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IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 9, NO. 8, AUGUST 1998 803

Improved Compressions ofCube-Connected Cycles Networks

Ralf Klasing

Abstract—We present a new technique for the embedding of large cube-connected cycles networks (CCC) into smaller ones, aproblem that arises when algorithms designed for an architecture of an ideal size are to be executed on an existing architecture of afixed size. Using the new embedding strategy, we show that the CCC of dimension l can be embedded into the CCC of dimension kwith dilation 1 and optimum load for any k, l ¶ N, k � 8, such 5

34 3

3 22, 2 3+ < ≤ = +

⋅c ck k k

lk

k , thus improving known results. Our

embedding technique also leads to improved dilation-1 embeddings in the case 32

53< ≤ +l

k ck .

Index Terms—Parallel computations, parallel architectures, interconnection networks, graph embedding, network simulation, cube-connected cycles network.

——————————��F��——————————

1 INTRODUCTION

VER the past few years, a lot of research has been donein the field of interconnection networks for parallel

computer architectures (for an overview, cf. [19]). Much ofthe work has been focused on the capability of certain net-works to simulate other network or algorithm structures inorder to execute parallel algorithms of a special structureefficiently on different processor networks (see, e.g., [5],[17], [25]). One problem that is of specific interest in thiscontext is that many existing algorithms are designed forarbitrarily large networks (see, e.g., [19]), whereas, in prac-tice, the processor network will be fixed and of smaller size.Thus, the larger network must be simulated in an efficientway on the smaller target network. There is an enormousamount of literature on this problem (see, e.g., [3], [8], [14],[15], [21], [23], [24], [26], [30]).

Customarily, the simulation problem is formalized as theembedding problem of one graph in another (for a formaldefinition of the embedding problem, see Section 2). The“quality” of an embedding is measured by the parametersload, dilation, and congestion. The importance of the differentparameters becomes apparent through the following result.

PROPOSITION 1 [20]. If there is an embedding of G into H withload ,, dilation d, and congestion c, then there is a simula-tion of G by H with slowdown O(, + d+ c).

As a consequence, the load ,, dilation d, and congestion chave been investigated for embeddings between manycommon network structures like hypercubes, binary trees,meshes, shuffle-exchange networks, deBruijn networks,cube-connected cycles, butterfly networks, etc. Most ofthe work was done on one-to-one embeddings (for anoverview, see, e.g., [25], [29]), but results on many-to-one

embeddings can also be found (see, e.g., [2], [6], [7], [9],[12], [13], [16], [18], [22], [26], [27]). In this paper, we focuson many-to-one embeddings of the cube-connected cyclesnetwork (CCC). The CCC was introduced as a network forparallel processing in [28]. It has fixed degree, small di-ameter, and good routing capabilities [19]. It can executethe important class of normal hypercube algorithms veryefficiently (see, e.g., [19]). In addition, there is also a strongstructural relationship to the deBruijn, shuffle-exchange,and butterfly networks [1], [10]. Hence, the efficient imple-mentation of algorithms on CCC networks (of fixed size) isof importance. According to Proposition 1, one way of exe-cuting algorithms designed for a CCC network of arbitrarysize efficiently on a CCC network of realistic (fixed) size, is tofind embeddings of large CCCs into small CCCs minimizingthe parameters load, dilation, and congestion. In this paper, wefocus on load and dilation. Using our embedding strategy,many important algorithms for large CCCs can be imple-mented very efficiently on a CCC network of realistic size.

Many-to-one embeddings of the CCC network have beeninvestigated in [2], [6], [12], [16], [27]. In [6], [12], [27], em-beddings with optimum dilation and load are presented inthe case of embedding CCCs of dimension l into k wherek|l. The authors also restrict themselves to special kinds ofembeddings of a very regular structure, like coverings [6],homogeneous emulations [12], and homomorphisms [27].Because of the very restricted nature, Bodlaender [6] andPeine [27] are also able to classify their embeddings com-pletely. In [2], a general procedure is described for mappingparallel algorithms into parallel architectures. This proce-dure is applied to the CCC network achieving dilation 1,but very high load. Also, only special kinds of embeddings,so-called contractions, are considered. In [16], the embed-ding problem for CCCs is investigated, taking into accountgeneral embedding functions and any possible networkdimension. More precisely, it is proved that the cube-connected cycles network of dimension l, CCC(l), can be em-bedded into CCC(k), l > k, with

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•� The author is with the Department of Computer Science, University ofWarwick, Coventry CV4 7AL, England.�E-mail: [email protected].

Manuscript received 20 May 1997.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number 105085.

O

804 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 9, NO. 8, AUGUST 1998

1)�dilation 2 and optimum load lk

l k⋅ −2 ,

2)�dilation 1 and load

lk

lk

pp p

pp

lk

pp

l k

l k

⋅�

"## ≥

−⋅

�

"## ∈

−− < ≤

−

%&KK

'KK

−

−

2 2

2 12 2 3

2 31

2 1

for

for such that

,

, , .K< A

In this paper, we present a new technique for the em-bedding of large cube-connected cycles networks intosmaller ones. Using the new embedding strategy, we show:

Let k, l ¶ N, k � 8, such that 53 2+ < ≤ck

lk , ck

kk= +

⋅4 3

3 22 3 .

Then, there is a dilation 1 embedding of CCC(l) into CCC(k)

with load lk

l k⋅ −2 .

This is optimal, and improves the results from [16]. Ourembedding technique also leads to improved dilation 1embeddings in the case 3

253< ≤ +l

k kc .The general strategy of the embeddings is the same as in

[16], namely to map 2l-k cycles in CCC(l) of length l ontoone cycle in CCC(k) of length k and to allocate the nodes ofthe guest cycles in as balanced a manner as possible on thehost cycle. But, in order to improve the results from [16], acompletely different way of allocating the guest nodes onthe host cycle is introduced.

The paper is organized as follows. Section 2 contains thedefinitions of the terms used in the paper. Section 3 pres-ents the new embedding strategy. Section 4 presents thederived results. The conclusion gives an outlook on furtherconsequences of the new embedding technique.

2 DEFINITIONS

(Most of the terminology is taken from [19], [25].) For anygraph G = (V, E), let V(G) = V denote the set of vertices of G,and E(G) = E denote the set of edges of G. Let a denote the

binary complement of a ¶ {0, 1}. For a = a0a1...am-1¶ {0, 1}m,let α( )i a a a a ai i i m= − + −0 1 1 1K K .

