Node-disjoint paths in incomplete WK-recursive networks

Node-disjoint paths in incomplete WK-recursive

networks

Ming-Yang Su

_{, Hui-Ling Huang}

_{, Gen-Huey Chen}

_,

Dyi-Rong Duh

a_{Department of Electronic Engineering, St. John's and St. Mary's Institute of Technology,}

Taipei, Taiwan, ROC

b_{Department of Information Management, Southern Taiwan University of Technology,}

Tainan, Taiwan, ROC

c_{Department of Computer Science and Information Engineering, National Taiwan University,}

Taipei 10764, Taiwan, ROC

d_{Department of Computer Science and Information Engineering, National Chi Nan University,}

Nantou, Taiwan, ROC

Received 23 July 1996; received in revised form 11 May 1998; accepted 19 April 2000

Abstract

The incomplete WK-recursive networks have been recently proposed to relieve the re-striction on the sizes of the WK-recursive networks. In this paper, a maximal set of node-disjoint paths is constructed between arbitrary two nodes of an incomplete WK-recursive network. The e€ectiveness of the constructed paths is veri®ed by both theoretic analysis and extensive experiments. A tight upper bound on the maximal length is suggested. On the other hand, experimental results show that for arbitrary two nodes, the expected maximal length is not greater than twice their distance and about equal to the diameter. When the two nodes are the farthest pair, the maximal length is not greater than twice the diameter and the expected maximal length is not greater than 1.5 times the diameter. Ó 2000 Elsevier Science B.V. All rights reserved.

Keywords: Graph-theoretic interconnection network; Incomplete WK-recursive network; Node-disjoint paths; WK-recursive network

*_{Corresponding author. Tel.: +886-2-23625336; fax: +886-2-23628167.}

E-mail address: ghchen@csie.ntu.edu.tw (G.-H. Chen).

0167-8191/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 8 1 9 1 ( 0 0 ) 0 0 0 6 5 - X

1. Introduction

The WK-recursive networks [23] own two attractive topological properties: ex-pansibility and equal degree. A network is expansible if no changes to node con-®guration and link connection are necessary when it is expanded, and of equal degree if its nodes have the same degree no matter what its size is. A network with these two properties will gain the advantages of easy implementation and low cost when it is manufactured. A VLSI implementation of a 16-node WK-recursive network has been realized at the Hybrid Computing Research Center [23]. This prototype net-work was further extended to 64 nodes later [24]. Recently two variants of the WK-recursive networks have been proposed in [5,6].

Although the WK-recursive networks own many favorable properties (see [1,3,4,7±9,11,23,24]), there is a rigorous restriction on their sizes. As will become clear in Section 2, the number of nodes contained in a WK-recursive network must satisfy dt_{, where d > 1 is the size of the basic building block and t P 1 is the level of}

expansion. Thus, as d ˆ 4, extra 3  47_{ˆ 49152 nodes are required to expand from a}

7-level WK-recursive network to an 8-level one. Almost all announced networks have su€ered from the same restriction. In order to relieve this restriction, some incomplete networks have been proposed recently. Among them, incomplete hy-percubes [12], incomplete star networks [14,17], clustered-star graphs [13], incom-plete rotator graphs [16], and incomincom-plete WK-recursive networks [19] are some examples. Previously several results on the incomplete WK-recursive networks were obtained; topological properties were investigated in [21], a shortest-path routing algorithm appeared in [22], and a broadcasting algorithm was proposed in [20].

Given a network, it is both theoretically interesting and practically important to ®nd node-disjoint paths (disjoint paths for short) between any two of its nodes. With disjoint paths, transmission rate can be accelerated and transmission reliability can be enhanced. In the past, a maximal set of disjoint paths was constructed for many (complete) networks, e.g., hypercubes [18], star graphs [2], and WK-recursive works [3]. However, the same problem remained unsolved for all incomplete net-works but the clustered-star graphs (see Ref. [10]). In this paper, a maximal set of disjoint paths is constructed for the incomplete WK-recursive networks.

In the next section, the incomplete WK-recursive networks are ®rst reviewed. A prerequisite step for constructing disjoint paths is presented in Section 3. By its aid a maximal set of disjoint paths is constructed in Section 4. Moreover, a tight upper bound on their maximal length is suggested. In Section 5, extensive experiments are further made to verify their e€ectiveness. Finally, we conclude this paper with some remarks in Section 6.

2. Incomplete WK-recursive networks

The WK-recursive networks can be built incrementally with basic building blocks. Any complete graph can serve as a basic building block. Throughout this paper we use graph and network interchangeably. Let K…d; t† denote a WK-recursive network

of level t whose basic building block is a d-node complete graph, where d > 1 and t P 1. K…d; 1†, which is the basic building block, is the d-node complete graph, and K…d; t† for t P 2 is a d-supernode complete graph, where each supernode is a K…d; t ÿ 1†. Each node of a K…d; t† is assigned with a unique identi®er which consists of a d-ary sequence of length t. The following de®nition is due to Chen and Duh [1]. De®nition 2.1. The node set of a K…d; t† is denoted by fatÿ1atÿ2


a1a0j

ai2 f0; 1; . . . ; d ÿ 1g for 0 6 i 6 t ÿ 1g. Node adjacency is de®ned as follows:

atÿ1atÿ2


a1a0is adjacent to (1) atÿ1atÿ2


a1b, where 0 6 b 6 d ÿ 1 and b 6ˆ a0, and

(2) atÿ1atÿ2


aj‡1ajÿ1ajjif aj6ˆ ajÿ1and ajÿ1ˆ ajÿ2ˆ


ˆ a0for some 1 6 j 6 t ÿ 1,

where ajjrepresents j consecutive ajs. The links of (1) are called substituting links, and

are labeled 0. The link of (2), if existing, is called j-flipping link (or simply flipping link), and is labeled j. Besides, if atÿ1ˆ atÿ2 ˆ


ˆ a0, there is a link, called open

link, incident to node atÿ1atÿ2


a1a0. The open link, which is labeled t, is reserved

for further expansion.

