Fault-Tolerant Path Embedding in Folded Hypercubes Under Conditional Vertex Constraints

Abstract

The n-dimensional folded hypercube, denoted as ${FQ}_n$, is an extended version of the n-dimensional hypercube $Q_n$, constructed by adding edges between opposite vertices in $Q_n$, i.e., vertices with complementary addresses. Folded hypercubes outperform traditional hypercubes in various metrics such as the fault diameter, connectivity, and path length. It is known that ${FQ}_n$ has bipartite characteristics in odd $n\ge 3$ and non-bipartite in even $n\ge 2$. In this paper, let ${FF}_v$ represent the set of faulty vertices in ${FQ}_n$ and suppose that each vertex is adjacent to at least four fault-free vertices in ${FQ}_n-{FF}_v$. Then, we consider the following path embedding properties: (1) For every odd $n\ge 3$, ${FQ}_n-{FF}_v$ contains a fault-free path with a length of at least $2^n-2\left|{FF}_v\right|-1$ (respectively, $2^n-2\left|{FF}_v\right|-2$) between any two fault-free vertices of odd (respectively, even) distance if $\left|{FF}_v\right|\le 2n-5$; (2) For every even $n\ge 4$, ${FQ}_n-{FF}_v$ contains a fault-free path with a length of at least $2^n-2\left|{FF}_v\right|-1$ between any two fault-free vertices if $\left|{FF}_v\right|\le 2n-6$.

1. Introduction

The main problem in designing parallel computers is the efficient connection between processors to maximize performance. This is the problem of interconnection networks. For a network, each node may be a processor or a computer, or even another network, but our concerns often focus on its topological properties. This is because from the topology characteristics, we can see the transmission delay and reliability of the entire network and then understand the advantages and disadvantages of the network and its applicable occasions. The hypercube is a well-known, versatile and efficient machine architecture, and it has quite a few outstanding properties, such as a recursive structure, regularity, symmetry, a low branching degree, and so on. Hypercubes are important for designing a massively parallel and decentralized system [1]. A large number of network topologies have been extensively designed by researchers in the past to facilitate the communication of processors in multiprocessor systems through interconnected networks (networks for short). In the proposed network topologies, the folded hypercube [2] is formed by adding edges between pairs of vertices with complementary addresses, i.e., by connecting opposite vertices in the hypercube. In many studies, folded hypercubes have been shown to improve the performance of hypercubes based on many metrics [3,4,5,6,7].

The service quality of a network is determined by the reliability or transmission delay time. Therefore, reliability and maximum transmission delay time are the two most important points in network design. In order to measure the quality of a network in network design, topological characteristics such as reliability, connectivity, and diameter are often evaluated [8]. When the network is ready to apply, the faulty network must be evaluated to avoid situations where edges and/or vertices may fail. Generally, there are two models, namely the standard fault model and Latifi’s conditional fault model [9], for the fault-tolerant embedded properties in faulty networks. The standard fault model is characterized by an unrestricted distribution of faulty edges and/or faulty vertices. In Latifi’s conditional failure model, each vertex in the network is adjacent to at least g fault-free vertices, where g ≥ 2.

Folded hypercubes, as important variants of hypercube interconnection networks, have attracted considerable attention due to their desirable topological properties and strong fault-tolerant capabilities. Foundational research laid the groundwork by exploring essential structural characteristics [10], while subsequent efforts expanded the understanding of edge-fault tolerance [11]. Building on these foundations, researchers have developed strategies for embedding paths that remain functional even when both vertices and edges experience failures. Other studies have explored advanced connectivity properties such as extended bipanconnectivity, panconnectivity, and the ability of folded hypercubes to preserve pancyclic and bipancyclic structures in the presence of multiple faults. Additional investigations have focused on embedding fault-free cycles under various fault models, constructing Hamiltonian cycles despite the presence of highly conditional edge faults, and ensuring that every edge can still lie on a cycle even when adjacent vertices are faulty [12]. These findings emphasize the structural resilience and robustness of folded hypercubes under diverse fault scenarios. Moreover, algorithms have been proposed to construct vertex-disjoint shortest paths efficiently, contributing to improved routing and communication performance in fault-prone environments [13]. The structural relationship between perfect matchings and sub-hypercubes has also been analyzed, offering deeper insights into the internal organization of folded hypercubes [14]. Parallel studies have also investigated long paths under conditional vertex faults in the more general hypercube, which informs and supports efforts to develop similar fault-tolerant strategies in folded hypercube variants [15]. Based on this comprehensive body of work, which covers topological properties [10], edge-fault resilience [11], efficient routing [13], and structural analysis [14], it is evident that significant effort has been made to ensure reliable communication under various fault models. However, relatively few studies have focused on path embedding under conditional vertex faults that more closely reflect real-world constraints. This motivates the present research, which investigates the path embedding properties of folded hypercubes under specific vertex-fault conditions.

