Node-Disjoint Paths and Related Problems on Hierarchical Cubic Networks

Node-Disjoint Paths and Related Problems on Hierarchical Cubic Networks

Jung-Sheng Fu

Takming College, Taipei, TAIWAN Gen-Huey Chen

Department of Computer Science and Information Engineering, National Taiwan University,

and Dyi-Rong Duh

Department of Computer Science and Information Engineering, National Chi Nan University, Nantou, TAIWAN

Abstract

An n-dimensional hierarchical cubic network (denoted by HCN(n)) contains 2ⁿ n-dimensional hypercubes. The diameter of the HCN(n), which is equal to n+(n+1)/3+1, is about two-thirds the diameter of a comparable hypercube, although it uses about half as many links per node. In this paper, a maximal number of node-disjoint paths are constructed between every two distinct nodes of the HCN(n). Their maximal length is bounded above by n+n/3+4, which is nearly optimal. The (n+1)-wide diameter and n-fault diameter of the HCN(n) are shown to be n+

n/3+3 or n+n/3+4, which are about two-thirds those of a comparable hypercube. Our results reveal that the HCN(n) has a smaller wide diameter and fault diameter than a comparable hypercube.

Index Terms: Container, fault diameter, hierarchical cubic network, node-disjoint paths, wide diameter

Correspondence Address: Professor Gen-Huey Chen

Department of Computer Science and Information Engineering, National Taiwan University,

Taipei, TAIWAN 10764

e-mail: ghchen@csie.ntu.edu.tw

Tel: (886)-(2)-23625336 Ext. 427

Fax: (886)-(2)-23628167

1 Introduction

The hierarchical cubic network (HCN for short), which was proposed in [9] as an alternative to the hy- percube, consists of 2ⁿ basic components named clusters. Each cluster is an n-dimensional hypercube (n-cube for short). If each cluster is viewed as a single node, then the HCN appears as a 2ⁿ-node complete graph. The HCN can emulate a hypercube of the same size in constant time, but with only about half as many links per node. The average internode distances in the HCN under random and localized traffic patterns are the same as a comparable hypercube. When message generation rates are moderate, the average message transit delays in the HCN are slightly better than a comparable hypercube. This is a consequence of the fact that the HCN has a smaller maximal routing distance than a comparable hy- percube.

Previous works related to the HCN can be found in the literature [3], [9], [18], [19]. A shortest-path routing algorithm was presented in [3], [18], [19]. A broadcasting algorithm appeared in [3]. Some par- allel algorithms were designed in [9]. The diameter, which is about two-thirds the diameter of a com- parable hypercube, was computed in [18], [19]. A Hamiltonian cycle was constructed in [3], [18].

Suppose that A and B are two distinct nodes of an interconnection network (network for short) W.

An (A, B)-container in W is a set of disjoint paths between A and B. Throughout this paper, "disjoint paths" always means "internally node-disjoint paths". The width of a container is the number of paths it contains. The length of a container is the maximal length of paths it contains. A container is the best if its length is minimum.

The length of a best (A, B)-container is the x-wide distance between A and B, where x is the width of the container. The maximal x-wide distance in W is the x-wide diameter of W. The maximal diameter in W with at most y nodes removed is the y-fault diameter of W. When x=1 (y=0), the x-wide diameter (y-fault diameter) is identical with the diameter. Apparently, the x-wide diameter is the maximal length of best containers of width x, and the y-fault diameter is bounded above by the (y+1)-wide diameter.

The concepts of container, wide diameter, and fault diameter arose naturally from the study of routing (such as Rabin's Information Dispersal Algorithm (IDA) [15]), reliability, fault tolerance, and communication protocols (such as Byzantine algorithms) in parallel architectures and distributed com- puter networks (see [10]). Containers can be used to accelerate the transmission rate and to enhance the transmission reliability. In [15], the IDA was proposed on the hypercube which involved the construction

of disjoint paths. The IDA has numerous potential applications to secure and fault-tolerant storage and transmission of information.

On the other hand, the wide diameter and fault diameter are two generalizations of the diameter.

For all pairs of nodes, the diameter measures the maximal length of shortest paths, while the wide di- ameter measures the maximal length of best containers. In practical networks, node faults may happen.

The fault diameter, which was first introduced in [12], estimates the maximal increment of the diameter when there are node faults. It is both practically and theoretically important to compute the wide diameter and fault diameter. Previous works related to container, wide diameter, and fault diameter can be found in the literature [2]-[8], [10]-[12], [14], [16], [17].

According to Menger’s theorem [1], there are kw disjoint paths between any two nodes of W, where kw denotes the connectivity of W. The x-wide diameter and y-fault diameter in W are infinity whenever x>kw and y>kw−1, respectively. For theoretical interest, most of previous works computed for W con- tainers of width kw (e.g., [2], [3], [5]-[8], [11], [17]), kw-wide diameters (e.g., [7], [8], [11]), and (kw−1)-fault diameters (e.g., [3], [4], [7], [8], [11], [12], [16]).

We use HCN(n) to represent the HCN that contains 2ⁿ n-cubes. The connectivity and diameter of the HCN(n) are n+1 (see [3]) and n+(n+1)/3+1 (see [19]), respectively. In [3], containers of width n+1 were proposed in the HCN(n) whose lengths are 2n+6 at most. In this paper, we improve on the work of [3] by constructing new containers of width n+1 in the HCN(n) whose lengths are n+n/3+4 at most. The construction of new containers makes use of shortest paths of the HCN(n) and best containers of the hypercube. In addition, the (n+1)-wide diameter and n-fault diameter of the HCN(n) are shown to be n+

n/3+3 or n+n/3+4.