The structures of K…4; 1† and K…4; 3† are illustrated in Fig. 1. The links within basic building blocks are substituting links, and those connecting two embedded K…d; j†s are j-¯ipping links. For example, the link between nodes 311 and 133 is a 2-¯ipping link, and the other links incident to node 311 are all substituting links. The open links are incident to nodes 000, 111, 222, and 333.

De®nition 2.2. De®ne ctÿ1ctÿ2


cm
K…d; m† to be the subgraph of a K…d; t† induced

by fctÿ1ctÿ2


cmamÿ1


a1a0jaj2 f0; 1; . . . ; d ÿ 1g for 0 6 j 6 m ÿ 1g, where

1 6 m 6 t ÿ 1 and ctÿ1; ctÿ2; . . . ; cmare all integers from f0; 1; . . . ; d ÿ 1g.

For example, refer to Fig. 1(b), where 31
K…4; 1† is the subgraph of K…4; 3† in-duced by {310, 311, 312, 313}.

De®nition 2.3. Node atÿ1atÿ2


a1a0 is a k-frontier if akÿ1ˆ akÿ2ˆ


ˆ a1ˆ a0,

where 1 6 k 6 t.

By de®nition a k-frontier is automatically an l-frontier for 1 6 l 6 k ÿ 1. Both end nodes of a k-¯ipping link are k-frontiers. For 1 6 m 6 t ÿ 1, an embedded K…d; m† contains one …m ‡ 1†-frontier and d ÿ 1 m-frontiers. These d frontiers are 2m_{ÿ 1}

distant from each other.

The incomplete WK-recursive networks, which were originally de®ned in [19], are induced subgraphs of the WK-recursive networks. If we number the nodes of a K…d; t† according to their lexicographical order, then an N-node incomplete WK-recursive network is the subgraph of the K…d; t† induced by the ®rst N nodes. Throughout this paper we use IK…d; t† to denote an N-node incomplete WK-recur-sive network, where dtÿ1_{< N < d}t_{and N is a multiple of d.}

Associated with an N-node IK…d; t† is a coefficient vector (btÿ1; btÿ2; . . . ; b1), where

0 6 bj6 d ÿ 1 for all 1 6 j 6 t ÿ 1 and N ˆ btÿ1dtÿ1‡ btÿ2dtÿ2‡


‡ b1d. It

embedded K…d; j†s: btÿ1btÿ2


bj‡10
K…d; j†; btÿ1btÿ2


bj‡11
K…d; j†; . . . ; btÿ1btÿ2

bj‡1…bjÿ 1†
K…d; j†. For example, the structure of an IK…4; 3† with coecient

vector (3, 2) is shown in Fig. 2. It contains three embedded K…4; 2†s, i.e., 0
K…4; 2†; 1
K…4; 2†, and 2
K…4; 2†, and two embedded K…4; 1†s, i.e., 30
K…4; 1†

Fig. 1. The structrue of: (a) K…4; 1† and (b) K…4; 3†. This ®gure also shows a heuristic routing path and the shortest routing path between nodes 033 and 133.

and 31
K…4; 1†. In the rest of this paper, coecient vector (btÿ1; btÿ2; . . . ; b1) is

written as (btÿ1; btÿ2; . . . ; bi; ), provided bi6ˆ 0 and biÿ1ˆ biÿ2ˆ


ˆ b1ˆ 0, where

1 6 i 6 t ÿ 1. For example, (2, 0, 4, 0, 0) is written as (2, 0, 4, ).

Given an IK…d; t† with coecient vector (btÿ1; btÿ2; . . . ; bi; ), let Sm represent the

subgraph induced by the nodes of btÿ1btÿ2


bm‡10
K…d; m†; btÿ1btÿ2


bm‡11

K…d; m†; . . . ; btÿ1btÿ2


bm‡1…bmÿ 1†
K…d; m†, where i 6 m 6 t ÿ 1. For example,

given an IK…5; 7† with coecient vector (4, 2, 4, 3, 1, 4, ), S6 contains

0
K…5; 6†; 1
K…5; 6†; 2
K…5; 6†, and 3
K…5; 6†; S5 contains 40
K…5; 5† and

41
K…5; 5†, and so on. We note that the embedded K…d; m†s within Sm join one

another through m-¯ipping links. That is, Sm is a bm-supernode complete graph

with each supernode being a K…d; m†. If each Sm is regarded as a stage, then the

structure of the IK…d; t† forms a (t ÿ i)-stage graph, denoted by Stÿ1‡ Stÿ2‡


‡ Si. Refer to Fig. 3, where three examples are shown. For the

sake of simplicity, each embedded K…d; m† within Sm is drawn as a circle, and the

one btÿ1btÿ2


bm‡1j
K…d; m† is denoted by Cmj, where 0 6 j 6 bmÿ 1. All the links

within Sm are omitted.

