Dual Protection Routing Trees on Graphs

1. Introduction

In IP networks, Intra-domain routing has traditionally relied on distributed computing among routers. Packet forwarding is destination-based and hop-by-hop, and routes are created when needed by a centralized unit. Kwong et al. [1] introduced a reactive routing scheme called protection routing, which employs the multi-paths technique for packet forwarding. If the routing is a protection routing, packet delivery to the destination node can proceed uninterrupted in the presence of any single node or link failure. All routers have an alternate path they can use in case of failure to forward packets.

An undirected simple graph G = (V(G), E(G)), where the vertex set V(G) and the edge set E(G) represent the set of nodes and the set of communication links between nodes, respectively. Let k ≥ 2 be an integer and T₁, T₂, ..., T_k be spanning trees of a graph G. A vertex in a tree T_i is a leaf if its degree is one, and an inner vertex otherwise. Two paths from u to v are internally vertex-disjoint if they have no common vertices other than u and v. Next, two spanning trees T_i and T_j are edge-disjoint if they share no common edge. Then, The spanning trees T₁, T₂, ..., T_k are completely independent spanning trees, abbreviated as CISTs, if they are pairwise edge-disjoint and internally vertex-disjoint. The k-CISTs problem was first posed in 2001 by Hasunuma [2]. Then, Tapolcai [3] provided sufficient conditions for protection routing in IP networks according to two CISTs. He gave a deterministic polynomial construction, which builds up a protection routing for an arbitrary destination node by two CISTs.

Previous related works are described below. Hasunuma first worked on the theoretical study of CISTs and showed that the underlying graph of any k-connected line-directed graph recognizes k CISTs [2], and there exist two CISTs in a 4-connected maximal planar graph [4] and the Cartesian product of two 2-connected graphs [5]. Cheng et al. construct two CISTs in crossed cubes [6]. Darties et al. determine three CISTs in some Cartesian products of three cycles [7]. Pai et al. proposed several known CISTs, such as complete graphs, complete bipartite and complete tripartite graphs [8], chord rings with specific conditions [9], and variants of hypercubes [10,11,12,13]. In the past ten years, data center networks have received much attention. Qin et al. constructing dual-CISTs of DCell data center networks [14]. Li et al. provide multi-CISTs on data center Networks [15], BCCC data center networks [16], and data center networks with heterogeneous edge-core servers [17].

Tapolcai [3] showed that two CISTs have an application on configuring a protection routing. According to this, Pai et al. construct protection routings via CISTs on some Cayley networks [18], crossed cubes [13], möbius cubes [12], pancake graphs [19], and dense Gaussian on-chip networks [20]. Li et al. involve multi-CISTs in the protection routing scheme on data center Networks [15], BCCC data center networks [16], and data center networks with heterogeneous edge-core servers [17].

Recently, we found that the protection routing can still be constructed by reducing some of the edge-disjoint requirements of two CISTs. Therefore, in this paper, we will define new trees called dual protection routing trees (abbreviated as dual-PRTs) and propose an algorithm that configures a protection routing via them. Taking complete graphs, complete bipartite graphs, hypercubes, locally twisted cubes as examples, we provide a recursive algorithm to find dual-PRTs on them and configure protection routings. Furthermore, we supplement some analysis on simulation to evaluate the corresponding performance. The performance evaluation shows that dual-PRTs are better than two CISTs.

The rest of the paper is organized as follows: Section 2 introduces the necessary definitions and theorems for protection routing and CISTs. In section 3, we first propose definitions of dual-PRTs, and a transformation algorithm that uses dual-PRTs to configure the protection routing. Taking complete graphs, complete bipartite graphs, hypercubes Qn, locally twisted cubes as examples, we provide a recursive method to construct dual-PRTs on them. Next, in order to compare with previous related research, we make the performance assessment of protection routings on hypercubes and locally twisted cubes in section 4. Finally, section 5 is the conclusion of this paper.