In the next section, we formally define the HCN(n) in graph-theoretic terms. The shortest-path routing algorithm of the HCN(n) and best containers of the hypercube are reviewed. New containers are proposed in Section 3, and their lengths are analyzed in Section 4. In Section 5, a lower bound on the n-fault diameter is suggested and the main result of this paper is summarized. Finally, this paper con- cludes with some remarks in Section 6.

2 Preliminaries

The following is a formal definition of the HCN(n) in graph-theoretic terms.

Definition 1. The node set of the HCN(n) is {(X, Y) | X and Y are binary sequences of length n}. Each node (X, Y) is adjacent to (1) (X, Y ^(k)) for all 1≤k≤n, where Y ^(k) differs from Y at the kth bit position, (2) (Y, X) if X≠Y, and (3) ( X ,Y ) if X=Y, where X andY are the bitwise complements of X and Y, respectively.

The cluster where a node (X, Y) resides is denoted by X, and its location in the cluster is denoted by Y. Links (1) are inside clusters, whereas links (2) and (3) connect two clusters. Links (2) and (3) are re- ferred to as nondiameter links and diameter links, respectively. The HCN(n) is regular of degree n+1.

Since the HCN(1) and the HCN(2) are easy, we assume n≥3 throughout this paper. Refer to Figure 1 for the HCN(3).

Suppose that I=(X, Y) and I'=(X', Y') are two distinct nodes of the HCN(n), where X≠X'. It was shown in [19] that any shortest path from I to I' contains (1) one nondiameter link (without diameter links) or (2) two nondiameter links (without diameter links) or (3) one diameter link. The shortest path for (1), denoted by P , can be expressed as follows. ₁^*

*

P : 1 (X, Y) ⇒* (X, X') → (X', X) ⇒* (X', Y'),

where → denotes a link and ⇒* denotes a shortest path (inside a cluster). The length of P , denoted by ₁^*

|P |, is equal to d₁^* H(Y, X')+dH(X, Y')+1, where dH() is the Hamming distance function.

Let P2 and P3 denote the paths for (2) and (3), respectively, which can be expressed as follows.

P2: (X, Y) ⇒* (X, Z) → (Z, X) ⇒* (Z, X') → (X', Z) ⇒* (X', Y');

P3: (X, Y) ⇒* (X, T) → (T, X) ⇒* (T, T) → (T ,T ) ⇒* (T , X') → (X',T ) ⇒* (X', Y'),

where Z∉{X, X'}, (X, T) → (T, X) ⇒* (T, T) degenerates to (X, X) if T=X, and (T ,T ) ⇒* (T , X') → (X', T ) degenerates to (X', X') if T= X' .

If Z belongs to a shortest path from Y to Y' in the n-cube, then P2 is a shortest path for (2), denoted by P . Clearly, |₂^* P |=d₂^* H(Y, Y')+dH(X, X')+2. On the other hand, P3 is a shortest path for (3), denoted by

*

P , if T=T3 ^* can minimize |P3|. T^* can be determined as described below.

We have |P3|=dH(Y, T)+dH(X, T)+dH(T , X')+dH(T , Y')+δ, where δ=1 if T=X= X' , δ=2 if T∈{X, X' } and X≠ X' , and δ=3 else. Define Qmin={T|dH(Y, T)+dH(X, T)+dH(T , X')+dH(T , Y') is minimum}.

Let X=x1x2…xn, Y=y1y2…yn, X'=x'1x'2…x'n, Y'=y'1y'2…y'n, and T=t1t2…tn. Then dH(Y, T)+dH(X, T)+dH(T , X')+dH( T , Y')=

∑

operation. We have T∈Qmin if and only if (y_i⊕t_i)+(x_i⊕t_i)+(t_i⊕ x'_i)+(t_i⊕y'_i) is minimum for all 1≤i≤n. According to [19], T^*=X if X∈Qmin, T^*= X' if X∉Qmin and X'∈Qmin, and T^* can be any element of Qmin else.We have |P |=d₃^* H(Y, T^*)+dH(X, T^*)+dH(T^*, X')+dH(T^*, Y')+δ. A shortest path from I to I' can be determined as the shortest one of P ,₁^* P , and₂^* P . ₃^*

In [19], bit patterns of X, Y, X', and Y' were examined in order to compute the diameter of the HCN(n). We use F1, F2, …, F8 to denote the sets of dimensions having the same bit patterns, where

F1={i | (xi, yi, x'i, y'i)=(0, 0, 0, 0) or (1, 1, 1, 1)}; F2={i | (xi, yi, x'i, y'i)=(0, 1, 1, 0) or (1, 0, 0, 1)};

F3={i | (xi, yi, x'i, y'i)=(0, 1, 0, 1) or (1, 0, 1, 0)}; F4={i | (xi, yi, x'i, y'i)=(0, 0, 1, 1) or (1, 1, 0, 0)};

F5={i | (xi, yi, x'i, y'i)=(0, 1, 0, 0) or (1, 0, 1, 1)}; F6={i | (xi, yi, x'i, y'i)=(0, 0, 0, 1) or (1, 1, 1, 0)};

F7={i | (xi, yi, x'i, y'i)=(0, 0, 1, 0) or (1, 1, 0, 1)}; F8={i | (xi, yi, x'i, y'i)=(0, 1, 1, 1) or (1, 0, 0, 0)}.