2.1 Cube-Connected Cycles NetworkThe (wrapped) cube-connected cycles network of dimension m,denoted by CCC(m), has vertex-set Vm = {0, 1, ..., m - 1}� {0, 1}m, where {0, 1}m denotes the set of length-m binarystrings. For each vertex v = (i, a) ¶ Vm, i ¶ {0, 1, ..., m - 1},a�� ¶ {0, 1}m, we call i the level and a the position-within-level(PWL) string of v. The edges of CCC(m) are of two types: For

each i ¶ {0, 1, ..., m - 1} and each a = a0a1...am-1 ¶ {0, 1}m, thevertex (i, a) on level i of CCC(m) is connected

•� by a cycle-edge with vertex ((i + 1) mod m, a) on level(i + 1) mod m and

•� by a cross-edge with vertex (i, a(i)) on level i.

For each a ¶ {0, 1}m, the cycle

(0, a) � (1, a) � ¤ � (m - 1, a) � (0, a)

of length m will be denoted by Ca(m) or Ca.

CCC(m) has m2m nodes, 3m2m-1 edges, and degree 3. Anillustration of CCC(3) is shown in Fig. 1.

2.2 Graph EmbeddingsLet G and H be finite undirected graphs. An embedding of Ginto H is a mapping f from the nodes of G to the nodes of H.G is called the guest graph and H is called the host graph ofthe embedding f. The load of the embedding f is the maxi-mum number of vertices of the guest graph G that aremapped to the same host graph vertex. (The optimum loadachievable is the ratio Ñ|V(G)|/|V(H)|á of the number ofnodes in G and H.) The dilation of the embedding f is themaximum distance in the host between the images of adja-cent guest nodes. A routing is a mapping r of G’s edges topaths in H, r(v1, v2) = a path from f(v1) to f(v2) in H. The con-gestion of the embedding f is the maximum number ofedges that are routed through a single edge of H.

2.3 Lexicographic OrderingsLet Lex : {0, ¤, m - 1} � {0, 1}n � N0, Lex(i, a0 ¤ an-1) = i2n +a02

n-1 + a12n-2 + ¤ + an-12

0. Then, the lexicographic order on{0, 1, ¤, m - 1} � {0, 1}n is defined by

i j i j, , , ,α β α β0 5 1 6 0 5 1 6< ⇔ <Lex Lex

and the lexicographic distance between (i, a) and (j, b) is de-fined as |Lex(i, a) - Lex(j, b)|.

2.4 Balanced AllocationsLet a1, b1, a2, b2 ¶ N0 such that b1 � a1, b2 � a2, b1 - a1 � b2 - a2.Let r ¶ N. A function

d : {a1, a1 + 1, ¤, b1} � {0, 1}r � {a2, a2 + 1, ¤, b2}

is called a balanced allocation of {a1, ¤, b1} � {0, 1}r among {a2,¤, b2} according to the lexicographic order on {a1, ¤, b1} � {0, 1}r

if d satisfies the following properties:

•� d(a1, 0r) = a2, d(b1, 1

r) = b2,•� d is monotonic nondecreasing in the lexicographic or-

dering of the arguments (i.e., d(i, b) � d(i�, b �), if (i, b) �

Fig. 1. The cube-connected cycles CCC(3).

KLASING: IMPROVED COMPRESSIONS OF CUBE-CONNECTED CYCLES NETWORKS 805

(i�, b�) according to the lexicographic order on {a1, ¤,b1} � {0, 1}r),

•� b ab a

r b ab a

rd j1 1

2 2

1 1

2 2

11

1 112 1 2− +

− +− − +

− +⋅ − ≤ ≤ ⋅( ) for all j ¶ {a2,

¤, b2}.

(Note that such an allocation function d can always be con-structed for the parameters a1, b1, a2, b2, r as above.)

3 THE GENERAL EMBEDDING STRATEGY

The basic idea of the embeddings presented here is to map

2l-k cycles C C Cl kα α α1 2 2

, , ,K−

in CCC(l) of length l onto one

cycle Cb of length k in CCC(k) and to allocate the l ¿ 2l-k nodes

of C Cl kα α1 2

, ,K−

appropriately among the k nodes of Cb .

FORMAL CONSTRUCTION:

Consider numbers p(0), p(1), ¤, p(k - 1), where each p(i)¶ {0, 1, ¤, l - 1}, and each p(i) < p(i + 1). Let

π π π

π π π

0 1 1 0 1 1

0 1 1

0 5 0 5 1 6 < A0 5 0 5 1 6< A

, , , , , , \

, , ,

K K

K

l k l

k

− − ∈ −

−

such that π π π( ) ( ) ( )0 1 1< < < − −K l k . (Note that

π π π π π π0 1 1 0 1 10 5 0 5 1 6< A 0 5 0 5 1 6< A, , , & , , ,K Kk l k− ∪ − −= −0 1 1, , ,K l< A .)

Let ap(0), ap(1), ¤, ap(k-1) ¶ {0, 1}. The cycles

C a a aa a a l kl0 1 1 0 1 1 0 1K K− − − ∈π π π0 5 0 5 1 5 < AJ L, , , ,

of CCC(l) are mapped onto the cycle Ca a a kπ π π0 1 10 5 0 5 1 6K − in

CCC(k) such that the nodes 0, 1, ¤, l - 1 of each Ca a al0 1 1K −

are allocated appropriately among the nodes of Ca a a kπ π π( ) ( ) ( )0 1 1K −.

The exact allocation of the nodes of

C a a aa a a l kl0 1 1 0 1 1 0 1K K− − − ∈π π π0 5 0 5 1 5 < AJ L, , , ,

on Ca a a kπ π π( ) ( ) ( )0 1 1K − is determined by an allocation function

d : {0, 1, ¤, l - 1} � {0, 1}l-k � {0, 1, ¤, k - 1}

which specifies, for each node number ¶ {0, 1, ¤, l - 1} onthe guest cycle Ca a al0 1 1K −

and each cycle index

a a a l kπ π π0 1 10 5 0 5 1 5K − − ,

the position on the host cycle Ca a a kπ π π( ) ( )0 1 10 5K −. (On each host

cycle Ca a a kπ π π0 1 10 5 0 5 1 6K −, a a a kπ π π0 1 1 0 10 5 0 5 1 5, , , { , }K − ∈ , the same

allocation function is used.) Formally, the embedding f :V(CCC(l)) � V(CCC(k)) is of the form

f i a a a d i a a a a

i l a a a

l l k k

ll

, : , ,

, , .