2. Preliminaries

The topology of a network is usually modeled as an undirected graph G = (V(G), E(G)). The neighborhood of a vertex v in a graph G, denoted by N_G(v), is the set of vertices adjacent to v in G. In a tree, an edge is called a leaf edge (respectively, a stem) if it is adjacent to at least one leaf (respectively, two inner vertices). For convenience, the terms "networks" and "graphs", "nodes" and "vertices", "links" and "edges" are often used interchangeably in this paper. For a destination node d ∈ V(G), the route of traffic destined for d is a directed acyclic graph R_d = (V(G), E_d(G)), where the underlying graph of R_d is the spanning subgraph of G and each node u ∈ V(G) \ {d} has at least one outgoing link in R_d. We use <u, v> to denote a directed link from u to v. If <u, v> ∈ E_d(G), node v is called the primary next-hop of node u, or PNH for short, and the link <u, v> is called a primary link of u. Furthermore, node u is called upstream of node v if there exists a directed path from u to v in R_d. In contrast, node v is downstream of node u. Except for the destination node d, when a single component fails f ∈ V(G) ∪ E_d(G) \ {d}, let G – f and R_d – f be the residual network and routing obtained from G and R_d, respectively. Kwong et al. [1] gave the following two definitions.

Definition 1.

([1]) A node u∈ V(G) \ {d} in a routing R_d is said to be protected with respect to d if after any single component failure f∈ V(G)∪E_d(G) \ {d} that affects node u’s PNH, then there exists a node x∈ N_G(u) − f(u) such that the following two conditions (1) x is not upstream of u in R_d – f; (2) x and all its downstream nodes, except d, have at least one PNH in Rd − f.

Definition 2.

([1]) A routing R_d is a protection routing if every node u∈ V(G) \ {d} is protected in R_d. A network G is protectable if there exists a protection routing R_d for all d∈ V(G), otherwise, G is unprotectable.

In fact, x is called the second next-hop of node u, SNH for short, and is denoted by SNH(u). We notice that when a failure occurs, the candidate SNHs of node u are not unique. However, the first condition is to avoid forwarding loops in case of failure, and the second condition guarantees that packets are delivered to d through node x and its downstream nodes in R_d – f. For example, Figure 1 shows a protection routing in hypercube Q₃ (we will formally introduce it in Section 3.3) for the destination node 0, and Table 1 gives a candidate SNH of node u ∈ V(Q₃) \ {0}.

Then, Tapolcai provided sufficient conditions for protection routing in IP networks according to two CISTs. The following definitions are about CISTs.

Definition 3.

Two spanning trees T₁ and T₂ are inner-vertex-disjoint if any vertex can only be an inner node in T₁ or T₂.

Definition 4.

Two spanning trees T₁ and T₂ are edge-disjoint if they share no common edge.

Definition 5.

([2]) The spanning trees T₁ and T₂ are two completely independent spanning trees if they follow definitions 3 and 4.

3. Main Results

In [3], Tapolcai uses two CISTs to configure a protection routing for a destination node d. Then we find another way to construct routing through two spanning trees T₁ and T₂ in which most of the leaf edges can be shared. We first give the following two definitions:

Definition 6.

Two spanning trees T₁ and T₂ are stem-disjoint if they share no common stem.

Definition 7.

The spanning trees T₁ and T₂ are dual protection routing trees based on destination node d, which are denoted as dual-PRT^ds, if Definitions 3, 6 and the following condition are met:

• Without loss of generality, we assume that node d is an inner vertex in T₁. There exists an inner node d’ in T₂ such that (d, d’) is a leaf edge in T₂ and d’ is adjacent to another inner node w (≠ d) in T₁.

For example, Figure 2 shows dual-PRT⁰s. The inner nodes of T₁ are vertices 0, 2, 3, 4, and the inner nodes of T₂ are vertices 1, 5, 6, 7. T₁ and T₂ follow Definition 3. Then the stem sets of T₁ and T₂ are {(0, 2), (0, 4), (2, 3)} and {(1, 5), (5, 7), (6, 7)} respectively, and them follow Definition 6. Finally, let d’, w be vertices 1 and 3 respectively. Since (0, 1) is a leaf edge in T₂ and vertex 1 is adjacent to inner node 3 in T₁, according to Definition 7, T₁ and T₂ are dual-PRT⁰s. In fact, Figure 1(b) shows the protection routing configured by T₁ and T₂. We will propose a formal construction algorithm later.