Define fk=|Fk|, where 1≤k≤8. Clearly, f1+f2+ … +f8=n. Fk and fk will be used to simplify the dis- cussion in Sections 3, 4, and 5. The following lemma expresses dH(Y, X'), dH(X, Y'), dH(Y, Y'), dH(X, X'), dH(X, Y), dH(X', Y'), dH( X , Y'), dH(Y , X'), and dH(Y, T)+dH(X, T)+dH(T , X')+dH(T , Y'), in terms of fk. They will be used very often in the rest of this paper.

Lemma 1. dH(Y, X')=f3+f4+f5+f7, dH(X, Y')=f3+f4+f6+f8, dH(Y, Y')=f2+f4+f5+f6, dH(X, X')=f2+f4+f7+f8, dH(X, Y)=f2+f3+f5+f8, dH(X', Y')=f2+f3+f6+f7, dH( X , Y')=f1+f2+f5+f7, dH(Y , X')=f1+f2+f6+f8, and dH(Y, T)+dH(X, T)+dH(T , X')+dH(T , Y')=2f1+2f2+2f3+f5+f6+f7+f8, where T∈Qmin.

Proof. We have dH(Y, X')=

∑

∑

2f2+2f3+f5+f6+f7+f8. …

Next, the best container of the hypercube is reviewed. Suppose that A=a1a2…an and B=b1b2…bn are two distinct nodes of an n-cube. A best (A, B)-container of width n was proposed by Saad and Schultz [17]. Let C=A⊕B. There are dH(A, B) 1 bits contained in C. Assume c=dH(A, B), and let ui and vj be the positions of the ith 1 bit and jth 0 bit, respectively, from the left in C, where 1≤i≤c, 1≤j≤n–c, 1≤ui≤n, and 1≤vj≤n. For example, if A=00001 and B=10011, then C=10010, (u1, u2)=(1, 4), and (v1, v2, v3)=(2, 3, 5).

Saad and Schultz's best (A, B)-container is shown in Figure 2, where both end nodes of a link labeled with ui (vj) differ at the uith (vjth) bit position. The upper c paths each of length c are obtained by cyclically shifting the vector (u1, u2, …, uc) left c–1 times. The other n–c paths each of length c+2 are obtained by prefixing and suffixing vj to the vector (u1, u2, …, uc). Saad and Schultz's best (A, B)-container has length dH(A, B) if dH(A, B)=n, and dH(A, B)+2 if dH(A, B)<n.

In the following, two properties of Saad and Schultz's containers are presented which will be used to show the disjoint property of the containers proposed in Section 3.

Lemma 2. Suppose that A, B, and H are three distinct nodes of an n-cube. There is a shortest path from A to H that has non-A common nodes with only one path, denoted by P, of Saad and Schultz's best (A, B)- container (the shortest path should not pass through B). Furthermore, |P|=3 if dH(A, B)=1.

Proof. Without loss of generality, suppose that A and H differ at the leftmost h bit positions, where h=

dH(A, H). Let D=a1a2…ah⊕b1b2…bh contain d 1 bits, where A=a1a2…an and B=b1b2…bn. The shortest path from A to H that corresponds to (v1, v2, …, vh–d, u1, u2, …, ud) meets our requirement, where ui and vj

have the same meanings as above. If dH(A, B)=1, then d=0 or 1. Since H≠B, we have h>d. Thus, a shortest path from A to H corresponds to (v1, v2, …, vh) if d=0 or (v1, v2, …, v_h−1, u1) if d=1. Both these paths

intersect the container path corresponding to (v1, u1, v1), i.e., |P|=3. …

Lemma 3. Suppose that A and B are two distinct nodes of an n-cube and dH(A, B)=c. The c shortest paths of Saad and Schultz's best (A, B)-container are disjoint with the n−c shortest paths of Saad and Schultz's best (A, B )-container.

Proof. Suppose C=A⊕B. The c shortest paths of Saad and Schultz's best (A, B)-container can be obtained by cyclically shifting the vector (u1, u2, …, uc) left c–1 times, where ui and vj have the same meanings as above. The n−c shortest paths of Saad and Schultz's best (A, B )-container can be obtained by cyclically shifting the vector (v1, v2, …, vn−c) left n−c–1 times. Hence they are disjoint. …

3 Containers of width n+1

Suppose that I=(X, Y) and I'=(X', Y') are two distinct nodes of the HCN(n). It is not easy to construct a best (I, I')-container because of diameter links and nondiameter links. In [3], an (I, I')-container was proposed whose length is not greater than n+5 if X=X', and 2n+6 if X≠X'. In this section, we improve on the work of [3] by constructing an (I, I')-container for X≠X' whose length is n+n/3+4 at most. The construction of

the (I, I')-container makes use of P ,₁^* P ,₂^* P , and Saad and Schultz's best containers. Throughout this ₃^* section, we assume that X≠X' and each (I, I')-container has width n+1.

The construction of a best (I, I')-container is closely related to the construction of the shortest path from I to I'. As described in Section 2, three shortest paths, i.e., P ,₁^* P , and₂^* P , obeying some con-₃^* straints need to be generated, in order to obtain the shortest path from I to I'. It appears impossible to construct a best (I, I')-container by a single construction method. The (I, I')-container to be proposed is obtained using a main construction method accompanied by six auxiliary construction methods. Actually, these construction methods correspond to P ,₁^* P , and₂^* P . The worst-case length of the (I, I')-container ₃^* is nearly optimal.