To effectively illustrate the main contributions of this study, the remainder of this paper is divided into three sections. In Section 2, entitled Preliminaries, we first introduce some necessary definitions and notations. In Section 3, entitled Path Embedding Strategies under Conditional Fault Constraints, we mainly provide the main proof and cases of embedding a fault-free path in ${FQ}_n$. In Section 4, entitled Conclusions, we summarize the results of this study.

2. Preliminaries

A graph G is usually used to denote an interconnection network. Let $G=\left(V\right(G),E(G\left)\right)$ be a finite and simple graph. $V\left(G\right)$ and $E\left(G\right)$ represent the vertex set and the edge set of G, respectively. Let $Fe$ and $Fv$ represent the faulty edge set and faulty vertex set in G, where $F_e\subseteq E\left(G\right)$ and $F_v\subseteq V\left(G\right)$, respectively. $G-F_e-F_v$ can be denoted as the subgraph generated by deleting $F_v$ and $F_e$ from G. Path $P\left[v_0,v_m\right]=〈v_0,v_1,\dots ,v_m〉$ is a sequence of adjacent vertices such that the vertices of $v_0,v_1,\dots ,v_m$ are all not the same. Furthermore, a path may contain a subpath, denoted as $〈v_0,v_1,\dots ,v_i,P\left[v_i,v_j\right],v_j,v_{j+1},\dots ,v_m〉$, where $P\left[v_i,v_j\right]=〈v_i,v_{i+1},\dots ,v_{j-1},v_j〉$. For graph-theoretic terminologies and notations, please refer to [16].

A graph of an n-dimensional hypercube ($Q_n$ for short) contains $2^n$ vertices. All vertices in $Q_n$ are symbolized by an n-bit binary string $V\left(Q_n\right)=\left\{x_n{x}_{n-1}\dots x_i\dots x_1|x_i\in \left\{0,1\right\},1\le i\le n\right\}$ from $0^n$ to $1^n$. The edge $e=(u,v)\in E\left(Q_n\right)$ links two vertices u and v if and only if u and v have different labels in one bit position. Thus, $u=b_n{b}_{n-1}\dots b_k\dots b_1$ and $v=b_n{b}_{n-1}\dots \overline{b_k}\dots b_1$, where $\overline{b_k}=1-b_k$ is the complement of $b_k$. The edge e can be regarded as dimension k.

For two n-bit binary strings $u=u_n{u}_{n-1}\dots u_i\dots u_1$ and $v=v_n{v}_{n-1}\dots v_i\dots v_1$, if $v_k=1-u_k$ and $v_i=u_i$ for all $i\ne k$, $1\le i\le n$, we denote $v=u^{\left(k\right)}$. Furthermore, if $v_i=1-u_i$ for all $1\le i\le n$, we denote $v=\overline{u}$. The Hamming distance of vertices u and v is denoted as $d_H(u,v)$; that is, the number of different bits in the corresponding strings of u and v.

An n-dimensional folded hypercube (${FQ}_n$ for short) consists of $Q_n$ augmented complementary edges, with an edge connecting to every pair of vertices that are the farthest apart in $Q_n$. Consequently, ${FQ}_n$ has $2^n-1$ more edges than $Q_n$. $E_c$ is used to represent the set of these complementary edges. Examples of ${FQ}_2$ and ${FQ}_3$ are shown in Figure 1.