Lemma 1.

There are no two CISTs in hypercube Q₃.

Proof of Lemma 1.

As shown in Figure 1(a), hypercube Q₃ has eight nodes and twelve edges. A tree with eight nodes has seven edges. By Definition 4, we need fourteen edges to find two edge-disjoint spanning trees, but Q₃ has only twelve edges. □

According to Lemma 1, if there are no two CISTs in Q₃, the protection routing cannot be configured by Tapolcai’s method [3]. That is why we propose dual-PRT^ds. Now we present the construction method in which the protecting routing is configured by dual-PRT^ds, as follows.

Algorithm 1: Configuring a protection routing via dual-PRTds

Input: Dual-PRTds T1, T2 of a network G where d is the destination node.

Output: A protection routing Rd = (V(G), PL, AL) where PL is the primary links set and AL is the alternative links set

Step 1 T1’ ← T1 which takes all links directed to root d; Step 2 T2’ ← T2 which takes all links directed to root d; Step 3 PL ← all stems in T1’ ∪ all stems in T2’ ∪ {<d’, d>}; Step 4 AL ← all leaf edges in T1’ ∪ all leaf edges in T2’ except <d’, d>; Step 5 If a vertex is a leaf in both T1 and T2, then add its T1 leaf edge to PL and add its T2 leaf edge to AL; Step 6 Return Rd = (V(G), PL, AL)

For example, taking Figure 2 (a) and (b) as input, according to steps 1, 2 of Algorithm 1, we have T₁’ and T₂’, as shown in Figure 3 (a) and (b). Next, classify the stems and leaf edges of the two trees into primary links or alternative links according to steps 3 and 4, then a protection routing R₀ can be obtained, as shown in Figure 3 (c).

Theorem 1.

The routing constructed by Algorithm 1 is a protection routing with respect to node d.

Proof of Theorem 1.

According to steps 1 and 2 in Algorithm 1, all edges of T₁ and T₂ are directed to destination node d. Obviously, stems of T₁ and stems of T₂ respectively form 2 primary routes by step 3. Then <d’, d> connect the second route to destination d. Figure 3 can be used as an illustration. By Definition 6, stem-disjoint can ensure that two primary routes will not overlap. Also according to Definition 3, each vertex can only be an inner node in T₁ or T₂, which ensures that each vertex has a primary link and an alternative link. If a vertex is a leaf in both T₁ or T₂, it still has both links at step 5. Since <d’, d> is a primary link, d’ needs another link <d’, w> as an alternative link. Now, let us consider a component failure, it could be a node or a link. If it is a node (respectively, a link), we assume it is the node z (respectively, <u, z>) and z = PNH(u). Regardless of whether the failure is z or <u, z>, the link <u, z> is unavailable. If <u, z> is in T₁ (respectively, T₂), then node u’s alternative link is in T₂ (respectively, T₁), so it forwards the packet to the primary routes of T₂ (respectively, T₁) to the destination node d. Therefore, the routing constructed by Algorithm 1 is a protection routing with respect to node d. □

In [8], we construct CISTs on complete graph K_n where n ≥ 4, complete bipartite graph K_n,m where m ≥ n ≥ 4. Therefore, we will construct dual-PRT^ds on them, and make some comparisons.

3.1. Dual-PRT^ds on complete graphs

A complete graph with n vertices denoted as K_n, is a graph in which every pair of distinct vertices is connected by a unique edge.

Theorem 2.

There exist dual-PRT^ds in K_n where d is any vertex of K_n and n ≥ 4.

Proof of Theorem 2.