We use (A), (B), (C), (D), (E), and (F) to denote the six auxiliary construction methods. They are applicable under some conditions. In fact, the main construction method corresponds to P . (A) and (B) ₂^* correspond to P₁^*and P , respectively. On the other hand, (C) is the combination of (A) and (B), (D) is ₃^* the combination of the main construction method and (A), and (E) is the combination of the main con- struction method and (B). (F) deals with a special situation for n=3.

3.1 Main construction method

Suppose that Y≠Y' and Q1, Q2, …, Qn are the n paths of Saad and Schultz's best (Y, Y')-container. Without loss of generality, we assume |Q1|≥|Q2|≥ … ≥|Qn|. If there exists Wi ∈Qi−{X, X', Y, Y'}, then let Ri be the path P2 with Z=Wi. Refer to Figure 3. The construction of Ri is in accordance with Qi. That is, the com- bination of (X, Y) ⇒* (X, Wi) and (X', Wi) ⇒* (X', Y') is the same as Qi, disregarding X and X'. We have

|Ri|=dH(X, X')+dH(Y, Y')+2 if i>n−dH(Y, Y'), and |Ri|=dH(X, X')+dH(Y, Y')+4 if i≤n−dH(Y, Y'). Ri and Rj are disjoint if i≠j. There are at least n−2 paths Qi such that Qi−{X, X', Y, Y'}≠φ. They are assumed to be Q1, Q2, …, Qn−2. From each of these paths we choose a Wi ∈Qi−{X, X', Y, Y'}. Further, we assume Qn−1−{X, X', Y, Y'}≠φ if Qn−1−{X, X', Y, Y'}≠φ or Qn−{X, X', Y, Y'}≠φ. So, when Qn−{X, X', Y, Y'}≠φ , R1, R2, …, Rn

can be obtained.

On the other hand, if Y=Y', then let Si be the path (X, Y) → (X, Y ⁽ⁱ⁾) → (Y ⁽ⁱ⁾, X) ⇒* (Y ⁽ⁱ⁾, X') → (X', Y ⁽ⁱ⁾) → (X', Y'), where Y ⁽ⁱ⁾∉{X, X'}. We have |Si|=dH(X, X')+4. Si and Sj are disjoint if i≠j. There are at least n−2 nodes Y ⁽ⁱ⁾∉{X, X'} in an n-cube, and they are assumed to be Y ⁽¹⁾, Y ⁽²⁾, …, Y ⁽ⁿ⁻²⁾. If Y ⁽ⁿ⁻¹⁾∉{X, X'} or Y ⁽ⁿ⁾∉{X, X'}, we assume Y ⁽ⁿ⁻¹⁾∉{X, X'}. So, when Y ⁽ⁿ⁾∉{X, X'}, S1, S2, …, Sn can be obtained.

We use P ,₁^M P , …,₂^M P_n^M₊₁to represent the n+1 disjoint paths that are obtained by the main con- struction method. Theycan be constructed as follows. If Y≠Y', then let P =R_i^M i for all 1≤i≤n−2. If Y=Y', then let P =S_i^M i for all 1≤i≤n−2. The construction of P ,_n^M₋₁ P , and_n^M P_n^M₊₁ depends on whether X≠Y and X'≠Y' or not, as discussed below.

Case 1. X≠Y and X'≠Y'. The construction further depends on whether X'≠Y, X≠Y', and Y≠Y' or not.

Case 1.1. X'≠Y, X≠Y', and Y≠Y'. If {Qn−1, Qn}={Y → X → Y', Y → X' → Y'}, then let P_n^M₋₁= (X, Y) → (X, X') → (X', X) → (X', Y'), P_n^M= (X, Y) → (X, X) → (X, Y') → (Y', X) ⇒* (Y', X') → (X', Y'), and P_n^M₊₁= (X, Y)

→ (Y, X) ⇒* (Y, X') → (X', Y) → (X', X') → (X', Y'). If {Qn−1, Qn}≠{Y → X → Y', Y → X' → Y'}, then let

M

Pn = (X, Y) → (Y, X) ⇒* (Y, X') → (X', Y) ⇒* (X', Y') and P_n^M₊₁= (X, Y) ⇒* (X, Y') → (Y', X) ⇒* (Y', X')

→ (X', Y'), where (X', Y) ⇒* (X', Y') and (X, Y) ⇒* (X, Y') are the same as Qn. P_n^M₋₁ can be determined as follows.

If Rn−1 exists, then let P_n^M₋₁=Rn−1. If Rn−1 does not exist, then dH(Y, Y')=1, |Qn−1|=3, and either Qn−1= Y → X → X' → Y' or Q_n−1= Y → X'→ X → Y'. P_n^M₋₁ can be obtained in accordance with Qn−1 by letting

M

−1

Pn = (X, Y) → (X, X) → (X, X') → (X', X) → (X', X') → (X', Y') if Q_n−1= Y → X → X' → Y', and (X, Y) → (X, X') → (X', X) → (X', Y') if Qn−1= Y → X'→ X → Y'.