There are minfbm; bmÿ1g m-¯ipping links between Smand Smÿ1that connect Cmj and

Cj

mÿ1 for all 0 6 j 6 minfbm; bmÿ1g ÿ 1. Besides, there may exist a u-¯ipping link

between Suand Sv, where i 6 v < u 6 t ÿ 1 and u ÿ v > 1. Such a link, if it exists, is

called a jumping u-flipping link. A necessary and sucient condition for the existence of jumping ¯ipping links is presented below.

Theorem 2.1. Given an IK…d; t† with coefficient vector (btÿ1; btÿ2; . . . ; bi; ), one

jumping u-flipping link exists between Su and Sv if and only if bu> buÿ1ˆ

buÿ2ˆ


ˆ bv‡1< bv, where i 6 v < u 6 t ÿ 1 and u ÿ v > 1. Moreover, this jumping

flipping link connects Ce

uand Cev, where e ˆ buÿ1ˆ buÿ2ˆ


ˆ bv‡1.

Proof. (() According to the de®nition of IK…d; t†, there are e u-¯ipping links between Suand Suÿ1that connect Cujand Cuÿ1j for all 0 6 j 6 buÿ1ÿ 1. Besides, there exists one

jumping u-¯ipping link connecting Ce

u and Cve whose two end nodes are

btÿ1btÿ2


bu‡1ebuu2 Cue and btÿ1btÿ2


bu‡1bueuˆ btÿ1btÿ2


bu‡1buee


eev‡12 Cve.

For buÿ1< j 6 buÿ 1, the jumping ¯ipping link …btÿ1btÿ2


bu‡1jbuu; btÿ1btÿ2

bu‡1buju) does not exist because btÿ1btÿ2


bu‡1buju is not a node of the IK…d; t†.

()) Without loss of generality, assume the jumping u-¯ipping link emits from Ca u

to Sv for some 0 6 a 6 buÿ 1. We ®rst show bu> buÿ1 by contradiction. Suppose

bu6 buÿ1. There is a u-¯ipping link between Cju and Cuÿ1j for all 0 6 j 6 buÿ 1.

Re-garding Suÿ1‡ Suÿ2‡


‡ Sias an embedded IK…d; u†, there are two u-¯ipping links

between Ca

uand the embedded IK…d; u†: one is between Cuaand Cuÿ1a and the other is

between Ca

u and Sv. This is a contradiction because at most one u-¯ipping link may

exist between any two embedded K…d; u†s and an IK…d; u† is a subgraph of a K…d; u†. Similarly, a P buÿ1 can be proved.

We continue to show that a ˆ buÿ1; buÿ1ˆ buÿ2ˆ


ˆ bv‡1< bv, and the jumping

u-¯ipping link connects Ca

uand Cav. According to the de®nition of u-¯ipping link, the

end node of the jumping u-¯ipping link in Su(actually in Cua) is btÿ1btÿ2


bu‡1abuu. Thus

the other end node in Sv is btÿ1btÿ2


bu‡1buau. It is not dicult to see that

btÿ1btÿ2


bu‡1buaudoes not belong to the IK…d; t† if a > buÿ1. Consequently, we have

a ˆ buÿ1 and the end node in Sv is btÿ1btÿ2


bu‡1bubuuÿ1ˆ btÿ1btÿ2

bu‡1bubuÿ1buÿ1


buÿ1…buÿ1†v‡1. The latter further implies that buÿ1ˆ buÿ2ˆ

ˆ bv‡1< bvand the end node is located in Cav ˆ Cbvuÿ1. 

In the rest of this paper we use Je

u;v to denote the jumping u-¯ipping link that

connects Ce

uand Cve(refer to Fig. 3 for illustration). Theorem 2.1 provides a fast way

to determine all jumping ¯ipping links from the coecient vector …btÿ1; btÿ2; . . . ; bi; †. We only need to examine …btÿ1; btÿ2; . . . ; bi; † from the left to

the right so that Je

u;vexists if bu> buÿ1ˆ buÿ2ˆ


ˆ bv‡1< bv, where u ÿ v > 1 and

e ˆ buÿ1ˆ buÿ2ˆ


ˆ bv‡1.

We note that for i 6 n 6 m 6 t ÿ 1; bm6ˆ 0, and bn6ˆ 0; Sm‡ Smÿ1‡


‡ Sn

forms an embedded IK…d; m ‡ 1† with coecient vector …bm; bmÿ1; . . . ; bn; † whose

each node has its identi®er pre®xed with btÿ1btÿ2


bm‡1. For example, refer to

Fig. 3(c), where S3‡ S2‡ S1forms an embedded IK…5; 4† with coecient vector (3, 1,

4, ) whose each node has its identi®er pre®xed with 424. Theorem 2.1 can be applied to Sm‡ Smÿ1‡


‡ Snas well.

3. A prerequisite step

Suppose X and Y are arbitrary two nodes of an IK…d; t† with coecient vector …btÿ1; btÿ2; . . . ; bi; †. Without loss of generality, we assume X 2 Cma and Y 2 Cbn,

where i 6 n 6 m 6 t ÿ 1; 0 6 a 6 bmÿ 1 and 0 6 b 6 bnÿ 1. In this section, an

algo-rithm that groups stages Sm; Smÿ1; . . . ; Siinto blocks is proposed. Each block contains

one or more consecutive stages, and every two adjacent blocks intersect with one stage. The union of all blocks is the set of all stages. The algorithm will be invoked when we construct disjoint paths between X and Y in Section 4.

