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Parallel Routing in Hypercube Networks with Faulty Nodes

Eunseuk Oh Jianer ChenDepartment of Computer Science, Texas A&M University

College Station, TX 77843-3112, USA

eunseuko, chen

@cs.tamu.edu

Abstract

The concept of strong fault-tolerance was introduced to

characterize the property of parallel routing [15]. A net-

work of degree is said strongly fault-tolerant if with at

most

faulty nodes, any two nodes

and

in

are

connected by

node-disjoint paths,

where

and

are the numbers of non-faulty

neighbors of the nodes and in , respectively. We show

that the hypercube networks are strongly fault-tolerant and

develop an algorithm that constructs the maximum number

of node-disjoint paths in a hypercube network with faults.

Our algorithm is optimal in terms of time and length of

node-disjoint paths.

1. Introduction

Parallel routing on large size networks with faults is an

important issue in the study of computer interconnection

networks, which allows networks to have alternative routes

to tolerate faulty nodes. The new concept of the network

strong fault tolerance was introduced to measure fault toler-

ance for interconnection networks and has been studied for

the star networks [15]. In the current paper, we continue the

study of the strong fault tolerance of interconnection net-

works, particularly, of the popular hypercube networks.

The

-dimensional hypercube

has been studied ex-

tensively by many researchers as an interconnection net-

work topology for multicomputer systems. Parallel routing

on hypercube networks without faulty nodes was first stud-

ied in [18]. An algorithmthat constructsnode-disjoint paths

between disjoint source-destination pairs was proposed in

[8, 14]. The problem of determining the diameter of hyper-

cube networks with faults was considered in [10, 11]. Many

This work is supported in part by the National Science Foundation

under Grant CCR-0000206.

fault-tolerant communication algorithms concentrating on

one-to-one routing or broadcasting in hypercube networks

have been proposed [2, 4, 6, 7, 12, 13, 16, 17].

A network of degree is said strongly fault-tolerant

[15] if with at most faulty nodes, any two nodes

and

in

are connected by

node-disjoint paths, where

and

are the numbers

of non-faulty neighbors of the nodes and in , respec-

tively.

The study of strong fault tolerance in the star networks

showed that node-disjoint paths can be constructed effi-

ciently based on the orthogonal partition of the star net-

works with faults, which decomposes the

-star network

into -dimensional substar networks and an

independent set of nodes [3]. Roughly speak-

ing, a path from a non-faulty neighbor of the source node

to a non-faulty neighbor of the destination node is con-

structed in a separated -dimensional substar, and theindependent set

helps the paths to enter the substar from

a proper node. The algorithm proposed in [15] constructs

the maximum number of node-disjoint paths of nearly opti-

mal length in the

-star networks with at most

faulty

nodes.

We observe that the techniques used in the previous

studying for star networks [15] are not applicable to the

case for hypercube networks. Specifically, the hypercube

networks do not seem to have similar orthogonal decompo-

sition structure. Parallel routing in the

-dimensional hy-

percube networks may require constructing node-disjoint

paths, while an -dimensional hypercubes can be decom-

posed into at most

-dimensional subcubes. There-

fore, there may be no extra nodes available that can help to

distribute the paths into the subcubes.

We develop new techniques that construct node-disjoint

paths between pairs of neighbors of the source node

and

the destination node . First, a prematching process pairs

non-faulty neighbors of and in

. For given pairs

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of neighbors of and , we introduce three procedures to

construct paths by permutations of edge sequences between

them. Node-disjoint paths are constructed by searching

properpaths, ensuring that each node in a path is notused by

other paths. Our algorithm constructs node-disjointpaths in

optimal time and the length of paths is also optimal in the

hypercube network

: For any two non-faulty nodes and

in

, the algorithm constructs min

node-disjoint paths of minimum length plus 4 between

and in time .