We have |P_n^M₋₁|≤dH(X, X')+dH(Y, Y')+2 if dH(Y, Y')>1, and |P_n^M₋₁|≤dH(X, X')+dH(Y, Y')+4 if dH(Y, Y')=

1. Both|P_n^M| and |P_n^M₊₁| are at most dH(X, X')+dH(Y, Y')+2.

Case 1.2. X'≠Y, X≠Y', and Y=Y'. We let P_n^M₊₁= (X, Y) → (Y, X) ⇒* (Y, X') → (X', Y) (=(X', Y')). The construction of P_n^M and P_n^M₋₁ depends on whether {Y ⁽ⁿ⁻¹⁾, Y ⁽ⁿ⁾}∩{X, X'} is empty or not. Recall that if there is one more adjacent node of Y that does not belong to {X, X'}, it is Y ⁽ⁿ⁻¹⁾. If {Y ⁽ⁿ⁻¹⁾, Y ⁽ⁿ⁾}∩{X, X'}

is empty, then let P_n^M₋₁=S_n−1 and P_n^M=Sn. If Y ⁽ⁿ⁻¹⁾∉{X, X'} and Y ⁽ⁿ⁾=X, then let P_n^M₋₁=S_n−1 and P_n^M= (X, Y)

→ (X, X) ⇒* (X, X') → (X', X) → (X', Y'), where (X, X) ⇒* (X, X') does not contain (X, Y ⁽¹⁾), (X, Y ⁽²⁾), …, (X, Y ⁽ⁿ⁻¹⁾). If Y ⁽ⁿ⁻¹⁾∉{X, X'} and Y ⁽ⁿ⁾=X', then let P_n^M₋₁=S_n−1 and P_n^M= (X, Y) → (X, X') → (X', X) ⇒* (X', X') → (X', Y').

If Y ⁽ⁿ⁻¹⁾=X and Y ⁽ⁿ⁾=X', then dH(X, X')=2 and there exists Z≠Y so that dH(X, Z)=1 and dH(X', Z)=1.

Let P_n^M₋₁= (X, Y) → (X, X) → (X, Z) → (Z, X) → (Z, Z) → (Z, X') → (X', Z) → (X', X') → (X', Y) and P_n^M= (X, Y) → (X, X') → (X', X) → (X', Y). The discussion is similar if Y ⁽ⁿ⁻¹⁾= X' and Y ⁽ⁿ⁾=X.

We have |P_n^M₋₁|,|P_n^M|, and |P_n^M₊₁| at most max{8, dH(X, X')+4}.

Case 1.3. X'≠Y and X=Y' (Y≠Y' is implied because X≠Y). Wn−1 can be determined and we let P_n^M₋₁=Rn−1. By Lemma 2, there is a shortest path from Y to X' that intersects with Qr for some 1≤r≤n, but does not intersect with Qj for all 1≤j≤n and j≠r.

If Rn does not exist, then either Qn= Y → X' → Y' or Qn= Y → Y'. If Qn= Y → X' → Y', then let P_n^M= (X, Y) → (Y, X) ⇒* (Y, X') → (X', Y) → (X', X') → (X', Y') and P_n^M₊₁= (X, Y) → (X, X') → (X', X) (=(X', Y')).

If Qn= Y → Y', then let P_n^M= (X, Y) → (Y, X) ⇒* (Y, X') → (X', Y) → (X', Y') and P_n^M₊₁= (X, Y) ⇒* (X, X')

→ (X', X) (=(X', Y')), where (X, Y) ⇒* (X, X') is the same as the shortest path from Y to X' above. Since

M +1

Pn and P_r^M conflict, P_r^M is changed as follows. By Lemma 2, we have |Qr|=3. Without loss of generality, we assumeQr= Y → Y ^(s) → Y' ^(s) → Y', where 1≤s≤n. P_r^M is changed as (X, Y) → (X, Y') → (X, Y' ^(s)) → (Y' ^(s), X) ⇒* (Y' ^(s), X') → (X', Y' ^(s)) → (X', Y') whose length is dH(X, X')+5=dH(X, X')+dH(Y, Y')+4.

If Rn exists, then letP_n^M=Rn. The construction of P_n^M₊₁ is the same as above (Qn= Y → Y'), and P_r^M is changed as (X, Y) → (Y, X) ⇒* (Y, X') → (X', Y) ⇒ (X', Y'), where ⇒ denotes a path (inside a cluster) and (X', Y) ⇒ (X', Y') is the same asQr.

We have |P_n^M₋₁| and |P_n^M| at most dH(X, X')+dH(Y, Y')+4, and |P_n^M₊₁|=dH(Y, X')+1≤dH(Y, X)+dH(X, X')+

1=dH(Y, Y')+dH(X, X')+1.

Case 1.4. X'=Y and X≠Y' (Y≠Y' is implied because X'≠Y'). Similar to Case 1.3.

Case 1.5. X'=Y and X=Y' (Y≠Y' is implied because X≠Y). W_n−1 can be determined and we let P_n^M₋₁=R_n−1. Let P = (X, Y) → (Y, X) (=(X', Y')). If d_n^M₊₁ H(Y, Y')>1, then Wn can be determined and we let P_n^M=Rn. If dH(Y, Y')=1, then let P_n^M= ((X, Y)=) (Y', Y) → (Y', Y') → (Y' ,Y' ) → (Y' ,Y ) → (Y ,Y' ) → (Y , Y ) → (Y, Y) → (Y, Y') (=(X', Y')). We have |P_n^M₋₁|, |P_n^M|, and |P_n^M₊₁| at most max{7, dH(X, X')+dH(Y, Y')+4}.

Case 2. X=Y and X'≠Y'. P_n^M₋₁, P_n^M, and P_n^M₊₁ can be obtained according to the value of dH(Y, Y').