If n is even (respectively, odd), let the vertices of K_n be labeled 1, 2, 3, …, n (respectively, 0, 1, 2, 3, …, n). For the case that n is even, let all odd vertices be inner nodes in T₁ and all even vertices be inner nodes in T₂. Stems are (1, 3), (1, 5), …, (1, n – 1) (respectively, (2, 4), (2, 6), …, (2, n)) and leaf edges are (3, 2), (3, 4), (5, 6), …, (n – 1, n) (respectively, (1, 2), (3, 4), …, (n – 1, n)) in T₁ (respectively, T₂). Clearly, T₁ and T₂ obey Definitions 3 and 6. Let d = 1, d’ = 2, w = 3, then Definitions 7 holds. T₁ and T₂ are dual-PRT¹s in K_n. For the case that n is odd, the trees are similar to the ones in the case that n is even. We let additional vertex 0 be adjacent to vertex 1 and 2 in T₁ and T₂ respectively. Clearly, T₁ and T₂ still obey Definitions 7, and they are dual-PRT¹s in K_n. It is widely known that K_n is vertex-symmetric, so there exist dual-PRT^ds where d can be any vertex in K_n. □

Figure 4 (a) and (b) illustrate the proof of Theorem 2. The tree diameter is the length of the shortest path between the most distanced nodes, which affects the length of the routing in practice. In [8], the diameters of CISTs in K_n are all 3, and the diameters of our dual-PRT^ds in K_n are 4, which is slightly worse than the former.

Theorem 3.

K_n is protectable for n ≥ 4.

Proof of Theorem 3.

According to Theorem 2, there exist dual-PRT^ds in K_n for n ≥ 4. Then, by Algorithm 1, we can use dual-PRT^ds to configure the protection routing as shown in Figure 4 (c). According to Definition 2, this theorem holds. □

3.2. Dual-PRT^ds on Complete bipartite graphs

A complete bipartite graph, denoted by K_m,n is a graph whose vertices can be partitioned into two subsets V₁ and V₂ such that |V₁|= m, |V₂|= n and two vertices u and v are adjacent if and only if u ∈ V₁ and v ∈ V₂.

Theorem 4.

There exist dual-PRT^ds in K_m,n where d is any vertex in V₁ (or V₂) and m ≥ n ≥ 3.

Proof of Theorem 4.

Without loss of generality, we assume that |V₁|= m ≥ |V₂|= n. Let the vertices of V₁ (respectively, V₂) be labeled a₁, a₂, a₃, …, a_m (respectively, b₁, b₂, b₃, …, b_n). For the case that d ∈ V₁, let vertices a₂, a₃, …, a_n-1, b_n be inner nodes in T₁ and vertices b₁, b₂, …, b_n-1 be inner nodes in T₂. Stem edges are (a₂, b_n), (a₃, b_n), …, (a_n-1, b_n) (respectively, (b₁, a_n), (b₂, a_n), …, (b_n-1, a_n)) and leaf edges are (b₁, a₂), (b₂, a₂), (b₃, a₃), …, (a_n, b_n), (a₁, b_n), (a_n+1, b_n), (a_n+2, b_n), …, (a_m, b_n) (respectively, (a₁, b₁), (a₂, b₂), …, (b_n, a_n), (a_n+1, b_n-1), (a_n+2, b_n-1), …, (a_m, b_n-1)) in T₁ (respectively, T₂). Clearly, T₁ and T₂ obey Definitions 3 and 6. Let d = a₁, d’ = b₁, w = a₂, then Definitions 7 holds. Since a₁ can be replaced by any vertex in V₁, T₁ and T₂ are dual-PRT^ds in K_m,n where d ∈ V₁. For the case that d ∈ V₂, we can swap V₁ and V₂, and prove it in a similar way. □

Figure 5 is used to illustrate the proof of Theorem 4. In [8], there are no 2 CISTs in K_3,3, because two edge-disjoint spanning trees require a total of 10 edges while K_3,3 only has 9 edges. But K_3,3 has dual-PRT^ds, which can be used to configure a protection routing. Next, when m ≥ n ≥ 4 the diameter of CISTs in K_m,n are all 5 in [8], and the diameters of our dual-PRT^ds in K_m,n are 4, which is slightly better than the former.