0 1 1 0 1 0 1

0 1 10 1 0 1

K K K

K

− − − −

−

= �� ≤ ≤ − ∈

2 7 4 9< A

0 5 1 5 0 5 1 5π π π π

for all

The load of f is determined by the allocation function d.Therefore, d should allocate the guest nodes in as balanceda manner as possible on each host cycle. In the sequel, dwill be chosen such that

d(p(i), b) = i for all 0 � i � k - 1, b ¶ {0, 1}l-k.

This guarantees that all the cross-edges

(i, a) � (i, a(i)), i ¶ {p(0), p(1), ¤, p(k - 1)}

of CCC(l) are mapped onto a corresponding cross-edge inCCC(k). All the other edges of CCC(l) are mapped onto a

path on a single cycle Cb in CCC(k). So, in this case, the di-lation is directly dependent on the allocation d of the guestnodes on the host cycle and stands partly in contrast to thedesired balancedness of the allocation as explained above.

For low dilation, the values of p(0), p(1), ¤, p(k - 1)should be allocated in a relatively balanced manner among

0, 1, ¤, l - 1, and the nodes (i, a0a1 ¤ al-1) and (j, b0b1 ¤ bl-1)of the cycles C C C

l kα α α1 2 2, , ,K

− of CCC(l) with a small

lexicographical distance between ( , )i a a l kπ π0 10 5 1 5K − − and

( , )j b b l kπ π0 10 5 1 5K − − should be mapped close together on the

cycle Cb in CCC(k).In [16], for 1 < l/k � 2, it was shown that the values of

p(0), p(1), ¤, p(k - 1) can be specified such that the followingholds:

1)�p(i + 1) - p(i) � 2 for all 0 � i < k - 1.2)�The nodes

π π π πi a a a a a al l k0 52 7 < AJ L0 5 0 5 1 5, , , , ,0 1 1 0 1 1 0 1K K− − − ∈

are mapped onto (i, ap(0)ap(1)¤ap(k-1)) for 0 � i � k - 1,ap(0), ap(1), ¤, ap(k-1) ¶ {0, 1}.

3)�The nodes

π π π πi a a a i l k a a al l k0 52 7J< AB

0 5 0 5 1 5, , , , ,

,

0 1 1 0 1 10 1

0 1

K K− − −≤ ≤ − −

∈

can be allocated in a balanced manner in certain sec-tions of the host cycle

C a a aa a a kkπ π π π π π0 1 1 0 1 1 0 10 5 0 5 1 6 0 5 0 5 1 5 < AK K− − ∈, , , , , ,

while maintaining dilation 1 at the same time.

Here, for 53

4 33 2

2 2 3+ < ≤ = +⋅

c cklk k

kk, , we show that p(0),

p(1), ¤, p(k - 1) can be specified such that the followingholds:

1)�p(i + 1) - p(i) � 3 for all 0 � i < k - 1.2)�The nodes

π π π πi a a a a a al l k0 52 7 < AJ L0 5 0 5 1 5, , , , ,0 1 1 0 1 1 0 1K K− − − ∈

are mapped onto (i, ap(0)ap(1)¤ap(k-1)) for 0 � i � k - 1,ap(0), ap(1), ¤, ap(k-1) ¶ {0, 1}.

3)�The nodes

π π π πi a a a i l k a a al l k0 52 7J< AB

0 5 0 5 1 5, , , , ,

,

0 1 1 0 1 10 1

0 1

K K− − −≤ ≤ − −

∈

can be allocated in a balanced manner on the com-plete host cycle

C a a aa a a kkπ π π π π π0 1 1 0 1 1 0 10 5 0 5 1 6 0 5 0 5 1 5 < AK K− − ∈, , , , , ,

while maintaining dilation 1 at the same time.


The main new technical contribution will be to show thatthe guest nodes

π π

π π π

i a a a i a a a

a a a

l l

l k

0 52 7 0 52 7>< AL0 5 0 5 1 5

+ +

∈

− −

− −

1 2

0 1

0 1 1 0 1 1

0 1 1

, , ,

, , , ,

K K

K

can be allocated in an appropriate way among the hostnodes {(j, ap(0)ap(1)¤ap(k-1)) | j ¶ {i - 1, i, i + 1, i + 2}} for 0 � i< k - 1 such that p(i + 1) - p(i) = 3, while maintaining dila-tion 1 at the same time.

4 IMPROVED DILATION 1 EMBEDDING OF THE CCCTHEOREM 1. Let k, l ¶ N, k � 8, such that 5

3 2+ < ≤cklk ,

ckk

k= +⋅4 3

3 22 3 . Then, there is a dilation 1 embedding of

CCC(l) into CCC(k) with load lk

l k⋅ −2 .

PROOF. (A) l - k evenWe show that the construction of Section 3 can be

adapted to yield an embedding of CCC(l) into CCC(k)

with dilation 1 and load lk

l k⋅ −2 .

For this, we specify the allocation d and the indicesp(i) for the embedding f in the construction of Section 3.

For 0 12≤ ≤ −−i l k , let

h ii ll k

kl k0 5 : =

⋅− −

�

"## +−

2 3

21.

(Then, h h ll k( ) ,0 1 1 32= − = −−3 8 ,.) For 0 � i � l - k - 1,

let

π i

hi

i

hi

i0 5 : =

��

��

!

#$#

��

�� +

%&KK

'KK

2

2 1

if is even,

if is odd.

Let

π π π π π π( ), ( ), , ( ) { , , , } \ { ( ), ( ), , ( )}0 1 1 0 1 1 0 1 1K K Kk l l k− ∈ − − −

such that p(0) < p(1) < ¤ < p(k - 1). (Note that

π π π π π π0 1 1 0 1 10 5 0 5 1 6< A 0 5 0 5 1 6< A, , , & , , ,K Kk l k− ∪ − −= −0 1 1, , ,K l< A .)

For the time being, we only construct the allocationd : {0, 1, ¤, l - 1} � {0, 1}l-k � {0, 1, ¤, k - 1} partially,namely we specify d(i, b) for i ¶ {p(0), p(1), ¤, p(k - 1)}.Let

d(p(i), b) := i for all 0 � i � k - 1, b ¶ {0, 1}l-k. (*)

(Later on, d(i, b) is specified for

i l k∈ − −{ , , , }π π π0 1 10 5 0 5 1 6K .

For the moment, d(i, b) may have an arbitrary valuefor i l k∈ − −{ , , , }π π π0 1 10 5 0 5 1 6K .)