$Q_n$ can be partitioned along dimension i into two subcubes $Q_{n-1}$, where $1\le i\le n$. The subcubes can be denoted as $Q_{n-1}^{0i}={＊}^{n-i}0{＊}^{i-1}$ and $Q_{n-1}^{1i}={＊}^{n-i}1{＊}^{i-1}$. We denote $Q_{n-1}^{0i}$ (respectively, $Q_{n-1}^{1i}$) as $Q_{n-1}^0$ (respectively, $Q_{n-1}^1$) if the dimension i is not ambiguous.

Definition 1

([17]). An i-partition on ${FQ}_n$, where $1\le i\le n$, partitions ${FQ}_n$ along dimension i into two subcubes $Q_{n-1}^{0i}$ (in brief, $Q_{n-1}^0$) and $Q_{n-1}^{1i}$ (in brief, $Q_{n-1}^1$). In addition, every complementary edge in $E_c$ joins $Q_{n-1}^0$ with $Q_{n-1}^1$.

Let $F_v$ and $Fe$ represent the faulty vertex set and faulty edge set in $Q_n$ and let $F{F}_v$ and $F{F}_e$ represent the fault vertex set and fault edge set in ${FQ}_n$. By Definition 1, we can execute an i-partition on ${FQ}_n$ to form $Q_{n-1}^0$ and $Q_{n-1}^1$, then $F_v^0=F{F}_v\cap V\left(Q_{n-1}^0\right)$ and $F_v^1=F{F}_v\cap V\left(Q_{n-1}^1\right)$. Hence, $F{F}_v=F_v^0\cup F_v^1$, where $1\le i\le n$. In the following, we give some reported properties in $Q_n$ or ${FQ}_n$, which are useful to develop our main result.

Lemma 1

([18]). $Q_n-F_v$ contains a fault-free path with a length of at least $2^n-2|F_v|-1$ (respectively, $2^n-2|F_v|-2$) between any two vertices of odd (respectively, even) distance, where $|F_v|\le n-2$ and $n\ge 3$.

Lemma 2

([15]). Assume that each vertex is adjacent to at least two fault-free vertices. $Q_n-F_v$ contains a fault-free path with a length of at least $2^n-2|F_v|-1$ (respectively, $2^n-2|F_v|-2$) between any two vertices of odd (respectively, even) distance, where $|F_v|\le 2n-5$ and $n\ge 3$.

Lemma 3

([19]). ${FQ}_n-F{F}_e-F{F}_v$ contains a fault-free path with a length of at least $2^n-2|F{F}_v|-1$ (respectively, $2^n-2|F{F}_v|-2$) between any two vertices of odd (respectively, even) distance, where $|F{F}_e|+|F{F}_v|\le n-1$ and $n\ge 1$ is odd; for every even $n\ge 2$ and $|F{F}_e|+|F{F}_v|\le n-2$, ${FQ}_n-F{F}_e-F{F}_v$ contains a fault-free path with a length of at least $2^n-2|F{F}_v|-1$ between any two vertices.

According to Lemma 3, Corollary 1 can be directly derived.

Corollary 1.

For every even $n\ge 2$ and $|F{F}_v|\le n-2$, ${FQ}_n-F{F}_v$ contains a fault-free path with a length of at least $2^n-2|F{F}_v|-1$ between any two vertices.

3. Path Embedding Strategies Under Conditional Fault Constraints

Assume each vertex is adjacent to at least four fault-free vertices in ${FQ}_n-F{F}_v$. We show that (1) for every odd $n\ge 3$, ${FQ}_n-F{F}_v$ contains a fault-free path with a length of at least $2^n-2|F{F}_v|-1$ (respectively, $2^n-2|F{F}_v|-2$) between any two fault-free vertices of odd (respectively, even) distance if $|F{F}_v|\le 2n-5$; and (2) for every even $n\ge 4$, ${FQ}_n-F{F}_v$ contains a fault-free path with a length of at least $2^n-2|F{F}_v|-1$ between any two fault-free vertices if $|F{F}_v|\le 2n-6$.

Lemma 4.