According to Theorem 4, there exist dual-PRT^ds in K_m,n where d ∈ V₁ (or V₂). Then they can be used to configure the protection routing via Algorithm 1. Finally, by Definition 2, we have the following theorem.

Theorem 5.

K_m,n is protectable where m ≥ n ≥ 3.

Theorem 5 improves the known result that K_m,n is protectable where m ≥ n ≥ 4.

3.3. Dual-PRT^ds on hypercubes

The n-dimensional hypercube, denoted by Q_n, is a graph with 2ⁿ vertices such that each vertex corresponds to an n-tuple (b_n, b_n−1, …, b₁) on the set {0, 1}ⁿ and two vertices are adjacent by an edge if and only if they differ in exactly one coordinate [21]. For example, Q₃ is shown in Figure 1(a). For conciseness, the labels of vertices are changed to their decimal. The hypercube is one of the most popular interconnection networks because of its attractive properties, including regularity, vertex symmetric, edge symmetric, small diameter, strong connectivity, recursive construction, partition capability, and small link complexity.

Lemma 2.

Q₃ has dual-PRT^ds with a diameter is 5 where d is any vertex in Q₃.

Proof of Lemma 2.

Let destination node d be vertex 0. dual-PRT⁰s are shown in Figure 2. Clearly, diameters of two trees are both 5. Vertices 0, 2, 3, 4 are inner nodes in T₁ and vertices 1, 5, 6, 7 are inner nodes in T₂. Stems are (0, 2), (0, 4),(2, 3) (respectively, (1, 5), (5, 7), (6, 7)) and leaf edges are (1, 3), (3, 7), (2, 6), (4, 5) (respectively, (0, 1), (2, 6), (3, 7), (4, 5)) in T₁ (respectively, T₂). Clearly, T₁ and T₂ obey Definitions 3 and 6. Let d’ be vertex 1 and w be vertex 3, then Definitions 7 is met. Since Q_n is vertex symmetric and there exists an automorphism of Q_n, vertex 0 can be replaced by any vertex in Q₃. This lemma holds. □

In [10], we provide and prove a simple unified approach to construct two CISTs in the n-dimensional hypercubes variants (abbreviated as QV_n) using two CISTs in QV_n-1. This approach is also applicable for dual-PRT^ds. For QV_n, it is constructed recursively from two QV_n-1s. Then (n-1)-dimensional dual-PRT^ds also exist in each QV_n-1. We can select a pair of port vertices and add an edge to connect two trees, where port vertices obey (1) removing the leftmost bit, their labels will be the same; (2) there exists an edge between them in QV_n. For example, Figure 6 shows dual-PRT⁰s of Q₄ which are constructed from dual-PRT⁰s of Q₃. Clearly, port vertices of T₁ (respectively, T₂) are vertices 0 and 8 (respectively, 5 and 13).

Using the unified approach, if we choose vertices near the center of the trees as their port vertices, then we can get a better result of the diameter of dual-PRT^ds for the construction. Suppose that T₁ and T₂ are dual-PRT^ds of QV_n-1 and let T’₁ and T’₂ be dual-PRT^ds of QV_n. If a vertex u is a center vertex of T₁ and v is a center vertex of T₂, then the choice of a pair (u, u+2^n-1) in T’₁ and a pair (v, v+2^n-1) in T’₂ as port vertices can build two dual-PRT^ds of QV_n by induction on n. Since we can always choose the center vertices of T₁ and T₂ as port vertices, it follows that D(T’_j ) = 2 ∙ 「 D(T_j ) / 2 」 + 1 where j ∈ {1, 2} and D(T) denotes the diameter of the tree T. For example, as shown in Figure 2 and Figure 6, the diameters of dual-PRT^ds of Q₃ are both 5 and the diameters of dual-PRT^ds of Q₄ are both 7. Then, by the unified approach and Lemma 2, we have the following theorem.

