Fault-free Path Research Articles

Ant Colony Optimization-Based Fault-Aware Routing in Mesh-Based Network-on-Chip Systems

The advanced deep submicrometer technology increases the risk of failure for on-chip components. In advanced network-on-chip (NoC) systems, the failure constrains the on-chip bandwidth and network throughput. Fault-tolerant routing algorithms aim to alleviate the impact on performance. However, few works have integrated the congestion-, deadlock-, and fault-awareness information in channel evaluation function to avoid the hotspot around the faulty router. To solve this problem, we propose the ant colony optimization-based fault-aware routing (ACO-FAR) algorithm for load balancing in faulty networks. The behavior of an ant colony while facing an obstacle (failure in NoC) can be described in three steps: 1) encounter; 2) search; and 3) select. We implement the corresponding mechanisms as: 1) notification of fault information; 2) path searching mechanism; and 3) path selecting mechanism. With proposed ACO-FAR, the router can evaluate the available paths and detour packets through a less-congested fault-free path. The simulation results show that this paper has higher throughput than related works by 29.1%–66.5%. In addition, ACO-FAR can reduce the undelivered packet ratio to 0.5%–0.02% and balance the distribution of traffic flow in the faulty network.

Secure Efficient On-Demand Insider Attacks Multicast Routing Protocol in Wireless Networks

Wireless multicast routing send and receives the data source to destination. High error rates, unfixed and changeable self of the signal power and broadcast change with time and environment regularly result in not effective links. These services more weak to internal attacks coming from compromised nodes that behave randomly to disrupt the network, also referred to as Inside attacks. Our method ensures that as long as a fault-free path exists between two node or multi nodes in multicast group they can communicate reliably even if an destroy majority of the network acts in a complex mode. Multicast Group is the link on different Multicast Group’s Group Leader in multi hops networks.

Maximally local connectivity and connected components of augmented cubes

The connectivity of a graph is an important issue in graph theory, and is also one of the most important factors in evaluating the reliability and fault tolerance of a network. It is known that the augmented cubeAQn is maximally connected, i.e. (2n-1)-connected, for n⩾4. By the classic Menger’s Theorem, every pair of vertices in AQn is connected by 2n-1vertex-disjoint paths for n⩾4. A routing with parallel paths can speed up transfers of large amounts of data and increase fault tolerance. Motivated by research on networks with faults, we obtained the result that for any faulty vertex set F⊂V(AQn) and |F|⩽2n-7 for n⩾4, each pair of non-faulty vertices, denoted by u and v, in AQn-F is connected by min{degf(u),degf(v)} vertex-disjoint fault-free paths, where degf(u) and degf(v) are the degree of u and v in AQn-F, respectively. Moreover, we demonstrate that for any faulty vertex set F⊂V(AQn) and |F|⩽4n-9 for n⩾4, there exists a large connected component with at least 2n-|F|-1 vertices in AQn-F, which improves on the results of Ma et al. (2008) who show this for n⩾6.

Extended fault-tolerant bipanconnectivity and panconnectivity of folded hypercubes

The folded hypercube ( for short) is a well-known variation of hypercube structure and can be constructed from a hypercube by adding a link to every pair of vertices with complementary addresses. for any odd n is known to be bipartite. Let f be a faulty vertex in for In addition, let u and v be any two fault-free vertices in It has been shown that: 1) contains a fault-free path of every length l with and where 2) contains a fault-free path of every length l with where is even. In this paper, we extend the above-mentioned result to obtain two further properties as follows: 1) contains a fault-free path of every length l with and for ; 2) contains a fault-free path of every length l with for is even.

Paths and cycles embedding on faulty enhanced hypercube networks

Let Qn,k (n ≥ 3, 1 ≤k ≤n−1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges, fv and fe be the numbers of faulty vertices and faulty edges, respectively. In this paper, we give three main results. First, a fault-free path P[u,v] of length at least 2n−2fv−1 (respectively, 2n−2fv−2) can be embedded on Qn,k with fv+fe ≤n−1 when dQn,k(u,v) is odd (respectively, dQn,k(u,v) is even). Secondly, an Qn,k is (n−2) edge-fault-free hyper Hamiltonian-laceable when n (≥ 3) and k have the same parity. Lastly, a fault-free cycle of length at least 2n−2fv can be embedded on Qn,k with fe ≤n−1 and fv+fe ≤ 2n−4.