For every odd $n\ge 3$, ${FQ}_n-F{F}_v$ contains a fault-free path with a length of at least $2^n-2|F{F}_v|-1$ (respectively, $2^n-2|F{F}_v|-2$) between any two fault-free vertices of odd (respectively, even) distance if $|F{F}_v|\le 2n-5$, and each vertex is adjacent to at least four fault-free vertices in ${FQ}_n-F{F}_v$.

Proof. 

By the definition of ${FQ}_n$, ${E(FQ}_n)={E(Q}_n)\cup E_c$ and ${V(FQ}_n)={V(Q}_n)$. If we eliminate all complementary edges in $E_c$, then ${FQ}_n-E_c\cong Q_n$. Since each vertex in ${FQ}_n-F{F}_v$ is adjacent to at least four fault-free vertices, we have that each vertex in ${FQ}_n-E_c-F{F}_v\cong Q_n-F_v$ would be adjacent to at least three fault-free vertices. Furthermore, since $|F{F}_v|=|F_v|\le 2n-5$ and $n\ge 3$, by Lemma 2, we have that $Q_n-F_v\subseteq {FQ}_n-F{F}_v$ contains a fault-free path with a length of at least $2^n-2|F{F}_v|-1$ (respectively, $2^n-2|F{F}_v|-2$) between any two vertices of odd (respectively, even) distance. □

Lemma 5.

For every even $n\ge 4$, ${FQ}_n-F{F}_v$ contains a fault-free path of a length of at least $2^n-2|F{F}_v|-1$ between any two fault-free vertices if $|F{F}_v|=|F_v|\le 2n-6$, and each vertex is adjacent to at least four fault-free vertices in ${FQ}_n-F{F}_v$.

Proof. 

We consider the cases for $n=4$ and every even $n\ge 6$.

Case 1. For $n=4$, since $|F{F}_v|\le 2\times 4-6=2$, by Corollary 1, ${FQ}_4-F{F}_v$ contains a fault-free path with a length of at least $2^4-2\times 2-1=11$ between any two vertices.

Case 2. For every even $n\ge 6$, let u and v be any two fault-free distinct vertices in ${FQ}_n$. Since $d_H(u,v)\ge 1$, by Definition 1, we can execute an i-partition on ${FQ}_n$ into two subcubes $Q_{n-1}^0$ and $Q_{n-1}^1$ such that one subcube contains u and the other subcube contains v, where $i\in \{1,2,\dots ,n\}$. Without loss of generality, let $i=n$, $u\in V\left(Q_{n-1}^0\right)$ and $v\in V\left(Q_{n-1}^1\right)$. Furthermore, we assume that $|F_v^0|\ge |F_v^1|$, then $|F_v^0|\le 2n-6$ and $|F_v^1|\le n-3$. According to the distribution of faulty vertices in ${FQ}_n$, we have the following scenarios.

Case 2.1. $|F_v^0|\le 2n-6$ and $|F_v^1|=0$. Since each vertex is adjacent to at least four fault-free vertices in ${FQ}_n-F{F}_v$, we have that each vertex in $Q_{n-1}^0-F_v^0$ is adjacent to at least two fault-free vertices in $Q_{n-1}^0$. (As a supplementary explanation, assume that a fault-free vertex $u\in V({FQ}_n-F{F}_v)$ is adjacent to four fault-free vertices. Under an i-partition of ${FQ}_n$, suppose $u\in V\left(Q_{n-1}^0\right)$. Then, at most two of its counterparts, $u^{\left(i\right)}$ and $\overline{u}$, belong to $V\left(Q_{n-1}^1\right)$. Consequently, the vertex u can be adjacent to two fault-free vertices within $Q_{n-1}^0$.) Let f be a faulty vertex in $F_v^0$ such that f is adjacent to at least two fault-free vertices in $Q_{n-1}^0$. (Assuming that for any vertex $f\in F_v^0$, vertex $f$ is adjacent to at most one fault-free vertex in $Q_{n-1}^0$, it immediately follows that $|F_v^0|>2n-6$, which contradicts the assumption that $|F{F}_v|\le 2n-6$.) Without loss of generality, we may assume that f is temporarily fault-free in $Q_{n-1}^0$. Since $\left|F_v^0-\left\{f\right\}\right|=2n-6-1\le 2\left(n-1\right)-5=2n-7$, by Lemma 2, there exists a fault-free path $P_0[u,f]$of a length of at least $2^{n-1}-2(|F_v^0|-1)-1$ (respectively, $2^{n-1}-2(|F_v^0|-1)-2$) if $d_H(u,f)$ is odd (respectively, even). Let p be a vertex in $P_0[u,f]$ such that p is adjacent to f. The following subcases need to be considered.