The paper is organized as follows. Notations and termi-

nology are introduced in section 2. In section 3, we dis-

cuss parallel paths between two non-faulty nodes. In case

there are no faulty neighbors for both sourceand destination

nodes, we pre-pair the neighbors of the source and the des-

tination nodes by a process, called Prematch-I. There is a

special situation that may block all possible sets of parallel

paths between two neighbors of the source and the destina-

tion nodes induced from Prematch-I. In this situation, we

use a different process, called Prematch-II instead. The

third process Prematch-III covers the case in which there

is at least one faulty neighbor of the source or the destina-

tion. The algorithm is presented and discussed in section 4

and the final section concludes the paper.

2. Preliminaries

An

-dimensional hypercube

is an undirected graph

consisting of nodes represented by binary numbers from

to

, and

edges connecting nodes whose bi-

nary representations differ in exactly one bit. An edge is

called an -edge if two nodes connected by it differ in the

th bit (the first bit is the leftmost bit). The Hamming dis-

tance between two nodes and , is the length

of the shortest path from to . Actually, is the

number of bits in which binary representations of

and

differ. Since the hypercube

is vertex-symmetric, a

set of node-disjoint paths from a node to a node can

be mapped to a set of node-disjoint paths from the node

to the node in a straightforward way,

where . Therefore, we will concentrate on

the construction of node-disjoint paths from the node

to

the node in

.

The node connected from the node by an -edge is

denoted by

, and the node connected from the node

by a -edge is denoted by

. A path from the node

to the node can be uniquely specified

by a sequence of labels of the edges on

in the order of

traversal. In particular, a path from the node to the node

that uses an

-edge, an

-edge, , an

-edge, in that or-

der, will be denoted by

. For example, for

the nodes and , spec-

ifies the path

. We extend this notation for a single permutation to

a set of permutations, as follows. Let be a set of permu-

tations, then the notation

denotes the set of paths:

is a permutation in

For example, suppose , ,

, , then consists of four paths

from

to

:

,

,

, and

.

We say that an edge

does not lead to a shortest

path to a node

if

. The

following fact can be easily verified.

Fact 2.1 If an edge

in

does not lead to a short-

est path to

, then

. In

general, if in a path from a node to a node , thereare exactly

edges that do not lead to a shortest path to

,

then the length of the path is equal to

.

It is known [18] that for any two nodes

and

in

, there exist node-disjoint paths such that

of these paths are of length , and the remaining

paths are of length

.

3. Parallel paths between two non-faulty nodes

In this section, we show how a set of paths between two

non-faulty nodes

and

in the hypercube network

canbe constructed.

Our parallel routing algorithm is based on an effective

pairing of the neighbors of the node

and

. We first assume that the nodes and have no

faulty neighbors. We pair the neighbors of

and

by the

following strategy.

Prematch-I

Assumption: and have no faulty neighbors.

1. pair

with

for 1 ;

2. pair

with

for

;

Under the pairing given by Prematch-I, we constructparallel paths between the paired neighborsof

and

using

the following procedure.

1The calculation for indices between 1 and

can be given by a rather

lengthy formula based on modular operation. For simplicity, we only need

to remember the following three special cases: Let be an index between

1 and . (1) for , is interpreted as and is interpreted as

; (2) for , is interpreted as ; and (3) for , is

interpreted as 1.

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u = 1110

1010

1000

1011

1001 v = 0000

1111

0110

0111

0101

0001

0100

0010 0011

1100 1101

2

4

3

Figure 1. Parallel paths between and

in

with two faulty(dark) nodes.

Procedure-I

1. for , and the paired neighbors

and

, we construct node-disjoint

paths between

and

, which consist of

paths of the form

(1)

where

is the set ofall cyclicpermutations

of the sequence ,

and of paths of the form

(2)

for all

,

.

2. for , and the paired neighbors

and

, we construct

node disjoint

paths between and , which consist of

paths of the form

(3)

where

is the set ofall cyclicpermutations

of the sequence , and of

paths of the form

(4)

for all , and .

The paths constructed by cyclic permutations of a se-

quence are pairwisely disjoint(see, for example [18]). It is

easy to verify that for each pair of neighbors of

and

,

the paths constructed between them are pairwisely disjoint.

Figure 1 shows an example of parallel paths in

with two

faulty(dark) nodes. For a path

from

to

, we define

as the node on the path

starting from

and

following the edge labels in .