Theorem 6.

Q_n has dual-PRT^ds with a diameter is 2n - 1 where d ∈ V(Q_n) and n ≥ 3.

By Theorem 6, there exist dual-PRT^ds in Q_n where d ∈ V(Q_n). According to Algorithm 1, two trees can be used to configure the protection routing. Then, by Definition 2, we have the following theorem.

Theorem 7.

Q_n is protectable where n ≥ 3.

As stated in Lemma 1, Theorem 7 improves on the known result that Q_n is protectable when n ≥ 4. As far as I know, the diameters of two CISTs of Q₄ are both 8 [5]. The diameters of dual-PRT^ds of Q_n are one less than the diameters of two CISTs of Q_n while n ≥ 4.

3.4. Dual-PRT^ds on LTQs

The n-dimensional locally twisted cube, denoted by LTQ_n is defined recursively as follows (see [22]):

(1)

LTQ₁ is the complete graph on two vertices labeled by 0 and 1. LTQ₂ is a graph consisting of four vertices with labels 00, 01, 10, 11 together with four edges (00, 01), (00, 10), (01, 11), and (10, 11).

(2)

For n ≥ 3, LTQ_n is composed of two subcubes LTQ⁰_n−1 and LTQ¹_n−1 such that each vertex x = 0b_n-1b_n-2···b₁ ∈ V(LTQ⁰_n−1) is connected with the vertex 1(b_n-1⊕b₁)b_n-2···b₁ ∈ V(LTQ¹_n−1) by an edge where ⊕ represents exclusive or.

LTQ_n is a variant of Q_n, and one advantage of LTQ_n is that the diameter is only about half of the diameter of Q_n. For example, LTQ₃ is shown in Figure 7(a), its diameter is 2, while the value of Q₃ is 3. In [23], LTQ_n is vertex-transitive if and only if n ≤ 3.

Lemma 3.

LTQ₃ has dual-PRT^ds with a diameter is 5 where d is any vertex in LTQ₃.

Proof of Lemma 3.

Let destination node d be vertex 0. dual-PRT⁰s are shown in Figure 7 (b) and (c). Clearly, the tree diameters are both 5. Vertices 0, 4, 5, 7 are inner nodes in T₁ and vertices 1, 2, 3, 6 are inner nodes in T₂. Stems are (0, 4), (4, 5), (5, 7) (respectively, (1, 3), (2, 3), (2, 6)) and leaf edges are (0, 2), (1, 7), (3, 5), (4, 6) (respectively, (0, 1), (1, 7), (3, 5), (4, 6)) in T₁ (respectively, T₂). Clearly, T₁ and T₂ obey Definitions 3 and 6. Let d’ be vertex 1 and w be vertex 7, then Definitions 7 holds. Since LTQ₃ is vertex symmetric and there exists an automorphism of LTQ₃, vertex 0 can be replaced by any vertex in LTQ₃. This lemma holds. □

In [23], the full automorphism group of LTQ_n with n ≥ 4 has exactly two orbits, and the odd and even vertices belong to the same group. Then, we will present dual-PRT^ds for d belonging to odd and even vertices respectively.

Lemma 4.

LTQ₄ has dual-PRT^ds with diameters 6 and 8 where d is any even vertex in LTQ₄.

Proof of Lemma 4.

Let destination node d be vertex 0. dual-PRT⁰s are shown in Figure 8. Clearly, the diameters of the two trees are 8 and 6 respectively. Vertices 0, 3, 4, 5, 8, 9, 12, 15 are inner nodes in T₁ and vertices 1, 2, 6, 7, 10, 11, 13, 14 are inner nodes in T₂. Stems are (0, 4), (0, 8), (3, 5), (4, 5), (4, 12), (8, 9), (9, 15) (respectively, (1, 7), (1, 13), (2, 6), (6, 7), (6, 14), (7, 11), (10, 11)) and leaf edges are (0, 2), (1, 3), (4, 6), (5, 7), (8, 10), (9, 11), (12, 14), (13, 15) (respectively, (0, 1), (2, 3), (4, 6), (5, 7), (8, 10), (9, 11), (12, 14), (13, 15)) in T₁ (respectively, T₂). Clearly, T₁ and T₂ obey Definitions 3 and 6. Let d’ be vertex 1 and w be vertex 3, then Definitions 7 holds. Since all even vertices belong to the same automorphism group in LTQ₄, this lemma holds. □

Lemma 5.