Case 2.1.1. $d_H(u,v)$ is odd and $d_H(u,f)$ is odd. Since $|F_v^1|=0$, $p^{\left(n\right)}$ is fault-free in $Q_{n-1}^1$. Note that $d_H(p^{\left(n\right)},v)$ is even in $Q_{n-1}^1$. If $p^{\left(n\right)}\ne v$ in $Q_{n-1}^1$, by Lemma 1, there exists a fault-free path $P_1[p^{\left(n\right)},v]$ of a length of at least $2^{n-1}-2$ in $Q_{n-1}^1$. Therefore, $〈u,P_0\left[u,p\right],p,p^{\left(n\right)},P_1\left[p^{\left(n\right)},v\right],v〉$ forms a path $P[u,v]$ of a length of at least $2^{n-1}-2(|F_v^0\left|-1\right)-1)-1+1+\left(2^{n-1}-2\right)=2^n-2|F_v^0|-1=2^n-2|F{F}_v|-1$ (see Figure 2a). In addition, if $p^{\left(n\right)}=v$ in $Q_{n-1}^1$, we can find an edge $(r,s)$ in $P_0[u,f]$ such that $(r,s)\ne (p,f)$. Since $|F_v^1|=0$, by Lemma 1, there exists a fault-free path $P_1\left[r^{\left(n\right)},s^{\left(n\right)}\right]$ of a length of at least $2^{n-1}-3$ in $Q_{n-1}^1-\left\{v\right\}$. Therefore, $〈u,P_0\left[u,r\right],r,r^{\left(n\right)},P_1\left[r^{\left(n\right)},s^{\left(n\right)}\right],s^{\left(n\right)},s,P_0\left[s,p\right],p,v〉$ forms a path $P[u,v]$ of a length of at least $2^{n-1}-2(|F_v^0\left|-1\right)-1)-2+3+\left(2^{n-1}-3\right)=2^n-2|F_v^0|-1=2^n-2|F{F}_v|-1$ (see Figure 2b).

Case 2.1.2. $d_H(u,v)$ is odd and $d_H(u,f)$ is even. Since $|F_v^1|=0$, $p^{\left(n\right)}$ is fault-free in $Q_{n-1}^1$. Note that $d_H(p^{\left(n\right)},v)$ is odd and $p^{\left(n\right)}\ne v$ in $Q_{n-1}^1$. By Lemma 1, there exists a fault-free path $P_1[p^{\left(n\right)},v]$ of a length of at least $2^{n-1}-1$ in $Q_{n-1}^1$. Therefore, $〈u,P_0\left[u,p\right],p,p^{\left(n\right)},P_1\left[p^{\left(n\right)},v\right],v〉$ forms a path $P[u,v]$ of a length of at least $2^{n-1}-2(|F_v^0\left|-1\right)-2)-1+1+\left(2^{n-1}-1\right)=2^n-2|F_v^0|-1=2^n-2|F{F}_v|-1$.

Case 2.1.3. $d_H(u,v)$ is even and $d_H(u,f)$ is odd. Since $|F_v^1|=0$, $\overline{p}$ is fault-free in $Q_{n-1}^1$. Note that $d_H(\overline{p},v)$ is even in $Q_{n-1}^1$. Then, the remaining proof of the case is similar to that in Case 2.1.1 of Lemma 5.

Case 2.1.4. $d_H(u,v)$ is even and $d_H(u,f)$ is even. Since $|F_v^1|=0$, $\overline{p}$ is fault-free in $Q_{n-1}^1$. Note that $d_H(\overline{p},v)$ is odd and $\overline{p}\ne v$ in $Q_{n-1}^1$. Then, the remaining proof of the case is similar to that in Case 2.1.2 of Lemma 5.