Lemma 3.1 Let

and

be two pairs given by

Prematch-I. Then, there is at most one path in the path set

constructed by Procedure-I for the pair

that share

common nodes with a path in the path set constructed by

Procedure-I for the pair

.

PROOF. For two paths

and

such that

is for

the pair

and

is for the pair

,

,

assume that

and

have a common node. Then, the

same set of bits in

and

are different

from those of

. Since

th bit and

th bit in the common

node are different from those of , it must be of the form

. We show below that

this node must have the form

. Thus, the path

is uniquely determined by

and

.

Case 1. Suppose

and

.

Suppose the common node is of the form

, then

must

be

(if

),

(if

), or

for

some . If

then the node

has

th bit identical to that of

while the node

has th bit different from that of .

If

and then there is no node of form

. Thus, the index

does not exist, and

must be of the form

. If

then

must be of the form

.

In that case, the sequences in

and

are constructed

by cyclic permutations of a sequence except

the index . It has been known that paths constructed

by cyclic permutations of a sequence are disjoint. This

property still holds when the index is added such as

. It contradicts the assumption that

and

has a common node. Thus, the index

does not

exist.

Case 2. Suppose

and

, or

and .

First assume

and

. The se-

quence in the path

must be of the form

for some since . Since is the only index larger

than

in this sequence and

, we must have

.

Thus, the path

must be of the form

.

The case and can be proved by

symmetry.

Case 3. and .

The sequences in

and

cannot be a permutation of

because , . Thus,

must be of the form

and

is

for some

. In that case,

should be

and

should be

because and are the only indices larger than . Thus,

the path

is of the form

.

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Combining all cases, we complete the proof.

Fact 3.1 For a pair

,

given by

Prematch-I, a path of the form

has no common nodes with any other paths con-

structed by Procedure-I.

We have shown that for each paired nodes by Prematch-

I, the algorithm Procedure-I constructs at least dis-

joint paths between them. Since there may be up to

faulty nodes, in the worst case, there can be a pair

of nodes by Prematch-I, for which all paths con-

structed by Procedure-I are blocked. In this case, we pair

the neighbors of

and

by the following rule:

Prematch-II

Assumption: there is a pair

,

given by Prematch-I such that all

paths constructed by Procedure-I for

are blocked by faulty nodes.

1.

is paired with

;

2.

is paired with

;

3.

is paired with

;

4. forother neighborsof and , use Prematch-I

In Prematch-II, operations on indices between 1 and

are by mod

. For each pair given by Prematch-II, we con-

struct a path as follows.

Procedure-II

1. For a pair of form

, the path is

;

2. For a pair of form

, the path is

;

3. For a pair of form

, the path is

;

4. For other pairs, use Procedure-I to con-

struct paths between them: For pair

,

,

, the

path is

;

For pair

,

, the

path is

if

, and

if .

Lemma 3.2 Under the conditions ofPrematch-II, the al-

gorithm Procedure-II constructs fault-free parallel paths

of length

from

to

.

PROOF. It easy to see that Paths constructed by

Procedure-II have length bounded by . Ex-

cept paths of form

or

,

, whose length is

, other paths have length

.

Now we show that all

pathsconstructed by Procedure-

II are fault-free. After that, we show that these paths

are disjoint. Recall that all possible paths constructed by

Procedure-I for the pair

are blocked by faulty

nodes. Denote these paths by

.

The path

and

only share a node

because every node in

has its th bit identical to that of while nodes ex-

cept

in

has

th bit different that of

. Since

the node

is non-faulty, the path

is fault-free. The

path

and

have no common nodes because

th bits in nodes in

and

are not identical. Thus,

is fault-free. The

path

and

only share a node

because nodes except

(=

) have th bit identical to that of

while nodes in

have th bit different that of . A

path of form

,

has no common nodes with any other paths

constructed by Procedure-I by fact 3.1. Since

, the

path

is fault-free. Finally, consider a path

constructed

for a pair

, . If , all faulty

nodes are in paths between

and

and

is of the form

. Sincea node

in the path

can be identical to only a node of form

,

by lemma 3.1, the path

and

haveno common nodes.