LTQ₄ has dual-PRT^ds with diameters 6 and 8 where d is any odd vertex in LTQ₄.

Proof of Lemma 5. Let destination node d be vertex 1. Dual-PRT¹s are shown in Figure 9. Clearly, the diameters of the two trees are 6 and 8 respectively. Vertices 1, 2, 6, 7, 10, 11, 13, 14 are inner nodes in T₁ and vertices 0, 3, 4, 5, 8, 9, 12, 15 are inner nodes in T₂. Stems are (1, 7), (1, 13), (2, 6), (6, 7), (6, 14), (7, 11), (10, 11) (respectively, (0, 4), (0, 8), (3, 5), (4, 5), (4, 12), (8, 9), (9, 15)) and leaf edges are (0, 2), (1, 3), (4, 6), (5, 7), (8, 10), (9, 11), (12, 14), (13, 15) (respectively, (0, 1), (2, 3), (4, 6), (5, 7), (8, 10), (9, 11), (12, 14), (13, 15)) in T₁ (respectively, T₂). Clearly, T₁ and T₂ obey Definitions 3 and 6. Let d’ be vertex 0 and w be vertex 2, then Definitions 7 holds. Since all odd vertices belong to the same automorphism group in LTQ₄, this lemma holds. □

By using the unified approach [10] to configure dual-PRT^ds of LTQ_n from dual-PRT^ds of LTQ_n-1. It is better to choose vertices near the center of the tree as port vertices. According to the edge adjacent rules of LTQ_n, it is necessary to select an even vertex x to ensure that (x, x+2^n-1) exists. As a counterexample, (1, 1+2⁴) does not exist in LTQ₅. Then, we deal with the following theorem.

Theorem 8.

LTQ_n has dual-PRT^ds with the diameter 2n – 1 (respectively, 6 or 8) while n ≥ 3 and n ≠ 4 (respectively, n = 4) where d ∈ LTQ_n.

Proof of Theorem 8.

First, according to lemmas 3, 4, 5, this theorem holds when n = 3 or 4. Then we use the unified approach [10] to recursively construct dual-PRT^ds of LTQ_n+1 from dual-PRT^ds of LTQ_n while n ≥ 5. We consider the following two cases.

• Case 1. Destination node d is any even vertex in LTQ_n while n ≥ 5.

We prove by induction, based on n = 4. Since LTQ_n+1 consists of two LTQ_n, there are n-dimensional dual-PRT^ds in each LTQ_n. We adopt the dual-PRT^ds shown in Figure 8 as the induction base, by using the unified approach, we shoose vertex 0 of T₁ and vertex 6 of T₂ as port vertices. For LTQ_n+1, we connect a pair of port vertices (0, 0+2ⁿ) (respectively, (6, 6+2ⁿ))to link two T₁s (respectively, T₂s) in each subcube LTQ_n-1, then we have dual-PRT^ds T’₁ (respectively, T’₂) in LTQ_n. Because the length from vertex 0 (respectively, vertex 6) to the farthest leaf in T₁ (respectively, T₂) is 4, the diameter of recursively constructed dual-PRT^ds is both 2n – 1 while n ≥ 5.

• Case 2. Destination node d is any even odd in LTQ_n while n ≥ 5.

The proof is similar to that of Case 1. In this case, we adopt the dual-PRT^ds shown in Figure 9 as the induction base, and select vertex 6 of T₁ and vertex 0 of T₂ as port vertices. Other parts are the same as in case 1. □

For example, Figure 10 shows dual-PRT⁰s on LTQ₅, which can be used to illustrate the proof of Theorem 8. According to Theorem 8, there exist dual-PRT^ds in LTQ_n where d ∈ V(LTQ_n) while n ≥ 5. Then two trees can be used to configure the protection routing via Algorithm 1. By Definition 2, we have the following theorem.

