings: given a set of n data items shared among n nodes in a distributed network, all nodes want to access all (or a subset of) the items residing on different nodes in a conflict-free manner. In addition, items may move from one node to the other during access. Our goal is to design distributed protocols so that all nodes access all the desired items as quickly as possible, while at the same time not overloading the storage space of any one node. Using centralized coordination among the nodes it is easy to design an optimal scheme in which all nodes can access all the items in n−1 steps storing only one item at any time. We show that a simple randomized distributed protocol performs almost as well as the optimal (centralized) scheme but with no coordination overhead. Our protocol takes O(n) time with high probability to access all n items which is asymptotically as good as the optimal centralized scheme. The protocol guarantees that the maximum load (the maximum number of items stored in any node) at any time is at most O(log n/log log n) with high probability which is only slightly larger compared to the Ω(1) load of the optimal scheme. Our analysis involves a stochastic analysis of a “balls into bins” problem in a dynamic setting where balls (data items) move into bins (nodes) on request and we study the time and load requirements to move all the balls to the requested bins. A short version of this paper appeared in the Proceedings of the 24th Annual ACM Symposium on Principles of Distributed Computing (PODC), 2005.     

A note on the alternating group network

The class of alternating group networks was introduced in the late 1990’s as an alternative to the alternating group graphs as interconnection networks. Recently, additional properties for the alternating group networks have been published. In particular, Zhou et al., J. Supercomput (2009), doi:, was published very recently in this journal. We show that this so-called new interconnection topology is in fact isomorphic to the (n,n−2)-star, a member of the well-known (n,k)-stars, 1≤k≤n−1, a class of popular networks proposed earlier for which a large amount of work have already been done. Specifically, the problem in Zhou et al., J. Supercomput (2009), doi:, was addressed in Lin and Duh, Inf. Sci. 178(3), 788–801, 2008, when k = n−2.     

Wavelength Assignment for Realizing Parallel FFT on Regular Optical Networks

Routing and wavelength assignment (RWA) is a central issue to increase efficiency and reduce cost in Wavelength Division Multiplexing (WDM) optical networks. In this paper, we address the problem of wavelength assignment for realizing parallel FFT on a class of regular optical WDM networks. We propose two methods for sequential mapping and shift-reversal mapping of FFT communication pattern to the optical WDM networks concerned. By sequential mapping, the numbers of wavelengths required to realize parallel FFT with 2ⁿ nodes on WDM linear arrays, rings, 2-D meshes and 2-D tori are 2^{n − 1}, 2^{n − 1}, 2^{max (k,n − k) − 1} and 2^{max (k,n − k) − 1} respectively. By shift-reversal mapping, the numbers of wavelengths required are max (3× 2^{n − 3},2), 2^{n − 2}, max (3× 2^{max (k,n − k) − 3},2) and 2^{max (k,n − k) − 2}. These results show that shift-reversal mapping outperforms sequential mapping. Our results have a clear significance for applications because FFT represents a common computation pattern shared by a large class of scientific and engineering problems and WDM optical networks as a promising technology in networking has an increasing popularity.