Case 2.2. $|F_v^0|\le 2n-7$ and $|F_v^1|\le n-3$. Since each vertex is adjacent to at least four fault-free vertices in ${FQ}_n-F{F}_v$, we have that each vertex in $Q_{n-1}^0-F_v^0$ (respectively, $Q_{n-1}^1-F_v^1$) is adjacent to at least two fault-free vertices in $Q_{n-1}^0$ (respectively, $Q_{n-1}^1$). The following subcases need to be considered.

Case 2.2.1. $d_H(u,v)$ is odd. Let p be a fault-free vertex in $Q_{n-1}^0$ such that $d_H(u,p)$ is odd and $p^{\left(n\right)}$ is fault-free in $Q_{n-1}^1$. (If no such a vertex exists, then $\left|F{F}_v\right|\ge {\displaystyle \frac{2^n-1}{2}}=2^{n-2}>2n-6$ for every even $n\ge 6$, which contradicts the assumption that $\left|F{F}_v\right|\le 2n-6$.) Since $\left|F_v^0\right|\le 2n-7=2\left(n-1\right)-5$, by Lemma 2, there exists a fault-free path $p_0[u,p]$ of a length of at least $2^{n-1}-2|F_v^0|-1$ in $Q_{n-1}^0$. Note that because $d_H(p^{\left(n\right)},v)$ is odd and $\left|F_v^1\right|\le n-3=\left(n-1\right)-2$ in $Q_{n-1}^1$, by Lemma 1, there exists a fault-free path $P_1\left[p^{\left(n\right)},v\right]$ of a length of at least $2^{n-1}-2|F_v^1|-1$ in $Q_{n-1}^1$. Therefore, $〈u,P_0\left[u,p\right],p,p^{\left(n\right)},P_1\left[p^{\left(n\right)},v\right],v〉$ forms a path $P[u,v]$ of a length of at least $2^{n-1}-2(|F_v^0\left|-1\right)+1+\left(2^{n-1}-2\left|F_v^1\right|-1\right)=2^n-2|F{F}_v|-1$.

Case 2.2.2. $d_H(u,v)$ is even. Let p be a fault-free vertex in $Q_{n-1}^0$ such that $d_H(u,p)$ is odd and $\overline{p}$ is fault-free in $Q_{n-1}^1$. Note that $d_H(\overline{p},v)$ is odd. Then, the remaining proof of the case is similar to that in Case 2.2.1 of Lemma 5.

According to the above case studies, we complete the proof. □

Theorem 1.

Let $F{F}_v$ represent the faulty vertex set in ${FQ}_n$ and assume that each vertex is adjacent to at least four fault-free vertices in ${FQ}_n-F{F}_v$. For every odd $n\ge 3$, if $\left|F{F}_v\right|\le 2n-5$, then ${FQ}_n-F{F}_v$ contains a fault-free path with a length of $2^n-2|F{F}_v|-1$ between any two fault-free vertices of odd distance or a length of $2^n-2|F{F}_v|-2$ for an even distance; and for every even $n\ge 4$, if $\left|F{F}_v\right|\le 2n-6$ then ${FQ}_n-F{F}_v$ contains a fault-free path with a length of $2^n-2|F{F}_v|-1$ between any two fault-free vertices.

4. Conclusions

This study enhances the understanding of fault-tolerant interconnection networks by analyzing long path embeddings in folded hypercubes under a conditional vertex-fault model. We establish the following lower bounds for the lengths of fault-free paths between any two fault-free vertices:

• For odd $n\ge 3$, the path length in ${FQ}_n-{FF}_v$ is at least $2^n-2\left|{FF}_v\right|-1$ (or, $2^n-2\left|{FF}_v\right|-2$) when $\left|{FF}_v\right|\le 2n-5$;
• For even $n\ge 4$, the path length in ${FQ}_n-{FF}_v$ is at least $2^n-2\left|{FF}_v\right|-1$ in ${FQ}_n-{FF}_v$ when $\left|{FF}_v\right|\le 2n-6$.

These results demonstrate the strong path-embedding capabilities of folded hypercubes under the conditional vertex faults and offer theoretical support for their use in fault-tolerant parallel computing systems.