Now, the embedding f of CCC(l) into CCC(k) is de-fined as in the construction of Section 3:


i l a a a

l l k k

ll

, : , ,

, , .

0 1 1 0 1 0 1

0 1 10 1 0 1

K K K

K

− − − −

−

= �� ≤ ≤ − ∈

2 7 4 9< A

0 5 1 5 0 5 1 5π π π π

for all

Note that (*) guarantees that all the cross-edges

(i, a) � (i, a(i)), i ¶ {p(0), p(1), ¤, p(k - 1)},

of CCC(l) are mapped onto a corresponding cross-edge in CCC(k) (see Claim A1 below).

Now, we construct d(i, b) for

i l k∈ − −{ , , , }π π π0 1 10 5 0 5 1 6K .

Let ap(0), ap(1), ¤, ap(k-1) ¶ {0, 1}. For the time being, weallocate the guest nodes

i a a a i l k

a a a

l

l k

, , , , ,

, , , ,

0 1 1

0 1 1

0 1 1

0 1

K K

K

−

− −

∈ − −

∈

2 7 0 5 0 5 1 6< A>< AL0 5 0 5 1 5

π π π

π π π

in a balanced manner on the host cycle Ca a a kπ π π0 1 10 5 0 5 1 6K −

according to the lexicographical order on {0, 1, ¤, l - 1}

� {0, 1}l-k, i.e., we use an allocation function~

: , , , ,

, , ,

d l k

k

l kπ π π0 1 1 0 1

0 1 1

0 5 0 5 1 6< A < A< A

K

K

− − × →

−

−

such that

•�~

,d l kπ 0 0 00 54 9− = , ~

,d l k kl kπ − − = −−1 1 11 64 9 ,

•�~d is monotonic nondecreasing in the lexicographicordering of the arguments (i.e.,

~,

~,d i d iβ β1 6 1 6≤ ′ ′ if

(i, b) � (i�, b �) according to the lexicographical order

on {0, 1, ¤, l - 1} � {0, 1}l-k),

•� l kk

l k l kk

l kd j− − − − −⋅ − ≤ ≤ ⋅2 1 21~ 1 6 for all j = 0, 1,

¤, k - 1.

(At this point, we are not concerned with the obtaineddilation. We will see later on that the allocation

~d can

be changed into an allocation

d l k kl k

: , , , , , , ,π π π0 1 1 0 1 0 1 10 5 0 5 1 6< A < A < AK K− − × → −−

which guarantees dilation 1, while maintaining thebalancedness of the allocation.)

Let r(i) := h(i) - 2i - 1 for all 0 12≤ ≤ −−i l k . Then,according to Claim A2 below,

1)�~

( ( ) , ) ( )d h i r i− =1 β for all

0 2 1 0 1≤ ≤−

− ∈ −il k l k, { , }β ,

2)�~

( ( ) , ) ( )d h i r i+ = +2 1β for all

0 2 1 0 1≤ ≤−

− ∈ −il k l k, { , }β .

Also, according to Claim A3 below,

1)� r i d h i d h i r i( )~

( ( ), )~

( ( ) , ) ( )− ≤ ≤ + ≤ +1 1 2β βfor all 0 1 0 12≤ ≤ − ∈− −i l k l k, { , }β ,


2)�~

( ( ) ) {( ( ), ), ( ( ) , ) { , } }d r i h i h i l k− −− ∩ + ∈1 1 1 0 1β β β

≤ ⋅ − ≤ ≤ −− − −l kk

l k l ki2 1 0 12 for all ,

3)�~

( ( ) ) {( ( ), ), ( ( ) , ) { , } }d r i h i h i l k− −+ ∩ + ∈1 2 1 0 1β β β

≤ ⋅ − ≤ ≤ −− − −l kk

l k l ki2 1 0 12 for all .

(As

h i h i k

h i h i l k

0 5 0 5 0 5 0 5 1 6< A0 5 0 5 0 5 0 5 1 6< A

− + ∈ −

+ ∈ − −

1 2 0 1 1

1 0 1 1

, , , , ,

, , , , ,

π π π

π π π

K

K

the dilation of the embedding f (using the allocation ~d

for d i i l k( , ), { ( ), ( ), , ( )}β π π π∈ − −0 1 1K ) would be 2.)

Then, according to Claim A4 below, ~d can be

changed to an allocation d such that:

1)�Let 0 12≤ ≤ −−i l k . For 1 � j � 4, let

n d r i j h i h i

n d r i j h i h i

jl k

jl k

: , , , , ,

~ :~

, , , , .

= − + ∩ + ∈

= − + ∩ + ∈

− −

− −

1

1

2 1 0 1

2 1 0 1

0 52 7 0 52 7 0 52 7 < AJ L0 52 7 0 52 7 0 52 7 < AJ L

β β β

β β β

Then,

n n

n n n n

n n n

n n n n

n n n

n n

1 1

2 2 1 1

2 2 1

3 3 4 4

3 3 4

4 4

1 0

0

1 0

0

=

≤ + >

= =

≤ + >

= ==

~ ,

max ~ , ~ ~ ,~ ~ ,

max ~ , ~ ~ ,~ ~ ,~ .

< A

< A

if

if

if

if

2)�For 0 1 0 12 0 1 1≤ ≤ − = ∈−− −

−i b b bl kl k

l k, { , }β π π π0 5 0 5 1 5K ,

r i d h i r i

r i d h i r i

d h i d h i

d h i b b

d h i b b b b b

d h i b b

d h i b

l k

i i i l k

l k

0 5 0 52 7 0 50 5 0 52 7 0 5

0 52 7 0 52 70 54 9

0 54 90 54 9

0 5

0 5 1 5

0 5 0 5 0 5 0 5 1 5

0 5 1 5

− ≤ ≤ +

≤ + ≤ +

+ − ≤

− ≤

+

− +

− −

− + − −

− −

1 1

1 2

1 1

1

1

1

0 1

0 2 1 2 2 1 1

0 1

, ,

, ,

, , ,

,

, ,

,

,

β

β

β β

π π

π π π π π

π π

π

K

K K

K

0 2 2 1 2 2 1 10 5 0 5 0 5 0 5 1 54 9K Kb b b bi i i l kπ π π π+ + − − ≤ .

It follows that the final embedding f (using the al-

location d) has dilation 1 and load lk

l k⋅ −2 . o

CLAIM A1. The cross-edge (i, a) � (i, a(i)), i = p(m), 0 � m �k - 1, of CCC(l) is mapped by f onto a cross-edge in CCC(k).