Fault-tolerant path embedding in folded hypercubes with both node and edge faults

The folded hypercube FQn is a well-known variation of the hypercube structure. FQn is superior to Qn in many measurements, such as diameter, fault diameter, connectivity, and so on. Let Ṽ(FQn) (resp. Ẽ(FQn)) denote the set of faulty nodes (resp. faulty edges) in FQn. In the case that all nodes in FQn are fault-free, it has been shown that FQn contains a fault-free path of length 2n−1 (resp. 2n−2) between any two nodes of odd (resp. even) distance if |Ẽ(FQn)|≤n−1, where n≥1 is odd; and FQn contains a fault-free path of length 2n−1 between any two nodes if |Ẽ(FQn)|≤n−2, where n≥2 is even. In this paper, we extend the above result to obtain two further properties, which consider both node and edge faults, as follows: 1.FQn contains a fault-free path of length at least 2n−2|Ṽ(FQn)|−1 (resp. 2n−2|Ṽ(FQn)|−2) between any two fault-free nodes of odd (resp. even) distance if |Ṽ(FQn)|+|Ẽ(FQn)|≤n−1, where n≥1 is odd.2.FQn contains a fault-free path of length at least 2n−2|Ṽ(FQn)|−1 between any two fault-free nodes if |Ṽ(FQn)|+|Ẽ(FQn)|≤n−2, where n≥2 is even.

An efficient fault-tolerant routing algorithm in bijective connection networks with restricted faulty edges

In this paper, we study fault-tolerant routing in bijective connection networks with restricted faulty edges. First, we prove that the probability that all the incident edges of an arbitrary node become faulty in an n -dimensional bijective connection network, denoted by X n , is extremely low when n becomes sufficient large. Then, we give an O ( n ) algorithm to find a fault-free path of length at most n + 3 ⌈ log 2 ∣ F ∣ ⌉ + 1 between any two different nodes in X n if each node of X n has at least one fault-free incident edge and the number of faulty edges is not more than 2 n − 3 . In fact, we also for the first time provide an upper bound of the fault diameter of all the bijective connection networks with the restricted faulty edges. Since the family of BC networks contains hypercubes, crossed cubes, Möbius cubes, etc., all the results are appropriate for these cubes.

Panconnectivity and edge-pancyclicity of [formula omitted]-ary [formula omitted]-cubes with faulty elements

The class of k-ary n-cubes represents the most commonly used interconnection topology for parallel and distributed computing systems. In this paper, we investigate the fault-tolerant capabilities of the k-ary n-cubes for odd integer k with respect to the panconnectivity and pancyclicity. By studying first the fault panconnectivity of two-dimensional torus networks and then using an induction argument, we prove that in a k-ary n-cube Qnk with odd k≥3, every pair of healthy vertices of Qnk are connected by fault-free paths of lengths from n(k−1)−1 to |V(Qnk−F)|−1 and every healthy edge is contained in fault-free cycles of lengths from n(k−1) to |V(Qnk−F)| for any set F of faulty elements (vertices and/or edges) with |F|≤2n−3. Finally, examples show that our results are best possible in some sense.

One-to-one communication in twisted cubes under restricted connectivity

The dimensions of twisted cubes are only limited to odd integers. In this paper, we first extend the dimensions of twisted cubes to all positive integers. Then, we introduce the concept of the restricted faulty set into twisted cubes. We further prove that under the condition that each node of the n-dimensional twisted cube TQ n has at least one fault-free neighbor, its restricted connectivity is 2n ? 2, which is almost twice as that of TQ n under the condition of arbitrary faulty nodes, the same as that of the n-dimensional hypercube. Moreover, we provide an O(NlogN) fault-free unicast algorithm and simulations result of the expected length of the fault-free path obtained by our algorithm, where N denotes the node number of TQ n . Finally, we propose a polynomial algorithm to check whether the faulty node set satisfies the condition that each node of the n-dimensional twisted cube TQ n has at least one fault-free neighbor.