Case 2.1. dH(Y, Y')=0. We have Y ⁽ⁿ⁻¹⁾∉{X, X'}. Let P_n^M₋₁=Sn−1. If dH(Y, X')=1, then let P_n^M= (X, Y) → (X, X') → (X', X) (=(X', Y')) and P_n^M₊₁= ((X, Y)=) (X, X) → ( X , X ) → ( X , X' ) → ( X' , X ) → ( X' , X' ) → (X', X') → (X', X) (=(X', Y')).

If dH(Y, X')>1, then Y ⁽ⁿ⁾∉{X, X'} and let P_n^M=Sn. Also let P_n^M₊₁= ((X, Y)=) (X, X) → (X, X^(r)) ⇒* (X, X') → (X', X) (=(X', Y')), where dH(X, X')=1+dH(X^(r), X') for some 1≤r≤n. Since P_n^M₊₁ and P_r^M conflict, P_r^M is changed as ((X, Y)=) (X, X) → ( X , X ) → ( X , X^(r)) → (X^(r), X ) → (X^(r),X^(r)) → (X^(r), X^(r)) ⇒*

(X^(r), X') →(X', X^(r)) → (X', X) (=(X', Y')) if dH(X, X')<n, and ((X, Y)=) (X, X) → ( X , X ) (=(X', X')) ⇒*

(X', X^(r)) → (X', X) (=(X', Y')) if dH(X, X')=n. The new P_r^M has length not greater than n+5.

We have |P_n^M₋₁| and |P_n^M| at most dH(X, X')+4, and |P_n^M₊₁|=max{6, dH(X, X')+1}.

Case 2.2. dH(Y, Y')=1. We have |Q_n−1|=3. Without loss of generality, suppose Q_n−1= Y → U → V → Y', where U≠X' and V≠X'. If X ≠X' andY'≠X', then let P_n^M₋₁= (X, Y) → (X, Y') → (X, V) → (V, X) ⇒* (V, X')

→ (X', V) → (X', Y') andP_n^M= (X, Y ) → (X, U) → (U, X) ⇒* (U, X') → (X', U) → (X', Y) → (X', Y').

Besides, let P_n^M₊₁ be the shorter one of the following two paths: ((X, Y)=) (X, X) → ( X , X ) → ( X ,Y' ) → (Y' , X ) → (Y' ,Y' ) → (Y', Y') ⇒* (Y', X') → (X', Y') and ((X, Y)=) (X, X) → ( X , X ) ⇒* ( X ,Y') → (Y',

X ) ⇒* (Y', X') → (X', Y'), where X≠Y' because dH(X, Y')=dH(Y, Y')=1. The former has length dH(Y', X')+

6≤dH(Y', X)+dH(X, X')+6=dH(X, X')+dH(Y, Y')+6, and the latter has length dH( X , Y')+dH( X , X')+3.

If X =X' orY' =X', then let P_n^M₋₁= (X, Y ) → (X, Y' ) → (Y', X) ⇒* (Y', X') → (X', Y') and P_n^M= (X, Y )

→ (X, U) → (U, X) ⇒* (U, X') → (X', U) → (X', Y) → (X', Y'). Besides, let P_n^M₊₁= ((X, Y)=) (X, X) → ( X , X ) (=(X',Y )) ⇒* (X', V) → (X', Y') if X =X' , and ((X, Y)=) (X, X) → ( X , X ) → ( X ,Y' ) → (Y' , X ) (=(X',Y )) ⇒* (X', V) → (X', Y') if Y' =X'. P_n^M₊₁ is disjoint with P ,₁^M P , …,₂^M P_n^M provided (X',Y ) ⇒* (X', V) is disjoint with Q1, Q2, …, Qn−2 and does not contain Y and U. They are true because dH(Y , V)=n−2, dH(Y , U)=n−1, dH(Y , Y ⁽ⁱ⁾)=n−1, and dH(Y , Y' ⁽ⁱ⁾)≥dH(Y , Y')−1=n−2 for all 1≤i≤n.

We have |P_n^M₋₁| and |P_n^M| at most dH(X, X')+dH(Y, Y')+4, and |P_n^M₊₁| at most max{n+2, min{dH(X, X')+dH(Y, Y')+6, dH( X , Y')+dH(Y , X')+3}}, where dH(Y , X')=dH( X , X').

Case 2.3. dH(Y, Y')=2. Wn−1 can be determined and we let P_n^M₋₁=Rn−1. We have |Qn|=2. If Qn= Y → X' → Y', then let P_n^M= (X, Y) → (X, X') → (X, Y') → (Y', X) ⇒* (Y', X') → (X', Y') and P_n^M₊₁= ((X, Y)=) (X, X) → ( X , X ) → ( X , X' ) → ( X' , X ) → ( X' , X' ) → (X', X') → (X', Y').

Otherwise (Qn≠ Y → X' → Y'), Wn can be determined and we letP_n^M=Rn. If X ≠X' andY'≠X', then let P_n^M₊₁be the shorter one of the following two paths: ((X, Y)=) (X, X) → ( X , X ) ⇒* ( X ,Y' ) → (Y' , X )

⇒* (Y' ,Y' ) → (Y', Y') ⇒* (Y', X') → (X', Y') and ((X, Y)=) (X, X) → ( X , X ) ⇒* ( X , Y') → (Y', X ) ⇒*

(Y', X') → (X', Y'), where X ≠Y' because dH(X, Y')=dH(Y, Y')=2 and n≥3. The former has length dH( X , Y' )+dH( X ,Y' )+dH(Y', X')+4=dH(Y', X')+8≤dH(Y', X)+dH(X, X')+8=dH(X, X')+dH(Y, Y')+8 (dH( X ,Y' )=2 because X=Y and dH(Y, Y')=2), and the latter has length dH( X , Y')+dH( X , X')+3.