PROOF OF CLAIM A1. f maps (i, a) = (i, a0a1¤al-1) onto

d i a a a al k k, ,π π π π0 1 0 10 5 1 5 0 5 1 54 9K K− − −��

and (i, a(i)) onto

d i a a a a a a al k m m m k, ,π π π π π π π0 1 0 1 1 10 5 1 5 0 5 0 5 0 5 0 5 1 54 9K K K− − − + −�� .

From (*),

d i a a d m a a ml k l k, ,π π π ππ0 1 0 10 5 1 5 0 5 1 54 9 0 54 9K K− − − −= = .

Hence, there is a cross-edge in CCC(k) between thetwo image nodes of (i, a) and (i, a(i)). o

CLAIM A2.

1)�~

( ( ) , ) ( )d h i r i− =1 β for all 0 1 0 12≤ ≤ − ∈− −i l k l k, { , }β ,

2)�~

( ( ) , ) ( )d h i r i+ = +2 1β for all

0 1 0 12≤ ≤ − ∈− −i l k l k, { , }β .

PROOF OF CLAIM A2.

PROOF OF 1. First, notice that h(i) - 1 = p(r) for some r ¶ {0, 1,¤, k - 1}. Hence, according to (*), it suffices to showthat

r(i) = r.

For this purpose, observe that

r h i k

h i h i l k

h i i

r i

= − ∩ − −

= − − − ∩ − −

= − −

=

0 1 1 0 1 1 1

1 0 1 1 0 1 1

1 2

, , , , ( ),

, , , , , ,

.

K K

K K

0 5< A 0 5 1 6< A0 5 0 5< A 0 5 0 5 1 6< A0 50 5

π π π

π π π

PROOF OF 2. First, notice that h(i) + 2 = p(r) for some r ¶ {0, 1,¤, k - 1}. Hence, according to (*), it suffices to showthat

r(i) + 1 = r.

For this purpose, observe that

r h i k

h i h i l k

h i i

h i i

r i

= + ∩ − −

= + − + ∩ − −

= + − += −= +

0 1 2 0 1 1 1

2 0 1 2 0 1 1

2 2 2

2

1

, , , , , ,

, , , , , ,

.

K K

K K

0 5< A 0 5 0 5 1 6< A0 5 0 5< A 0 5 0 5 1 6< A0 5 0 50 50 5

π π π

π π π

o

CLAIM A3.

1)� r i d h i( )~

( ( ), )− ≤1 β for all 0 1 0 12≤ ≤ − ∈− −i l k l k, { , }β ,

~, , , ,

,

d r i h i h i

l kk for all i

l k

l k

l k

− −

−

− ∩ + ∈ ≤

−⋅

�

"## − ≤ ≤

−−

1 1 1 0 1

2 1 0 2 1

0 52 7 0 52 7 0 52 7 < AJ Lβ β β

2)�~

( ( ) , ) ( )d h i r i+ ≤ +1 2β for all

0 2 1 0 1≤ ≤−

− ∈ −il k l k, { , }β ,

~, , , ,

.

d r i h i h i

l kk for all i

l k

l k

l k

− −

−

+ ∩ + ∈ ≤

−⋅

�

"## − ≤ ≤

−−

1 2 1 0 1

2 1 0 2 1

0 52 7 0 52 7 0 52 7 < AJ Lβ β β


PROOF OF CLAIM A3.

PROOF OF 1. The number of nodes in

π β βi i l kl k0 52 7 < AJ L, , ,0 1 0 1≤ ≤ − − ∈ −

between (h(i), 0l-k) and ( ( ), )π l k l k− − −1 1 in

lexicographical order is (l - k - 2i) ¿ 2l-k. If this numberdoes not exceed the minimal capacity

k r il k

kl k− + ⋅

−⋅

�

"## −

��

��

−0 52 71 2 1

of the nodes between r(i) - 1 and k - 1, then 1) fol-lows. Hence, we have to check that

l k i k r il k

kl k l k− − ⋅ ≤ − + ⋅

−⋅

�

"## −

��

��

− −2 2 1 2 11 6 0 52 7 .

This is true, because

k r il k

k

k h i il k

k

ki ll k

ki

l kk

ki ll k

ki

l kk

l k

l k

l kl k

l kl k

− + ⋅−

⋅�

"## −

��

��

= − + + ⋅−

⋅�

"## −

��

��

= −⋅− −

�

"## + +

��

�� ⋅

−⋅

�

"## −

��

��

≥ −⋅− −

��

�� +

��

�� ⋅

−⋅ −

��

��

=

−

−

−−

−−

0 52 7

0 52 7

1 2 1

2 2 2 1

2 3

22 1 2 1

2 3

22 2 1

ki kl k

k l kk

l k i l k kk

l k ik

l k i k kk k

l k i k

l k i

l kl k

l k

c k

l k

l k k

l kk k

l k

l

k

−⋅− +

��

�� ⋅

−⋅ −

��

��

= − − ⋅ + ⋅ − − + − ⋅ −

≥ − − ⋅ + + ⋅+

−

≥ − − ⋅ +

≥ − − ⋅

−−

−

≥ + ⋅ ≥

−

− ≥

−

≥−

−

2 3

22 1

2 2 3 23

2

2 24 3

2

3

2

2 2

2 2

23 0 2 3

2 3 2 3

0

1 6 1 6�

1 6

1 61 6

4 9123 124 34

1 2444 3444

k .

PROOF OF 2. The number of nodes in

π β βi i l kl k0 52 7 < AJ L, , ,0 1 0 1≤ ≤ − − ∈ −

between (h(0), 0l-k) and (h(i) + 1, 1l-k) inlexicographical order is (2i + 2) ¿ 2l-k. If this numberdoes not exceed the minimal capacity

r il k

kl k0 52 7+ ⋅

−⋅

�

"## −

��

��

−3 2 1

of the nodes between 0 and r(i) + 2, then 2) follows.Hence, we have to check that

2 2 2 3 2 1i r il k

kl k l k+ ⋅ ≤ + ⋅

−⋅

�

"## −

��

��

− −0 5 0 52 7 .