Embedding cycles of various lengths into star graphs with both edge and vertex faults

The n-dimensional star graph S n is an attractive alternative to the hypercube graph and is a bipartite graph with two partite sets of equal size. Let F v and F e be the sets of faulty vertices and faulty edges of S n , respectively. We prove that S n − F v − F e contains a fault-free cycle of every even length from 6 to n! − 2∣ F v ∣ with ∣ F v ∣ + ∣ F e ∣ ⩽ n − 3 for every n ⩾ 4. We also show that S n − F v − F e contains a fault-free path of length n! − 2∣ F v ∣ − 1 (respectively, n! − 2∣ F v ∣ − 2) between two arbitrary vertices of S n in different partite sets (respectively, the same partite set) with ∣ F v ∣ + ∣ F e ∣ ⩽ n − 3 for every n ⩾ 4.

Long paths in hypercubes with a quadratic number of faults

A path between distinct vertices u and v of the n-dimensional hypercube Q n avoiding a given set of f faulty vertices is called long if its length is at least 2 n - 2 f - 2 . We present a function ϕ ( n ) = Θ ( n 2 ) such that if f ⩽ ϕ ( n ) then there is a long fault-free path between every pair of distinct vertices of the largest fault-free block of Q n . Moreover, the bound provided by ϕ ( n ) is asymptotically optimal. Furthermore, we show that assuming f ⩽ ϕ ( n ) , the existence of a long fault-free path between an arbitrary pair of vertices may be verified in polynomial time with respect to n and, if the path exists, its construction performed in linear time with respect to its length.

Long paths and cycles in hypercubes with faulty vertices

A fault-free path in the n -dimensional hypercube Q n with f faulty vertices is said to be long if it has length at least 2 n - 2 f - 2 . Similarly, a fault-free cycle in Q n is long if it has length at least 2 n - 2 f . If all faulty vertices are from the same bipartite class of Q n , such length is the best possible. We show that for every set of at most 2 n - 4 faulty vertices in Q n and every two fault-free vertices u and v satisfying a simple necessary condition on neighbors of u and v , there exists a long fault-free path between u and v . This number of faulty vertices is tight and improves the previously known results. Furthermore, we show for every set of at most n 2 / 10 + n / 2 + 1 faulty vertices in Q n where n ⩾ 15 that Q n has a long fault-free cycle. This is a first quadratic bound, which is known to be asymptotically optimal.

Embedded paths and cycles in faulty hypercubes

An important task in the theory of hypercubes is to establish the maximum integer f n such that for every set ? of f vertices in the hypercube ${\mathcal {Q}}_{n},$ with 0?f?f n , there exists a cycle of length at least 2 n ?2f in the complement of ?. Until recently, exact values of f n were known only for n?4, and the best lower bound available for f n with n?5 was 2n?4. We prove that f 5=8 and obtain the lower bound f n ?3n?7 for all n?5. Our results and an example provided in the paper support the conjecture that $f_{n}={n\choose 2}-2$ for each n?4. New results regarding the existence of longest fault-free paths with prescribed ends are also proved.

Long paths in hypercubes with conditional node-faults

Let F be a set of f⩽2n-5 faulty nodes in an n-cube Qn such that every node of Qn still has at least two fault-free neighbors. Then we show that Qn-F contains a path of length at least 2n-2f-1 (respectively, 2n-2f-2) between any two nodes of odd (respectively, even) distance. Since the n-cube is bipartite, the path of length 2n-2f-1 (or 2n-2f-2) turns out to be the longest if all faulty nodes belong to the same partite set. As a contribution, our study improves upon the previous result presented by [J.-S. Fu, Longest fault-free paths in hypercubes with vertex faults, Information Sciences 176 (2006) 759–771] where only n-2 faulty nodes are considered.

Fault-free cycles passing through prescribed paths in hypercubes with faulty edges

An n -dimensional hypercube, or n -cube, denoted by Q n , is well known as bipartite and one of the most efficient networks for parallel computation. In this work, we consider the problem of cycles passing through prescribed paths in an n -dimensional hypercube with faulty edges. We obtain the following result: For n ≥ 3 and 2 ≤ h < n , let F be a subset of E ( Q n ) with | F | < n − h . Then, every fault-free path P with length h lies on a fault-free cycle in Q n − F of every even length from d to 2 n inclusive where d = 2 h if h > | F | + 1 and d = 2 h + 2 otherwise. The result is optimal.