If X =X' orY' =X', then by Lemma 2 there is a shortest path fromY to Y' that intersects with Qr for some 1≤r≤n, but does not intersect with Qj for all 1≤j≤n and j≠r. Let P_n^M₊₁= ((X, Y)=) (X, X) → ( X , X ) (=(X',Y )) ⇒* (X', Y') if X =X', and ((X, Y)=) (X, X) → ( X , X ) (=(Y ,Y )) ⇒* (Y ,Y' ) → (Y' ,Y ) (=(X',Y )) ⇒* (X', Y') ifY' =X', where (X',Y ) ⇒* (X', Y') is the same as the shortest path fromY to Y' above. P_r^M is changed as (X, Y) ⇒ (X, Y') → (Y', X) ⇒* (Y', X') → (Y', X'), where (X, Y) ⇒ (X, Y') is the same as Qr.

We have |P_n^M₋₁|=|P_n^M|=dH(X, X')+dH(Y, Y')+2, and |P_n^M₊₁|≤max{n+2, min{dH(X, X')+dH(Y, Y')+8, dH( X , Y')+dH(Y , X')+3}, where dH(Y , X')=dH( X , X').

Case 2.4. dH(Y, Y')≥3. Wn−1 can be determined and we let P_n^M₋₁=Rn−1. Suppose, without loss of gener- ality, that Qn= Y → U ⇒* Y' does not contain X' andU ≠X'. Then Wn≠U (Wn ∈Qn−{X, Y, X', Y'}) can be determined and we let P_n^M=Rn. If X =X', then P_n^M₊₁ can be obtained all the same as the situation of X =X' in Case 2.3.

If X ≠X', then let P_n^M₊₁ be the shorter one of the following two paths: ((X, Y)=) (X, X) → ( X , X ) → ( X ,U ) → (U , X ) → (U ,U ) → (U, U) ⇒* (U, Y') →(Y', U) ⇒* (Y', X') → (X', Y') and ((X, Y)=) (X, X) → ( X , X ) ⇒* ( X , Y') → (Y', X ) ⇒* (Y', X') → (X', Y'), where ( X , X ) ⇒* ( X , Y') → (Y', X ) ⇒*

(Y', X') is replaced with ( X , X ) ⇒* ( X , X') if X =Y'. The former has length dH(U, Y')+dH(U, X')+7≤

dH(Y, Y')+dH(X, X')+7 (because dH(Y, Y')=1+dH(U, Y') and dH(X, X')=dH(U, X')±1), and the latter has

length dH( X , Y')+dH( X , X')+3. We have |P_n^M₋₁|=|P_n^M|=dH(X, X')+dH(Y, Y')+2, and |P_n^M₊₁|=min{dH(X, X')+

dH(Y, Y')+7, dH( X , Y')+dH(Y , X')+3}, where dH(Y , X')=dH( X , X').

Case 3. X≠Y and X'=Y'. Similar to Case 2.

Case 4. X=Y and X'=Y'. Since X≠X', we have Y≠Y'. Wn−1 can be determined and we let P_n^M₋₁=Rn−1. If dH(Y, Y')=1, then let P_n^M= ((X, Y)=) (X, X) → (X, X') → (X', X) → (X', X') (=(X', Y')). If dH(Y, Y')≥2, then Wn can be determined and we let P_n^M=Rn. Also, let P_n^M₊₁= ((X, Y)=) (X, X) → ( X , X ) ⇒* ( X , X' ) → ( X' , X )

⇒* ( X' , X' ) → (X', X') (=(X', Y')) if X≠X', and (X, X) → ( X , X ) (=(X', Y')) if X =X'. We have |P_n^M₋₁|,

|P_n^M|, and |P_n^M₊₁| at most dH(X, X')+dH(Y, Y')+4.

3.2 Construction method (A)

The construction method (A) can be applied when f2≥2. According to Lemma 1, we have dH(X, Y)≥2, dH(X', Y')≥2, and dH(Y, Y')≥2. We use P ,₁^A P , …,₂^A P_n^A₊₁to denote the resulting n+1 disjoint paths. Let Pi, j= (X, Y) → (X, Y⁽ⁱ⁾) → (Y⁽ⁱ⁾, X) ⇒* (Y⁽ⁱ⁾, Y'^(j)) → (Y'^(j), Y⁽ⁱ⁾) ⇒* (Y'^(j), X') → (X', Y'^(j)) → (X', Y') (refer to Figure 4), where 1≤i≤n, 1≤j≤n, and {Y⁽ⁱ⁾, Y'^(j)}∩{X, X', Y, Y'} is empty. If Y⁽ⁱ⁾=Y'^(j), then (Y⁽ⁱ⁾, X) ⇒*

(Y⁽ⁱ⁾, Y'^(j)) → (Y'^(j), Y⁽ⁱ⁾) ⇒* (Y'^(j), X') is replaced with (Y⁽ⁱ⁾, X) ⇒* (Y'^(j), X').