This is true, because

r il k

k

h i il k

k

i ll k

ki

l kk

i ll k

ki

l kk

il k

k

l k

l k

l kl k

l kl k

0 52 7

0 52 7

+ ⋅−

⋅�

"## −

��

��

= − + ⋅−

⋅�

"## −

��

��

=⋅− −

�

"## − +

��

�� ⋅

−⋅

�

"## −

��

��

≥⋅− − − +

��

�� ⋅

−⋅ −

��

��

= + ⋅−

−

−

−−

−−

≥

3 2 1

2 2 2 1

2 3

22 3 2 1

2 3

22 3 2 1

2 3

23 +

−

≤ ≤

−

≥

≤ −

−

−−

−

�

�

��

�

�

��⋅ − ⋅ − − − ⋅ + −

≥ + ⋅ ++

⋅

�

�

��

�

�

��

�

�

��

�

�

��⋅ − −

≥ + ⋅ ++

⋅��

��

��

�� ⋅ − −

= + ⋅

c

l k

k k

l k

k

k l k

l k

l kl k

l k

k

l kk

l k ik

ik

k

ik

k

i

123 123 124 34

124 34

2 3 23

23

2 323

4 3

3 22 4 3

2 323

4 3

3 22 4 3

2 2 2

0

2 3

2 3

1 6�

0 5 .o

CLAIM A4.

1)�Let 0 12≤ ≤ −−i l k . For 1 � j � 4, let

n d r i j h i h i

n d r i j h i h i

jl k

jl k

: , , , , ,

~ :~

, , , , .

= − + ∩ + ∈

= − + ∩ + ∈

− −

− −

1

1

2 1 0 1

2 1 0 1

0 52 7 0 52 7 0 52 7 < AJ L0 52 7 0 52 7 0 52 7 < AJ L

β β β

β β β

Then,

n n

n n n if n

n n if n

n n n if n

n n if n

n n

1 1

2 2 1 1

2 2 1

3 3 4 4

3 3 4

4 4

1 0

0

1 0

0

=

≤ + >

= =

≤ + >

= ==

~ ,

max ~ , ~ ~ ,~ ~ ,

max ~ , ~ ~ ,~ ~ ,~ .

< A

< A

2)�For 0 12≤ ≤ −−i l k ,

β π π π= ∈− −−

b b b l kl k

0 1 1 0 10 5 0 5 1 5 < AK , :

r i d h i r i

r i d h i r i

d h i d h i

d h i b b

d h i b b b b b

d h i b b

d h i b

l k

i i i l k

l k

0 5 0 52 7 0 50 5 0 52 7 0 5

0 52 7 0 52 70 54 9

0 54 90 54 9

0 5

0 5 1 5

0 5 0 5 0 5 0 5 1 5

0 5 1 5

− ≤ ≤ +

≤ + ≤ +

+ − ≤

− ≤

+

− +

− −

− + − −

− −

1 1

1 2

1 1

1

1

1

0 1

0 2 1 2 2 1 1

0 1

, ,

, ,

, , ,

,

, ,

,

,

β

β

β β

π π

π π π π π

π π

π

K

K K

K



PROOF OF CLAIM A4. Let n := l - k. d is constructed as follows:

1)�Consider the following lexicographical numberingon {0,1}n:

Lex

Lex

1

1

: , ,

.

0 1

2 2 2

2 2 2

2 2

0

0 1 1 01

12

2 12

2 22 1

2 32 2

12

2 11

20

< A2 7

n

nn n

in i

in i

in i

n

i i

b b b b b b

b b b

b b

→

= + + +

+ + + +

+ +

−− −

−−

+− −

+− −

−

+

N

K K

K

Now, define:

d h i r i n

d h i r i n

n

n

0 52 7 0 5 < A 1 60 52 7 0 5 < A 1 6

, , , ~ ,

, , , ~ .

β β β

β β β

= − ∈ ≤ ≤ −

+ = ∈ ≤ ≤ −

1 0 1 0 1

1 0 1 0 1

1

1

for all Lex

for all Lex

1

1

If ~n1 odd: d h i r i( ( ), ) ( )β = for all

β β∈ ={ , } , ( ) ~0 1 1n n Lex1 .

2)�Consider the following lexicographical numberingon {0,1}n:

Lex

Lex

2

2

: , ,

.

0 1

2 2 2

2 2 2

2 2

0

0 1 1 01

12

2 12

2 22 1

2 32 2

12

2 11

20

< A2 7

n

nn n

in i

in i

in i

n

i i

b b b b b b

b b b

b b

→

= + + +

+ + + +

+ +

−− −

−−

+− −

+− −

−

+

N

K K

K

Now, define:

d h i r i

n

d h i r i

n

n n n

n n n

0 52 7 0 5< A 1 6

0 52 7 0 5< A 1 6

,

, , ~ ,

,

, , ~ .

β

β β

β

β β

= +

∈ − ≤ ≤ −

+ = +

∈ − ≤ ≤ −

2

0 1 2 2 1

1 1

0 1 2 2 1

4

4

for all Lex

for all Lex

2

2

If ~n4 odd: d(h(i), b) = r(i) + 1 for all b ¶ {0, 1}n,

Lex2 β1 6 = − −2 14n n~ .

(Note that the definitions above are meaningful, be-cause if we consider the lexicographicalnumbering

Lex3(b0b1 ¤ b2i-1b2i+2 ¤ bn-1) = b02n-3 + b12

n-4 + ¤ +

b2i-12n-2i-2+ b2i+22

n-2i-3 + b2i+32n-2i-4 + ¤ + bn-12

0

on {0, 1}n-2, in the first case, we choose the first~n14 bitstrings and, in the second case, we choose

the last ~n44 bitstrings. To guarantee that these bit-

strings are different, we have to check that~ ~n n n1 44 4

22+ ≤ − . But, this is true for lk > 5

3 .)

3)�The nodes from {(h(i), b), (h(i) + 1, b) | b ¶ {0, 1}n}which have not been assigned a value for d in 1)and 2) are assigned arbitrary values from {r(i),r(i) + 1} such that the requirements for n2 and n3 arefulfilled.

From the definition of d, it follows immediately that dhas the claimed Properties 1 and 2. o

(B) l - k oddWe show that the construction of Section 3 can be

adapted to yield an embedding of CCC(l) into CCC(k)

with dilation 1 and load lk

l k⋅ −2 .

For this, we specify the allocation d and the indicesp(i) for the embedding f in the construction of Section 3.

For 0 112≤ ≤ −− −i l k , let

h ii ll k

kl k0 5 : =

⋅− −

�

"## +−

2 3

21.

(Then, h(0) = 1, h l ll k− − − ∈ − −12 1 6 53 8 { , }.) For 0 � i �

l - k - 2, let

π i

hi

i

hi

i0 5 : =

��

��

!