Fault-tolerant embedding of paths in crossed cubes

The crossed cube C Q n is an important variant of the hypercube Q n and possesses many desirable properties for interconnection networks. This paper shows that in C Q n with f v faulty vertices and f e faulty edges there exists a fault-free path of length ℓ between any two distinct fault-free vertices for each ℓ satisfying 2 n − 1 − 1 ≤ ℓ ≤ 2 n − f v − 1 provided that f v + f e ≤ n − 3 , where the lower bound of ℓ and the upper bound of f v + f e are tight for some n . Moreover, this result improves the known result that C Q n is ( n − 3 ) -Hamiltonian connected.

Panconnectivity and edge-pancyclicity of faulty recursive circulant [formula omitted

In this paper, we investigate a problem on embedding paths into recursive circulant G ( 2 m , 4 ) with faulty elements (vertices and/or edges) and show that each pair of vertices in recursive circulant G ( 2 m , 4 ) , m ≥ 3 , are joined by a fault-free path of every length from m + 1 to | V ( G ( 2 m , 4 ) ∖ F ) | − 1 inclusive for any fault set F with | F | ≤ m − 3 . The bound m − 3 on the number of acceptable faulty elements is the maximum possible. Moreover, recursive circulant G ( 2 m , 4 ) has a fault-free cycle of every length from 4 to | V ( G ( 2 m , 4 ) ∖ F ) | inclusive excluding 5 passing through an arbitrary fault-free edge for any fault set F with | F | ≤ m − 3 .

Path embedding in faulty hypercubes

There are some interesting results concerning longest paths or even cycles embedding in faulty hypercubes. This paper considers the embeddings of paths of all possible lengths between any two fault-free vertices in faulty hypercubes. Let f v (respectively, f e ) denote the number of faulty vertices (respectively, edges) in an n-dimensional hypercube Q n . We prove that there exists a fault-free path of length l between any two distinct fault-free vertices u and v in Q n with f v + f e ⩽ n - 2 for each l satisfying d Q n ( u , v ) + 2 ⩽ l ⩽ 2 n - 2 f v - 1 and 2 | ( l - d Q n ( u , v ) ) . The bounds on path length l and faulty set size f v + f e for a successful embedding are tight. That is, the result does not hold if l < d Q n ( u , v ) + 2 or l > 2 n - 2 f v - 1 or f v + f e > n - 2 . Moreover, our result improves some known results.

Robust network connectivity

This work analyzes the connectivity of large diameter networks where every link has an independent probability p of failure. We give a (relatively simple) topological condition that guarantees good connectivity between regions of such a network. Good connectivity means that the regions are connected by nearly as many disjoint, fault-free paths as there are when the entire network is fault-free. The topological condition is satisfied in many cases of practical interest, even when two regions are at a distance much larger than the expected "distance between faults", 1/p. We extend this result to networks with failures on nodes, as well as geometric radio networks with random distribution of nodes in a deployment area of a given topography.A rigorous formalization of the intuitive notion of "hole" in a (not necessarily planar) graph is at the heart of our result and our proof. Holes, in the presence of faults, degrade connectivity in the region "around" them to a distance that grows with the size of the hole and the density of faults. Thus, to guarantee good connectivity between two regions even in the presence of faults, the intervening network should not only sport multiple paths, but also not too many large holes.Our result essentially characterizes networks where connectivity depends on the "big picture" structure of the network, and not on the local "noise" caused by faulty or imprecisely positioned nodes and links.

The Conditional Fault-Diameter of the K-ary n-Cube

We obtain the conditional fault diameter of the k-ary n-cube interconnection network. It has been previously shown that under the condition of forbidden faulty sets (i.e. assuming each non-faulty node has at least one non-faulty neighbor), the k-ary n-cube, whose connectivity is 2n, can tolerate up to 4n-3 faulty nodes without becoming disconnected. We extend this result by showing that the conditional fault-diameter of the k-ary n-cube is equal to the fault-free diameter plus two. This means that if there are at most 4n-3 faulty nodes in the k-ary n-cube and if every non-faulty node has at least one non-faulty neighbor, then there exists a fault-free path of length at most the diameter plus two between any two non faulty nodes. We also show how to construct these fault-free paths. With this result the k-ary n-cube joins a group of interconnection networks (including the hypercube and the star-graph) whose conditional fault diameter has been shown to be only two units over the fault-free diameter.