1 1, j

Pi and

2 2, j

Pi are disjoint if {Y ,⁽ⁱ¹⁾ Y^(j1)}∩{Y⁽ⁱ²⁾,Y^(j2)} is empty. We have |Pi,j|=dH(X, Y'^(j))+dH(X', Y⁽ⁱ⁾)+5≤dH(X, Y')+dH(X', Y)+7 if Y⁽ⁱ⁾≠Y'^(j), and dH(X, X')+4≤dH(X, Y'^(j))+dH(X', Y⁽ⁱ⁾)+4<dH(X, Y')+dH(X', Y)+7 if Y⁽ⁱ⁾=Y'^(j). When i∈F4 and j∈F4, we have |Pi,j|=dH(X, Y')+dH(X', Y)+3 because xj≠y'j implies dH(X, Y'^(j))=dH(X, Y')−1 and x'i≠yi

implies dH(X', Y⁽ⁱ⁾)=dH(X', Y)−1.

A

P ,1 P , …,₂^A P_n^A can be obtained, depending on whether dH(X, Y')≠1 and dH(X', Y)≠1 or not.If dH(X, Y')≠1 and dH(X', Y)≠1, then {Y⁽ⁱ⁾, Y'^(j)}∩{X, X', Y, Y'} is empty for all 1≤i≤n and 1≤j≤n. For all 1≤k≤n, we let P =P_k^A k,u if Y^(k)=Y'^(u) for some 1≤u≤n, and P =P_k^A k,k otherwise. If dH(X, Y')≠1 and dH(X', Y)=1, then X'=Y^(r) for some 1≤r≤n. We let P = (X, Y) → (X, X') → (X', X) ⇒* (X', Y'_r^{A (r)}) → (X', Y'), and for all k∈{1, 2, …, n}−{r}, let P =P_k^A k,u if Y^(k)=Y'^(u) for some 1≤u≤n, and P =P_k^A k,k otherwise. If dH(X, Y')=1 and dH(X', Y)≠1, the discussion is similar. If dH(X, Y')=1 and dH(X', Y)=1, then X'=Y^(s) and X=Y'^(t) for some 1≤s≤n and 1≤t≤n. We let P = (X, Y) → (X, X') → (X', X) → (X', Y'),_s^A P =P_t^A t,s if t≠s, and for all k∈{1, 2, …,

n}−{s, t}, P =P_k^A k,u if Y^(k)=Y'^(u) for some 1≤u≤n, and P =P_k^A k,k otherwise. These paths have lengths at most dH(X, Y')+dH(X', Y)+7.

A +1

Pn can be obtained, depending on whether X'≠Y and X≠Y' or not. If X'≠Y and X≠Y', then let P = _n^A₊₁ (X, Y) → (Y, X) ⇒* (Y, Y') → (Y', Y) ⇒* (Y', X') → (X', Y'). If X'≠Y and X=Y', then let P = (X, Y) → (X, _n^A₊₁ Y^(q)) ⇒* (X, X') → (X', X) for some 1≤q≤n, which conflicts with P ._q^A P_q^A is changed as (X, Y) → (Y, X)

⇒* (Y, Y'^(q)) → (Y'^(q), Y) ⇒* (Y'^(q), X') → (X', Y'^(q)) → (X', Y') whose length is at most dH(X, Y')+dH(Y, X')+5. The discussion is similar if X'=Y and X≠Y'. If X'=Y and X=Y', then let P = (X, X') → (X', X). We _n^A₊₁ have |P |≤d_n^A₊₁ H(X, Y')+dH(X', Y)+5.

3.3 Construction method (B)

The construction method (B) can be applied when f4≥2. By P ,₁^B P , …,₂^B P_n^B₊₁ we denote the resulting n+1 disjoint paths. First we determine M as follows: M=Y if X=Y, M=Y' if X'=Y', and M is an arbitrary element of Qmin else. Suppose X=x1x2…xn, Y=y1y2…yn, X'=x'1x'2…x'n, Y'=y'1y'2…y'n, and M=m1m2…mn. When X=Y, we have(y_i⊕m_i)+(x_i⊕m_i)+(m_i⊕x'_i)+(m_i⊕y'_i)≤2 if mi=yi, and ≥2 if mi≠yi, where 1≤i≤n.

Hence M=Y∈Qmin. Similarly, when X'=Y', we haveM=Y'∈Qmin.

For all 1≤i≤n, let P_i^Bbe the path P3 with T=M⁽ⁱ⁾. Refer to Figure 5. As a consequence of Saad and Schultz's best (Y, M)-container (refer to Figure 2), there are n disjoint shortest paths from (X, Y) to (X, M⁽¹⁾), (X, M⁽²⁾), …, (X, M⁽ⁿ⁾) (and from (X', M⁽¹⁾), (X', M⁽²⁾ ), …, (X',M⁽ⁿ⁾) to (X', Y')), respectively. We have

|P | ≤ d_i^B H(Y, M⁽ⁱ⁾)+dH(X, M⁽ⁱ⁾)+dH(M , X')+d⁽ⁱ⁾ H(M , Y')+3 ⁽ⁱ⁾ = dH(Y, M)+dH(X, M)+dH( M , X')+dH( M , Y')+3+∆, where ∆=0 if i∈F1∪F2∪F3, ∆=4 if i∈F4, and ∆=2 if i∈F5∪F6∪F7∪F8.

B +1

Pn whose length is at most dH(Y, M)+dH(X, M)+dH( M , X')+dH( M , Y')+5 can be obtained, de- pending on whether X≠Y and X'≠Y' or not.

Case 1. X≠Y and X'≠Y'. The construction further depends on whether Y∉{M⁽¹⁾, M⁽²⁾, …, M⁽ⁿ⁾} and Y'∉ {M ,⁽¹⁾ M⁽²⁾ , …, M⁽ⁿ⁾} or not.