#$#

��

�� +

%&KK

'KK

2

2 1

if is even,

if is odd.

Let

π l k l− − = −1 21 6 : .

Let

π π ππ π π( ), ( ), , ( ) { , , , } \

{ ( ), ( ), , ( )}0 1 1 0 1 10 1 1

K K

K

k l

l k

− ∈ −− −

such that p(0) < p(1) < ¤ < p(k - 1). (Note that

{ ( ), ( ), , ( )} & { ( ), ( ), , ( )}π π π π π π0 1 1 0 1 1K Kk l k− ∪ − −= −{ , , , }0 1 1K l .)

For the time being, we only construct the allocationd : {0, 1, ¤, l - 1} � {0, 1}l-k � {0, 1, ¤, k - 1} partially,namely we specify d(i, b) for i ¶ {p(0), p(1), ¤, p(k - 1)}.Let

d(p(i), b) := i for all 0 � i � k - 1, b ¶ {0, 1}l-k. (*)

(Later on, d(i, b) is specified for

i l k∈ − −{ ( ), ( ), , ( )}π π π0 1 1K .

For the moment, d(i, b) may have an arbitrary valuefor i l k∈ − −{ ( ), ( ), , ( )}π π π0 1 1K .)

Now, the embedding f of CCC(l) into CCC(k) is de-fined as in the construction of Section 3:


i l a a a

l l k k

ll

, : , ,

, , .

0 1 1 0 1 0 1

0 1 10 1 0 1

K K K

K

− − − −

−

= �� ≤ ≤ − ∈

2 7 4 9< A

0 5 1 5 0 5 1 5π π π π

for all

Note that (*) guarantees that all the cross-edges

(i, a) � (i, a(i)), i ¶ {p(0), p(1), ¤, p(k - 1)},

of CCC(l) are mapped onto a corresponding cross-edge in CCC(k) (see Claim B1 below).

Now, we construct d(i, b) for

i l k∈ − −{ ( ), ( ), , ( )}π π π0 1 1K .

Let ap(0), ap(1), ¤, ap(k-1) ¶ {0, 1}. For the time being, weallocate the guest nodes


i a a a i l k

a a a

l

l k

, , , , ,

, , , ,

0 1 1

0 1 1

0 1 1

0 1

K K

K

−

− −

∈ − −

∈

2 7 0 5 0 5 1 6< A>< AL0 5 0 5 1 5

π π π

π π π

in a balanced manner on the host cycleCa a a kπ π π0 1 10 5 0 5 1 6K −

according to the lexicographical order on {0, 1, ¤, l - 1}

� {0, 1}l-k, i.e., we use an allocation function~

: , , , ,

, , ,

d l k

k

l kπ π π0 1 1 0 1

0 1 1

0 5 0 5 1 6< A < A< A

K

K

− − × →

−

−

such that

•�~

( ( ), )d l kπ 0 0 0− = , ~

( ( ), )d l k kl kπ − − = −−1 1 1,

•�~d is monotonic nondecreasing in the lexicographicordering of the arguments (i.e.,

~( , )

~( , )d i d iβ β≤ ′ ′ , if

(i, b) � (i�, b �) according to the lexicographical order

on {0, 1, ¤, l - 1} � {0, 1}l-k),

•� l kk

l k l kk

l kd j− − − − −⋅ − ≤ ≤ ⋅2 1 21~( ) for all j = 0, 1,

¤, k - 1.

(At this point, we are not concerned with the obtaineddilation. We will see later on that the allocation

~d can

be changed into an allocation

d l k kl k

: , , , , , , ,π π π0 1 1 0 1 0 1 10 5 0 5 1 6< A < A < AK K− − × → −−

which guarantees dilation 1, while maintaining thebalancedness of the allocation.)

Let r(i) := h(i) - 2i - 1 for all 0 112≤ ≤ −− −i l k . Then,

according to Claim B2 below,

1)�~

( ( ) , ) ( )d h i r i− =1 β for all

01

2 1 0 1≤ ≤− −

− ∈ −il k l k, { , }β ,

2)�~

( ( ) , ) ( )d h i r i+ = +2 1β for all

01

2 1 0 1≤ ≤− −

− ∈ −il k l k, { , }β ,

3)�~

( ( ) , )d l k kπ β− − − = −1 1 2 for all b ¶ {0, 1}l-k,

4)�~

( ( ) , )d l k kπ β− − + = −1 1 1 for all b ¶ {0, 1}l-k.

Also, according to Claim B3 below,

1)� r i d h i d h i r i( )~

( ( ), )~

( ( ) , ) ( )− ≤ ≤ + ≤ +1 1 2β β

for all 01

2 1 0 1≤ ≤− −

− ∈ −i

l k l k, ,β < A ,

2)�~

, , , ,

,

d r i h i h i

l kk i

l k

l k

l k

− −

−

− ∩ + ∈ ≤

−⋅

�

"## − ≤ ≤

− −−

1 1 1 0 1

2 1 01

2 1

0 52 7 0 52 7 0 52 7 < AJ Lβ β β

for all

3)�

~, , , ,

,

d r i h i h i

l kk i

l k

l k

l k

− −

−

+ ∩ + ∈ ≤

−⋅

�

"## − ≤ ≤

− −−

1 2 1 0 1

2 1 01

2 1

0 52 7 0 52 7 0 52 7 < AJ Lβ β β

for all

4)� k d l k k− ≤ − − ≤ −2 1 1~

( ( ), )π β for all b ¶ {0, 1}l-k.

(As

h i h i l k l k

k h i h i l k

l k

0 5 0 5 1 6 1 60 5 0 5 1 6< A 0 5 0 5 1 60 5 0 5 1 6< A

− + − − − − − + ∈

− + − − ∈

− −

1 2 1 1 1 1

0 1 1 1 1

0 1 1

, , ,

, , , , , ,

, , , ,

π π

π π π π

π π π

K

K

the dilation of the embedding f (using the allocation ~d

for d(i, b), i l k∈ − −{ ( ), ( ), , ( )}π π π0 1 1K ) would be 2.)

Then, according to Claim B4 below, ~d can be

changed to an allocation d such that:

1)�Let 0 112≤ ≤ −− −i l k . For 1 � j � 4, let

n d r i j h i h i

n d r i j h i h i

jl k

jl k

: , , , , ,

~ :~

, , , , .