With input …bm; bmÿ1; . . . ; bi; † and a, the algorithm produces a sequence of

integers m0; m1; . . . ; mk, where k P 0 and m P m0> m1>


> mkˆ i. These

integers de®ne k ‡ 1 blocks, i.e., Sm‡ Smÿ1‡


‡ Sm0; Sm0‡ Sm0ÿ1‡


‡ Sm1;


;

Smkÿ1‡ Smkÿ1ÿ1‡


‡ Smk. The algorithm, as shown below, takes O…m† time.

Algorithm (Stage_Grouping……bm; bmÿ1; . . . ; bi; †; a†: = 0 6 a 6 bmÿ 1 =.

(1). Scan …bm; bmÿ1; . . . ; bi; † from the left to the right and determine in sequence

Jx1

y1;z1; Jyx22;z2; . . . ; Jyxcc;zc, so that a > x1> x2>


> xc. That is, Jyx11;z1 is the ®rst jumping

¯ipping link encountered in the scanning which has x1< a. Each Ju;ve between Jyxjj;zj

and Jxj‡1

yj‡1;zj‡1 has e P xj, where 1 6 j < c, and each Ju;ve after Jyxcc;zc has e P xc. Let

L ˆ fx1; x2; . . . ; xcg. If no feasible jumping ¯ipping link is found in the scanning, L is

empty.

For example, refer to Fig. 3(c) again. If X 2 C3

6, then (4, 2, 4, 3, 1, 4, ) and 3 are

taken as input. Since J2

6;4 and J3;11 are found in the scanning, L ˆ f2; 1g. Similarly, if

X 2 C2

6, only J3;11 is found and thus L ˆ f1g. On the other hand, if X belong to C61or

C0

6; L is empty because no feasible jumping ¯ipping link can be found. We also note

that m P y1> z1P y2> z2P


P yc> zcP i. By the aid of Theorem 2.1, this step

can be completed in O…m† time.

(2). Determine m0ˆ minfr jbr> a and bjP a for all m P j > rg. If L is not

empty, determine m1; m2; . . . ; mc sequentially as follows: m1ˆ minfr jbr> x1 and

bjP x1 for all m0> j > rg, m2ˆ minfr jbr> x2 and bjP x2 for all m1> j > rg; . . . ;

mcˆ minfr jbr> xc and bjP xc for all mcÿ1> j > rg.

By examining …bm; bmÿ1; . . . ; bi; † from the left to the right, this step can be

completed in O…m† time. For example, refer to Fig. 3(c) again. If a ˆ 3, we have m0ˆ 6; m1ˆ 3, and m2ˆ 1. If a ˆ 2, we have m0ˆ 3 and m1ˆ 1. If a ˆ 1 or 0, we

have m0ˆ 1. We note that m P m0P yl> z1P m1P y2> z2P


P mcÿ1P

yc> zcP mcP i.

(3). Output m0; m1; . . . ; mk…ˆ i†, where k ˆ c or c ‡ 1, according to the following

four cases:

Case 1. L is empty and m0ˆ i. Output m0.

Case 2. L is empty and m0> i. Set m1ˆ i and output m0; m1.

Case 3. L is not empty and mc ˆ i. Output m0; m1; . . . ; mc.

Case 4. L is not empty and mc > i. Set mc‡1ˆ i and output m0; m1; . . . ; mc; mc‡1.

Refer to Fig. 3(c) again. The algorithm will output (6, 3, 1), (3, 1), (1), and (1) if a ˆ 3; 2; 1, and 0, respectively. The output …m0; m1; . . . ; mk† de®nes k ‡ 1 blocks,

denoted by B0; B1; . . . ; Bk, where B0ˆ Sm‡ Smÿ1‡


‡ Sm0 and Blˆ Smlÿ1‡ Smlÿ1ÿ1

‡


‡ Sml for all 1 6 l 6 k. Every two adjacent blocks Blÿ1 and Bl contain one

common stage Smlÿ1. Lemma 3.1 [22]. Let Jx1 y1;z1; J x2 y2;z2; . . . ; J xc

yc;zc and m0; m1; . . . ; mcbe defined as in Algorithm

Stage_Grouping. Then, for all 1 6 j 6 c, 1. mjÿ1P yj;

2. if mjÿ1> yj, then bmjÿ1 > …xjÿ1P †bmjÿ1ÿ1P


P byj;

4. if zj> mj, then bzj> xj; bmj > xj, and bqP xjfor all zj> q > mj;

5. xjˆ minfbmjÿ1; bmjÿ1ÿ1; . . . ; byj; . . . ; bzj; . . . ; bmjg;where x0ˆ a is assumed.

This lemma is illustrated in Fig. 4, where mjÿ1> yj and zj> mj are assumed.

According to Lemma 3.1, Jx1

y1;z1; Jyx22;z2; . . . ; Jyxcc;zc are the leftmost and upmost jumping

¯ipping links in B1; B2; : . . . ; Bc, respectively (the smaller the value xjis, the upper Jyxjj;zj

is). That is, for any Je

u;vin Bjwe have u 6 yjand e P xj(actually u 6 zjif Ju;ve 6ˆ Jyxjj;zj).

We note that B0 may or may not contain jumping ¯ipping links and Bc‡1, if it

exists, does not contain any jumping ¯ipping link. For Bc‡1 we have bmc >

…xcP †bmcÿ1P


P bmc‡1…mc‡1ˆ i†. We also note that B0 contains at least one

stage, Bjfor 0 < j 6 c contains at least three stages, and Bc‡1, if it exists, contains at

least two stages.