In case , all faulty nodes are in paths between

and

and

is of the form

. Similarly, we can

prove that

is fault-free sincea node

in

can

be identical to only a node of form

.

Therefore, all paths constructed by Procedure-II are

fault-free.

Now we show that paths constructed by Procedure-II

are disjoint.

It is easy to see that

and

have no common nodes

because of an index . Similarly,

and

have no

commonnodes because of an index . Also,

and

have no common nodes because nodes except

in

have th bit identical to that of while nodes in

have th

bit different that of , and the node

is not included in

. Thus, paths

,

, and

are disjoint. Each path

of the form

, is

disjoint by fact 3.1. To show that paths constructed for pairs

,

are disjoint with paths

,

, and

, suppose

is the sequence of the

path

. Since ,

becomes and the path

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1010

1000

1011

1001 v = 0000

0110

0111

0101

0001

0100

0010 0011

1100 1101

4

1110

u = 1111

3

2

1

Figure 2. Paths constructed by Procedure-II

between and in

.

cannot have common nodes with the path

. Suppose

is the sequenceof

, then since ,

becomes

and

and

have no common nodes.

Also if

is the sequence of

, then since

,

becomes

when

and

becomes

when

. For both cases, it is easy to see that

and

have no common nodes. Finally, each path of

the form

or

,

is disjoint with other paths because of a unique index .

Therefore, all paths constructed by Procedure-II are

pairwisely disjoint. Figure 2 shows paths constructed by

Procedure-II between

and

with two

faulty(dark) nodes in

.

So far, we have assume that all neighbors of the source

node and the destination node are non-faulty. Now we

relax such a restriction to deal with any faulty neighbors

of two nodes

and

. We introduce Prematch-III to pairthe edges incident on the neighbors of the nodes

and

,

instead of the neighbors of

and

.

Prematch-III

Assumption: or have at least one faulty

neighbor, and edges are paired only when all

nodes on them are non-faulty.

for each edge

where both

and

are non-faulty do

1. if and , then pair

with the edge

;

2. if and , then pair

with the edge

;

3. otherwise, pair

with

,

where the indices and are such that

Prematch-I pairs the node

with

, and

the node

with

.

Note that it is possible that an edge

with both

and

non-faulty is not paired with any edge because the

corresponding edge in Prematch-III contains faulty nodes.

For each pair of edges givenby Prematch-III, we construct

a path between them by thealgorithmcalled Procedure-III.

Basically, sequences in paths constructed in Procedure-III

follow Procedure-I if there is no comment for that. We

assume that for edges

, , their

pairs are given in the increasing order of

, and operations

on indices between 1 and are by mod .

Procedure-III

1. for and and paired

edges

,

, construct

the path

;

2. for and and

paired edges

,

,

construct a path by flipping

and

in the

path

;

3. otherwise, for paired edges

,

, if , construct a path by flip-

ping

and

in the path

; if

, construct a path by flipping and

in the path

.

Notice that Procedure-III forces the sequence between

a pair induced by Prematch-III to be obtained from a path

constructed by Procedure-I such that is of the form

, where

or when

.Thus, consider paths of the form

constructed

by Procedure-I such that or when .

The following fact shows that any two paths of the forms

and

as described above have no

nodes in common. It comes directly from lemma 3.1.

Fact 3.2 Let

be a path of the form

such

that

or

when

as given in Procedure-I.

For a path

of the form

, ,

has no

nodes in common.

Lemma 3.3 For a non-faulty node

, Procedure-III con-

structs at most fault-free one-to-many disjoint paths

from the node

to all non-faulty nodes

,

.

PROOF. For all non-faulty edges of form

,

, consider edges paired by Prematch-III. Sup-

pose . For ,

is paired with

when , and

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when . Otherwise,

is paired with

such that

is paired with

by Prematch-I.

Since

is uniquely paired with

and

by Prematch-I,

there are at most non-faulty edge pairs constructed

by Prematch-III. Suppose . Since for

, a non-faulty node

,

is uniquely

paired with a node

by Prematch-I, there are at most

non-faulty edge pairs constructed by Prematch-III.