Theorem 9.

LTQ_n is protectable where n ≥ 3.

Theorem 9 improves the known result that LTQ_n is protectable when n ≥ 4. As far as I know, the diameters of two CISTs of LTQ₄ are both 7 [10]. In each dimension of LTQ_n, the diameter of dual-PRT^ds is about the same as that of two CISTs.

4. Performance Evaluation

In this section, we present simulation results for evaluating the performance of the protection routings on Q_n and LTQ_n. The protection routings can be configured by dual-PRT^ds or two CISTs. For Q_n (respectively, LTQ_n), we use C programs to implement the routing algorithms by dual-PRT^ds in the previous section and two CISTs in [5] (respectively, [10]). To speed up the evaluation, we have carried out the simulation separately for each of Q_n and LTQ_n by using a 5.10 GHz Intel® Core™ i9-12900 CPU and 32GB RAM under the Linux operating system.

For each dimension 3, 4, 5, …, 9, we randomly generate 1,000,000 instances of vertex-list (s, d, f) with s ≠ d ≠ f for the two kinds of networks, where s, d and f are source, destination, failure node of traffic, respectively. The protection routing R_d is constructed by dual-PRT^ds (respectively, two CISTs) using Algorithm 1 (respectively, Tapolcai’s method [3]). We are interested in computing the path length from s to d under two scenarios: (i) no failure and (ii) a single node failure. For each instance, if no failure occurs, we compute the path length by tracking the primary links and PHNs alternately from s to d in R_d. For convenience, we call this specific routing is the default path P_s,d. Then, the length of P_s,d is defined by the number of links from s to d in R_d. Likewise, we are also concerned with computing path lengths in case of node failures in R_d. We assume that the appearance of node failure is uniformly distributed throughout the network. If failure node f = PNH(u) and f is contained in R_d, we calculate the path length of an alternate path P_s,u ∪ P_u,SNH(u) ∪ P_SNH(u),d. We then compute three statistical quantities related to path length: (a) average path length, (b) standard deviation of path length, and (c) maximum path length and the number of its occurrences. More descriptions and examples of the simulation process are available on the website [24].

Table 2 and Table 3 show some simulation results of the two scenarios (no failure and a single node failure) on Q_n and LTQ_n. Due to save space, four additional tables are available on the website [24]. In these tables, three quantities mentioned above are calculated by the usual way in statistics. They present simulation results for Q_n and LTQ_n when the routing is with two scenarios. Also, the ratio of invoking SNH under a single node failure is calculated (see the footer * in Table 2 and Table 3). It is obvious that the ratio tends to decrease gradually as the network’s dimension increases, and eventually, its effect on length becomes less pronounced. According to Lemma 1, there are no 2 CISTs in Q₃, so there are 2 missing values in Table 2. For the same reason, there are also 2 missing values in Table 3.

According to Table 2 and Table 3, we have Figure 11 and Figure 12. Obviously, the average path lengths of protection routing by dual-PRT^ds are smaller than the ones by two CISTs, no matter in which scenario.

5. Conclusions

In our previous research on using CISTs to construct the protection routing, we found that it is not easy to find k-CISTs on some graph classes even when k is equal to 2. We try to find trees that are less restrictive, but can still be used to configure the protection routing. Then we propose dual-PRT^ds in this paper. As mentioned earlier, there are no two CISTs on K_3,3, Q₃, and LTQ₃. But there are dual-PRT^ds on them, which can be used to construct the protection routing. These results improve on previous research. Also, as shown in the performance evaluation of protection routing through simulation results, the average path length of protection routing configured by dual-PRT^ds is shorter than that by two CISTs. For future research, it would be an interesting question to determine the existence of dual-PRT^ds in other graph classes.