4. Construction of disjoint paths

In this section, disjoint paths are constructed between arbitrary two nodes X and Y of an IK…d; t† with coecient vector …btÿ1; btÿ2; . . . ; bi; †. Without loss of

generality, we assume X 2 Ca

m and Y 2 Cnb, where i 6 n 6 m 6 t ÿ 1; 0 6 a

6 bmÿ 1; and 0 6 b 6 bnÿ 1. The disjoint paths have maximal length not greater

than 2m‡1_{‡ 2}m_{ÿ 1. The construction time is O…d  D}

m†, where Dmˆ 2mÿ 1 is the

diameter of a K…d; m†.

First we consider a trivial case of m ˆ n and a ˆ b. Since X and Y belong to the same embedded K…d; m†; d ÿ 1 disjoint paths between X and Y can be obtained by Duh and Chen's work [3]. These d ÿ 1 paths are all within Ca

m…ˆ Cbn†, and their

maximal length is not greater than 3Dmÿ1‡ 2 ˆ 2m‡ 2mÿ1ÿ 1. It should be

men-tioned that there may exist one more disjoint path between X and Y that goes outside Ca

m. This path, if it exists, is much longer than those d ÿ 1 paths within Cam. We

exclude this path from our discussion.

In the rest of this section, a maximal set of disjoint paths between X and Y is constructed within Sm‡ Smÿ1‡


‡ Sifor: (i) m ˆ n and a 6ˆ b and (ii) m > n. It is

still possible that there exists one additional disjoint path going outside Sm‡ Smÿ1‡


‡ Si. This path, if it exists, is excluded from our discussion because it

is too long as compared with those within Sm‡ Smÿ1‡


‡ Si.

To begin, we have to review Vecchia and Sanges' routing algorithm [23] for a K…d; t† because it is necessary to the construction of the disjoint paths. Suppose A and B are arbitrary two nodes of a K…d; t†. We de®ne A ˆrB if they belong to the

same embedded K…d; r†, and A 6ˆrB otherwise, where 1 6 r 6 t. For example, refer to

Fig. 1(b), where 033 ˆ3133, but 033 6ˆ2133. A routing path from A to B within a

K…d; t† can be obtained by the following procedure.

1. Determine the level r so that A ˆrB but A 6ˆrÿ1B, where 1 6 r 6 t.

2. Determine the ¯ipping link, say (W ; Z), so that A ˆrÿ1W and Z ˆrÿ1B.

3. Determine a routing path from A to W and a routing path from Z to B, recursively. A routing path from A to B is obtained by concatenating the routing path from A to W, the ¯ipping link (W ; Z), and the routing path from Z to B. For example, a routing path from node 033 to node 133 within K…4; 3† is shown with bold lines in Fig. 1(b). When a message is transmitted from A to B, it is ®rst routed to the nearest …r ÿ 1†-frontier, say Z, with Z ˆrÿ1B, then routed to the nearest …r ÿ 2†-frontier, say

Z0_{, with Z}0_ˆ

rÿ2B, and so on. In other words, when the message is going along the

routing path, the identi®ers of the traversed nodes are gradually equalized with B from the left to the right. For example, let us consider the routing path from node 033 to node 133 that is indicated with bold lines in Fig. 1(b). The left digit is equalized at node 100, the middle digit is equalized at node 130, and ®nally all the three digits are equalized at the destination node 133. The following observation is immediate.

Observation 4.1. When routing a message according to Vecchia and Sanges' algo-rithm, the identi®ers of the traversed nodes are gradually equalized with B from the left to the right.

Vecchia and Sanges' algorithm, although simple, does not guarantee the shortest path. For example, the shortest path from node 033 to node 133 is shown with dashed lines in Fig. 1(b). Let p…A; B† denote the routing path from node A to node B within a K…d; t† that is produced by Vecchia and Sanges' algorithm. The following two lemmas have been proved in [3].

Lemma 4.1 [3]. Suppose A and B are arbitrary two nodes of a K(d; t). If A ˆrB and

either of them is an r-frontier, then p…A; B† is the shortest, where 1 6 r 6 t. Moreover, it takes at most O…Dt† time to determine p…A; B†, where Dtˆ 2tÿ 1 is the diameter of the

Lemma 4.2 [3]. Suppose A is an arbitrary node of a K…d; t†, and let Vl;0; Vl;1; . . . ; Vl;dÿ1

be the dl-frontiers of an embedded K…d; l† that contains A, where 1 6 l 6 t. Then, the d paths p…A; Vl;0†; p…A; Vl;1†; . . . ; p…A; Vl;dÿ1† are mutually disjoint, exclusive of A.

By Lemma 4.1, p…A; Vl;0†; p…A; Vl;1†; . . . ; p…A; Vl;dÿ1† are all the shortest and they can

be determined in O…d  Dl† time. Now we are ready to construct disjoint paths

between X 2 Ca

m and Y 2 Cnb. First we consider the situation of m ˆ n and a 6ˆ b.

Within Smthere are bmÿ 1 disjoint paths between X and Y. Besides, there may exist

one more disjoint path that passes through the embedded IK…d; m† formed by Smÿ1‡ Smÿ2‡


‡ Si. These bm paths are pictorially expressed in Fig. 5, where

a 62 f0; bmÿ 1g; b 62 f0; bmÿ 1g, and a < b are assumed. In Fig. 5, each thin line

represents a ¯ipping (or jumping ¯ipping) link and each bold line represents a subpath obtained by Vecchia and Sanges' algorithm. According to Lemmas 4.1 and 4.2, these bm paths are mutually disjoint (exclusive of X and Y) and they can be

determined in O…bm Dm† 6 O…d  Dm† time.