Now, consider paths between above edge pairs. All paths

constructed by Procedure-III are obtained from paths con-

structed by Procedure-I such that these paths are of the

form

, or when , flipping

first two indices or last two indices. We have shown that

such paths of the form

,

or

when

, are disjoint in fact 3.2. Thus, flipping first two

indices on such paths make paths share the node

, and

flipping last two indices on these path make paths have all

different entering nodes to which are given by Prematch-

III. Thus, Procedure-III constructs at most fault-free

one-to-many disjoint paths from

to all non-faulty nodes

, .

Further, the following fact can be easily verified from the

above discussion.

Fact 3.3 For any two edge pairs

and

given by Prematch-III, paths

constructed by Procedure-III for these pairs have common

nodes of form

, where

and

.

4. Parallel routing algorithm on faulty hyper-cube networks

First, consider the lower bound of the length of the

min

node-disjoint paths from a node

to a node

in hypercube

, where

. Suppose a neighbor node of

,

be non-faulty, and we want to find a path from to

via

. Assume that all neighbors of

are faulty except

two nodes

and

. Then, a

fault-free path of the form from to has

length at least

. Thus, the length of the

min

disjoint paths from to is at least

.

We now ready to present our main algorithm for par-

allel routing in the hypercube networks

with at most

faulty nodes. For two non-faulty nodes

and

in

, our algorithm constructs

node-disjoint fault-free paths from

to such that the length of the paths is bounded by

Parallel-Routing

Input: non-faulty nodes

and

in

with at most

faulty nodes.

Output:

parallel fault-free paths of

length

from

to

.

1. case 1.

and

have no faulty neighbors

for each pair

given by Prematch-I do

1.1 if all paths for

by Procedure-I include faultynodes

then use Prematch-II and Procedure-II to construct

parallel paths from to ; STOP.

1.2 if there is a fault-free unused path from

to

by

Procedure-I

then mark the path as used by

;

1.3 ifall fault-free paths constructed for

include

used nodes

then pick any fault-free path for

, and for the

pair

that uses a node on , find a new

path;

2. case 2. and have at least one faulty neighbor

for each edge pair

given by

Prematch-III do

2.1 ifthere is a fault-free unused path from

to

by

Procedure-III

then mark the path as used by the pair

,

,

,

;

2.2 ifall fault-free paths constructed for the pair include

used nodes

then pick any fault-free path

for the edge pair, and

for the edge pair that uses a node on

, find a new

path;

Figure 3. Parallel routing on the hypercube

network with faulty nodes

. The algorithm called Parallel-Routing is

given in Figure 3.

Lemma 3.2 guarantees that step 1.1 of the algorithm

Parallel-Routing constructs

fault-free parallel paths oflength

from

to

. Step 1.3 of the algo-

rithm requires further explanation. In particular, we need to

show that for the pair

, we can always construct

a new fault-free path from

to

in which no nodes

are used by other paths. This is ensured by the following

lemma.

Lemma 4.1 Let

and

be two pairs given

by Prematch-I such that two paths constructed for

and

share a node. Then the algorithm Parallel-

Routing can always find fault-free paths for

and

, in which no nodes are used by other paths.

PROOF. We assume that we will search a fault-free and

unused path for each pair given by Prematch-I in order of

cyclic permutations as given in Procedure-I in the follow-

ing discussion. For example,

,

,

,

.

Suppose all fault-free paths constructed for

in-

clude used nodes, and one of fault-free path is picked for

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, which includes a node used for

, .

Then we want to show that we can always find a fault-free

unused path for

.

Since all fault-free paths constructed for

are

used, all unused paths must be blocked by faulty nodes. By

lemma 3.1, a node of form

can be identical to

a node of form

. Thus, all paths leaving with

from

are unused, and there are

such paths. Remain paths of form

,

are used or faulty. Since a used path means

that unused paths previously searched before it also should

be blocked by faulty nodes, when we search new path for

among paths unused by other pairs, there exists

at least one unused fault-free path which can be used for

. From the above discussion, we can easily show

that for a pair

, , there exist at least one unused

fault-free path between them. Therefore, step 1.3 of the

algorithm Parallel-Routing occurs once during the whole

execution.