We also note that the path passing the embedded IK…d; m† exists only if the two nodes btÿ1btÿ2


bm‡1bmam and btÿ1btÿ2


bm‡1bmbm belong to the embedded

IK…d; m† (refer to Fig. 5). According to Observation 4.1, p…btÿ1btÿ2


bm‡1bmam;

btÿ1btÿ2


bm‡1bmbm†, which is constructed within btÿ1btÿ2


bm‡1bm
K…d; m†, can be

expressed as follows: btÿ1btÿ2


bm‡1bmam!


! btÿ1btÿ2


bm‡1bmabmÿ1; ! btÿ1btÿ2


bm‡1bmbamÿ1!


! btÿ1btÿ2


bm‡1bmbabmÿ2; ! btÿ1btÿ2


bm‡1bmb2amÿ2!


! btÿ1btÿ2


bm‡1bmb2abmÿ3; ! ... ! btÿ1btÿ2


bm‡1bmbmÿ2a2! btÿ1btÿ2


bm‡1bmbmÿ2ab; ! btÿ1btÿ2


bm‡1bmbmÿ1a ! btÿ1btÿ2


bm‡1bmbm:

Each node in the subpath from btÿ1btÿ2


bm‡1bmam to btÿ1btÿ2


bm‡1bmabmÿ1

has leading digits btÿ1btÿ2


bm‡1bma, each node in the subpath from

btÿ1btÿ2


bm‡1bmbamÿ1 to btÿ1btÿ2


bm‡1bmbabmÿ2 has leading digits

btÿ1btÿ2


bm‡1bmba, and so on. That is, every node in p…btÿ1btÿ2


bm‡1bmam;

btÿ1btÿ2


bm‡1bmbm† precedes btÿ1btÿ2


bm‡1bmbm lexicographically if a < b,

and precedes btÿ1btÿ2


bm‡1bmam lexicographically if a > b. Hence,

p…btÿ1btÿ2


bm‡1bmam; btÿ1btÿ2


bm‡1bmbm† is entirely contained in the embedded

IK…d; m†, provided the two end nodes are contained in the embedded IK…d; m†. Then we consider the situation of m > n. With input …bm; bmÿ1; . . . ; bi; † and

a, Algorithm Stage_Grouping is ®rst executed to produce m0; m1; . . . ; mk so

‡


‡ Smk, where k P 0 and mk ˆ i. Suppose Y 2 Cbn belongs to Bl, where 0 6 l 6 k.

If Y 2 Sml, we consider Y 2 Bl but 62 Bl‡1. Two cases are discussed below.

Case 1 (l ˆ 0). We have m > n P m0. Let j ˆ minfbm; bmÿ1; . . . ; bng if

minfbm; bmÿ1; . . . ; bng ˆ bm or bnand j ˆ minfbm; bmÿ1; . . . ; bng ‡ 1 else. There are j

disjoint paths between X and Y, denoted by P0; P1; . . . ; Pjÿ1, where Pr passes

Ca

m; Crm; Crmÿ1; . . . ; Crn; Cbn in sequence for all 0 6 r 6 j ÿ 1…Cma ˆ Crm as a ˆ r, and

Cr

nˆ Cbn as r ˆ b). Besides, there is an additional disjoint path, denoted by Pj, if

j ˆ bn< bm and there is a jumping ¯ipping link, say Ju;vbn, under Sn, where

m > u > n > v. It is still possible that Pj contains other jumping ¯ipping links

Fig. 5. bm Disjoint paths between X and Y, where a 62 f0; bmÿ 1g; b 62 f0; bmÿ 1g, and a < b are

between Sm and Su. Without loss of generality, we assume Pj contains only Ju;vbn, and

so Pj passes Cam; Cjm; Cmÿ1j ; . . . ; Cuj; Cvj; Cvb; Cbv‡1; . . . ; Cbn, in sequence. According to

Lemmas 4.1 and 4.2, these Pjpaths are mutually disjoint (exclusive of X and Y) and

they can be determined in O…j  Dm† 6 O…d  Dm† time.

For example, let us consider X 2 C2

6 and Y 2 C51 in an IK…5; 7† with coecient

vector (4, 2, 4, 3, 1, 4, ) (refer to Fig. 6). The execution of Algorithm Stage_Grouping produces B0ˆ S6‡ S5‡ S4‡ S3 and B1ˆ S3‡ S2‡ S1. We have

j ˆ minf4; 2g ˆ 2. Since j ˆ b5< b6 and one jumping ¯ipping link goes under

S5, there are j ‡ 1 ˆ 3 disjoint paths between X and Y. These three paths are

shown in Fig. 6, where each thin line represents a ¯ipping (or jumping ¯ipping) link and each bold line represents a subpath obtained by Vecchia and Sanges' algorithm.

Case 2 (0 < l 6 k). We have mlÿ1> n P ml. Let …a >†x1> x2>


> xc be

de-®ned as in Algorithm Stage_Grouping, where c ˆ k or k ÿ 1. Recall the discussion in the last paragraph of Section 3. There is at least one jumping ¯ipping link in Bl unless l ˆ c ‡ 1. We ®rst assume l < c ‡ 1 and let Jyxll;zl represent the leftmost

and upmost jumping ¯ipping link in Bl. Three subcases have to be discussed

below.

Subcase 1 (mlÿ1> n P yl). There are bn disjoint paths between X and Y, denoted

by P0; P1; . . . ; Pbnÿ1, where Pr passes Cma; Crm; Cmÿ1r ; . . . ; Crn; Cnb in sequence for all

0 6 r 6 bnÿ 2 and Pbnÿ1 passes Cma; Cmÿ1a ; . . . ; Cma0; Cxm10; C x1 m0ÿ1; . . . ; Cmx11; Cxm21; C x2 m1ÿ1; . . . ; Cx2 m2; Cmx32; . . . ; Cmxlÿ1lÿ1; Cmbnlÿ1ÿ1; C bnÿ1 mlÿ1ÿ1; . . . ; Cbnnÿ1; Cnb in sequence.

For example, refer to Fig. 7, where an IK…5; 7† with coecient vector (4, 2, 4, 3, 1, 4, ) is shown. Suppose X 2 C3

6 and Y 2 C31. The execution of Algorithm

Stage_Grouping produces B0ˆ S6; B1ˆ S6‡ S5‡ S4‡ S3, and B2ˆ S3‡ S2‡ S1.

There are b3ˆ 3 disjoint paths between X and Y as shown in Fig. 7(a).

Subcase 2 (yl> n P ml and bnˆ xl). There are bn‡ 1 disjoint paths between X

and Y. Since bnˆ xlˆ minfbmlÿ1; bmlÿ1ÿ1; . . . ; bmlg (by Lemma 3.1), bmlÿ1 > xlÿ1> xl,

and bml > xl, there exists a jumping ¯ipping link under Sn, say Ju;vbn, where

mlÿ1P u > v P ml. We use P0; P1; . . . ; Pbnto denote the bn‡ 1 disjoint paths, where Pr

passes Ca

m; Crm; Cmÿ1r ; . . . ; Cnr; Cnb in sequence for all 0 6 r 6 bnÿ 1 and Pbn passes

Ca m; Camÿ1; . . . ; Cam0; C x1 m0; C x1 m0ÿ1; . . . ; C x1 m1; C x2 m1; C x2 m1ÿ1; . . . ; C x2 m2; C x3 m2; . . . ; C xlÿ1 mlÿ1; C xl mlÿ1; Cxl mlÿ1ÿ1; C xl mlÿ1ÿ2; . . . ; C xl u; Cvxl; Cvb; Cv‡1b ; . . . ; Cnb in sequence.

For example, suppose X 2 C3

6 and Y 2 C20belong to the same IK…5; 7†. There are

b2‡ 1 ˆ 2 disjoint paths between X and Y as shown in Fig. 7(b).

Subcase 3 (yl> n P ml and bn> xl). Actually we have zlP n P ml and bn> xl

because bylÿ1 ˆ bylÿ2 ˆ


ˆ bzl‡1ˆ xl can be assured by Theorem 2.1. There are

xl‡ 1 disjoint paths between X and Y, denoted by P0; P1; . . . ; Pxl, where Pr passes

Fig. 7. Disjoint paths between X and Y for Case 2. (a) mlÿ1> n P yl; (b) yl> n P mland bnˆ xlÿ 1;

Ca

m; Crm; Crmÿ1; . . . ; Cnr; Cbn in sequence for all 0 6 r 6 xlÿ 1 and Pxl passes

Ca m; Camÿ1; . . . ; Cm0 a; Cxm10; C x1 m0ÿ1; . . . ; Cxm11; Cmx21; Cm1ÿ1 x2; . . . ; Cmx22; Cxm32; . . . ; Cmxlÿ1lÿ1; Cmxllÿ1; Cxl mlÿ1ÿ1; C xl mlÿ1ÿ2; . . . ; Cxyll; Czxll; . . . ; Cxnl; Cbn in sequence.

For example, suppose X 2 C3

6 and Y 2 C12 belong to same IK…5; 7†. There are

xl‡ 1 ˆ 2 disjoint paths between X and Y as shown in Fig. 7(c).

On the other hand, if l ˆ c ‡ 1, we have bmc> …xcP †bmcÿ1P


P bnP


P

bmc‡1…mc‡1ˆ i†. There are bn disjoint paths between X and Y whose construction is

similar to Subcase 1.

According to Lemmas 4.1 and 4.2, the paths obtained for Case 2 are mutually disjoint (exclusive of X and Y), and they can be determined in O…maxfbn‡ 1;

xl‡ 1g  Dm† 6 O…d  Dm† time.

The following theorem holds as a consequence of our discussion above. Theorem 4.1. Suppose X 2 Ca

mand Y 2 Cbnbelong to an IK…d; t† with coefficient vector

…btÿ1; btÿ2; . . . ; bi; †, where i 6 n 6 m 6 t ÿ 1; 0 6 a 6 bmÿ 1, and 0 6 b 6 bnÿ 1. Then

the disjoint paths between X and Y can be determined in O…d  Dm† time.

Let len…X ; Y † be the maximal length of the disjoint paths between X and Y. In the following we show that len…X ; Y † has an upper bound of 2m‡1_{‡ 2}m_{ÿ 1.}

Theorem 4.2. Suppose X 2 Ca

mand Y 2 Cbnbelong to an IK…d; t† with coefficient vector

…btÿ1; btÿ2; . . . ; bi; †, where i 6 n 6 m 6 t ÿ 1; 0 6 a 6 bmÿ 1, and 0 6 b 6 bnÿ 1. Then

len…X ; Y † 6 2m‡1_{‡ 2}m_{ÿ 1.}

Proof. If m ˆ n, there are at most bm disjoint paths between X and Y. By the aid

of Lemma 4.1, the path passing the embedded IK…d; m† has length at most Dm‡ 1 ‡ Dm‡ 1 ‡ Dmˆ 2m‡1‡ 2mÿ 1, and the others each have length at most

Dm‡ 1 ‡ Dm‡ 1 ‡ Dmˆ 2m‡1‡ 2mÿ 1. Hence, len…X ; Y † 6 2m‡1‡ 2mÿ 1. In the

rest of the proof, we assume m > n and Y 2 Blfor some 0 6 l 6 k.