A similar analysis shows that step 2.2 of the algorithm

Parallel-Routing can always construct a new fault-free

path without nodes used by other paths.

Lemma 4.2 Let

,

and

,

, be edge pairs given by Prematch-III such that

two paths constructed for them share a node. Then the al-

gorithm Parallel-Routing always find fault-free paths be-

tween them, in which no nodes are used by other paths.

We summarize all these discussions in the following the-orem.

Theorem 4.3 If the hypercube network

has at most

faulty nodes, then for each pair of non-faulty nodes

and in

, in time the algorithm Parallel-Routing

constructs

node-disjoint fault-free

paths of length bounded by from to .

PROOF. We first discuss the length of

,

node-disjoint fault-free paths. It is easy to see

that paths constructed by Procedure-I is length of at most

. If paths are constructed by Procedure-II or

Procedure-III, the length is still at most be-

cause all paths constructed by Procedure-II or Procedure-

III are constructedbasedon Procedure-I, only flipping first

or last two indices in paths.

We now discuss the time complexity the algorithm

Parallel-Routing.

For each pair given by Prematch-I, a path is constructed

by the algorithm by searching a proper path in a set of paths

1010

1000

1011

1001 v = 0000

0110

0111

0101

0001

0100

0010 0011

1100 1101

4

11110

u = 1111

3

2

Figure 4. Paths constructed by Parallel-

Routing between and in

between them, which takes time

, where

is the number of faulty nodes in the set of paths for the pair

. If we find a fault-free and unused path of the form

for a pair

, , then mark a node

as an used node. In such a way, we can detect usedpaths in time

since thereare at most

used paths for

. If all fault-free paths for the pair

include

used nodes, we pick any fault-free path for

, and

for the pair

that uses a node on

, find a new path.

As we have discussed in previous lemma, this happens once

during the whole execution. Thus, the time complexity is

bounded by

since the

number

is bounded by .

If for a pair

given by Prematch-I, all possible

paths are blocked by faulty nodes, we simply ignore all

paths constructed for other pairs

, , and ap-

ply Procedure-II. Thus, it takes additional

time to

construct paths for pairs given by Prematch-II.

For pairs given by Prematch-III, paths are constructed

in the similar way to construct paths for pairs given by

Prematch-I. Thus, without detail explanations, we con-

clude that the time complexity for constructing paths be-

tween non-faulty neighbors of

and

is bounded by

. Figure 4 shows paths constructed by Parallel-

Routing between and in

.

5 Conclusion

Network strong fault tolerance is a natural extension of

the study of network fault tolerance and network parallel

routing. In particular, it studies the fault tolerance of large

size networks with faulty nodes. In this paper, we have

studied the strong fault tolerance of the popular hypercube

networks, and shown that hypercube networks are strong

fault tolerant. We developed an time algorithm that

for two given non-faulty nodes and in a -dimensional

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hypercube

with at most faulty nodes, constructs

node-disjoint fault-free paths from

to

such that the length of the paths is bounded by

. The time complexity of our algorithm is opti-

mal since each path from to may have length as large as

, and there can be as many as

node-disjoint paths from

to . Thus, even printing these paths should take time

. The length of the paths constructed by our algo-

rithm is also optimal, as we can construct pairs of nodes

and in the hypercube

with faulty nodes for which

any set of parallel paths connecting and has at least

one path of length

. Finally, our algorithm

does not require prior knowledge of the failures.

Strong fault tolerance for networks with bounded de-

gree, such as ring networks, mesh networks, and butterfly

networks, are relatively easier. On the other hand, strong

fault tolerance for unbounded degree networks, such as net-

works based on Cayley graphs, seems much more difficult.

The hypercube networks and the star networks are the first

two such classes of networks whose strongly faulty toler-

ant have been proved. For star networks, the strong fault

tolerance was proved based on the orthogonal partition of

the star networks, while for hypercube networks, the strong

fault tolerance was proved by careful pre-matching of the

neighbors of the source and destination nodes. It will be

interesting to study the strong fault tolerance of other hier-

archical networks with unbounded degree.

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