If l ˆ 0, there are at most j ‡ 1 disjoint paths P0; P1; . . . ; Pj between X and Y. The

path Pj, if it exists, has length at most Dm‡ 1 ‡ Dm‡ 1 ‡ Dmÿ1‡ 1 ‡


‡ Du‡1

‡1 ‡ Du‡ 1 ‡ Dv‡ 1 ‡ Dv ‡1 ‡ Dv‡1‡ 1 ‡


‡ Dn< 2m‡1‡ 2mÿ 1, and the

oth-ers each have length at most Dm‡ 1 ‡ Dm ‡1 ‡ Dmÿ1‡ 1 ‡ Dmÿ2‡ 1 ‡


‡

Dn‡1‡ 1 ‡ Dn‡ 1 ‡ Dnˆ 2m‡1‡ 2mÿ 1. Hence, len…X ; Y † 6 2m‡1‡ 2mÿ 1.

If 0 < l 6 k, there are bn or bn‡ 1 or xl‡ 1 disjoint paths between X and Y,

ac-cording to three subcases. Similarly, len…X ; Y † 6 2m‡1_{‡ 2}m_{ÿ 1. }

5. Experiments and results

In this section the e€ectiveness of the disjoint paths is veri®ed by extensive ex-periments. The following two algorithms were implemented for the need of our experiments.

· Su, Chen, and Duh's algorithm that computes the diameter of an IK…d; t† [21]. · Su, Chen, and Duh's algorithm that computes the shortest path between arbitrary

Remarks. Adopting the prune-and-search technique [15], the algorithm of Su et al. [21] can compute the diameter of an IK…d; t† and the farthest pair of nodes in O…t† time. Although the diameter of an IK…d; t† can be computed by Su, Chen, and Duh's algorithm, no formula is available for computing it.

First we compare len…X ; Y † with dis…X ; Y †, where dis…X ; Y † is the distance between X and Y. Fig. 8 shows the average ratios of len…X ; Y † to dis…X ; Y † for IK…d; t†s with 4 6 d 6 6 and 2 6 t 6 10. The values of dis…X ; Y † were computed by the algorithm of Su et al. [22]. For each pair of d and t; 106 _{random instances were run and their}

average ratio was computed. The averages got stable after running as many as 106

instances. A randomly generated coecient vector combined with two nodes, also randomly generated, of an IK…d; t† forms an instance. Experimental results showed that len…X ; Y † is not greater than twice dis…X ; Y † in average.

Then we compare len…X ; Y † with the diameter. Fig. 9 shows the average ratios of len…X ; Y † to the diameter for IK…d; t†s with 4 6 d 6 6 and 2 6 t 6 10. The diameters were computed by the algorithm of Su et al. [21]. Like Fig. 8, the average ratio for 106 _{random instances was taken for each pair of d and t. Experimental results}

showed that len…X ; Y † is smaller than the diameter in average both as d ˆ 4 and t P 2 and as d ˆ 5 and t P 5. Besides, len…X ; Y † tends to the diameter as d ˆ 6 and t increases.

When X and Y were selected to be the farthest pair of nodes (i.e., dis…X ; Y † is equal to the diameter of the IK…d; t†), the average ratios of len…X ; Y † to the diameter were shown in Fig. 10. The farthest pair of nodes can be determined by the algorithm of Su et al. [21]. The average ratios each were obtained by running 105 _random

in-stances because the averages got stable after running as many as 105 _instances.

Experimental results showed that most of the averages fall in the range of 1.2±1.3. The maximal average ratio does not exceed 1.5.

It is worth mentioning that for the experiments of Fig. 10, no instance has len…X ; Y † exceeding twice the diameter. The distributions of the 105_{ratios obtained}

for an IK…4; 8†, an IK…5; 8†, and an IK…6; 8† were shown in Figs. 11(a)±(c), respec-tively. For example, for the IK…4; 8† there are 54,566 ratios (about 54%) fall in the range of 1.0±1.1 and there are 4044 ratios (about 4%) fall in the range of 1.1±1.2.

Fig. 10. Average ratios of len…X ; Y † to dis…X ; Y † for IK…d; t†, where X and Y are the farthest pair nodes. Fig. 9. Average ratios of len…X ; Y † to diameter for IK…d; t†.

6. Concluding remarks

In this paper, we have constructed a maximal set of disjoint paths between X and Y, which are arbitrary two nodes of an IK…d; t†. The construction time is bounded by O…d  Dt†. We have shown that the disjoint paths have maximal length not greater

than 2m‡1_{‡ 2}m_{ÿ 1, where X 2 C}a

m; Y 2 Cnb, and i 6 n 6 m 6 t ÿ 1 are assumed. The

e€ectiveness of the disjoint paths was further veri®ed by extensive experiments. Experimental results showed that the disjoint paths have expected maximal length not greater than twice their distance and about equal to the diameter. Besides, when X and Y are the farthest pair, the disjoint paths have maximal length not greater than

twice the diameter and expected maximal length not greater than 1.5 times the di-ameter.

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