= − + ∩ + ∈

= − + ∩ + ∈

− −

− −

1

1

2 1 0 1

2 1 0 1

0 52 7 0 52 7 0 52 7 < AJ L0 52 7 0 52 7 0 52 7 < AJ L

β β β

β β β

Then,

n n

n n n n

n n n

n n n n

n n n

n n

1 1

2 2 1 1

2 2 1

3 3 4 4

3 3 4

4 4

1 0

0

1 0

0

=

≤ + >

= =

≤ + >

= ==

~ ,

max ~ , ~ ~ ,~ ~ ,

max ~ , ~ ~ ,~ ~ ,~ .

< A

< A

if

if

if

if

2)�For 0 112≤ ≤ −− −i l k ,

β π π π= ∈− −−b b b l k

l k0 1 1 0 10 5 0 5 1 5K { , } :

r i d h i r i

r i d h i r i

d h i d h i

d h i b b

d h i b b b b b

d h i b b

d h i b

l k

i i i l k

l k

0 5 0 52 7 0 50 5 0 52 7 0 5

0 52 7 0 52 70 54 9

0 54 90 54 9

0 5

0 5 1 5

0 5 0 5 0 5 0 5 1 5

0 5 1 5

− ≤ ≤ +

≤ + ≤ +

+ − ≤

− ≤

+

− +

− −

− + − −

− −

1 1

1 2

1 1

1

1

1

0 1

0 2 1 2 2 1 1

0 1

, ,

, ,

, , ,

,

, ,

,

,

β

β

β β

π π

π π π π π

π π

π

K

K K

K


3)�For k - 2 � j � k - 1:

d j l k

d j l k

l k

l k

− −

− −

∩ − − ∈

= ∩ − − ∈

1

1

1 0 1

1 0 1

1 6 1 62 7 < AJ L1 6 1 62 7 < AJ L

π β β

π β β

, ,

~, , .

For b ¶ {0, 1}l-k:

k d l k k− ≤ − − ≤ −2 1 1π β1 62 7, .


It follows that the final embedding f (using the al-

location d) has dilation 1 and load lk

l k⋅ −2 . o

CLAIM B1. The cross-edge (i, a) � (i, a(i)), i = p(m), 0 � m �k - 1, of CCC(l) is mapped by f onto a cross-edge in CCC(k).

PROOF OF CLAIM B1. Exactly like the proof of Claim A1. o

CLAIM B2.

1)�~

( ( ) , ) ( )d h i r i− =1 β for all

01

2 1 0 1≤ ≤− −

− ∈ −i

l k l k, ,β < A ,

2)�~

( ( ) , ) ( )d h i r i+ = +2 1β for all

01

2 1 0 1≤ ≤− −

− ∈ −i

l k l k, ,β < A ,

3)�~

( ( ) , )d l k kπ β− − − = −1 1 2 for all b ¶ {0, 1}l-k,

4)�~

( ( ) , )d l k kπ β− − + = −1 1 1 for all b ¶ {0, 1}l-k.

PROOF OF CLAIM B2. The proof of 1) and 2) is exactly like theproof of Claim A2. 3) and 4) follow directly from thefact that

π π π π( ) ( ), ( ) ( )l k k l k k− − − = − − − + = −1 1 2 1 1 1

and from (*). o

CLAIM B3.

1)� r i d h i( )~

( ( ), )− ≤1 β

for all il k l k

01

2 1 0 1≤ ≤− −

− ∈ −, ,β < A ,

~, , , ,

,

d r i h i h i

l kk for all i

l k

l k

l k

− −

−

− ∩ + ∈ ≤

−⋅

�

"## − ≤ ≤

− −−

1 1 1 0 1

2 1 01

2 1

0 52 7 0 52 7 0 52 7 < AJ Lβ β β

2)�~

( ( ) , ) ( )d h i r i+ ≤ +1 2β

for all il k l k

01

2 1 0 1≤ ≤− −

− ∈ −, ,β < A ,

~, , , ,

,

d r i h i h i

l kk for all i

l k

l k

l k

− −

−

+ ∩ + ∈ ≤

−⋅

�

"## − ≤ ≤

− −−

1 2 1 0 1

2 1 01

2 1

0 52 7 0 52 7 0 52 7 < AJ Lβ β β

3)� k d l k k− ≤ − − ≤ −2 1 1~

( ( ), )π β for all b ¶ {0, 1}l-k.

PROOF OF CLAIM B3. The proof of 1) and 2) is exactly like theproof of Claim A3. 3) follows directly from the defini-tion of

~d (and from the fact that l

k ≥ 53 ). o

CLAIM B4.

1)� Let 0 112≤ ≤ −− −i l k . For 1 � j � 4, let

n d r i j h i h i

n d r i j h i h i

jl k

jl k

: , , , , ,

~ :~

, , , , .

= − + ∩ + ∈

= − + ∩ + ∈

− −

− −

1

1

2 1 0 1

2 1 0 1

0 52 7 0 52 7 0 52 7 < AJ L0 52 7 0 52 7 0 52 7 < AJ L

β β β

β β β

Then,

n n

n n n if n

n n if n

n n n if n

n n if n

n n

1 1

2 2 1 1

2 2 1

3 3 4 4

3 3 4

4 4

1 0

0

1 0

0

=

≤ + >

= =

≤ + >

= ==

~

max ~ , ~ ~ ,~ ~ ,

max ~ , ~ ~ ,~ ~ ,~ .

< A

< A

2)�For 0 112≤ ≤ −− −i l k ,

β π π π= ∈− −−

b b b l kl k

0 1 1 0 10 5 0 5 1 5 < AK , :

r i d h i r i

r i d h i r i

d h i d h i

d h i b b

d h i b b b b b

d h i b b

d h i b

l k

i i i l k

l k

0 5 0 52 7 0 50 5 0 52 7 0 5

0 52 7 0 52 70 54 9

0 54 90 54 9

0 5

0 5 1 5

0 5 0 5 0 5 0 5 1 5

0 5 1 5

− ≤ ≤ +

≤ + ≤ +

+ − ≤

− ≤

+

− +

− −

− + − −

− −

1 1

1 2

1 1

1

1

1

0 1

0 2 1 2 2 1 1

0 1

, ,

, ,

, , ,

,

, ,

,

,

β

β

β β

π π

π π π π π

π π

π

K

K K

K


3)�For k - 2 � j � k - 1:

d j l k

d j l k

l k

l k

− −

− −

∩ − − ∈

= ∩ − − ∈

1

1

1 0 1

1 0 1

1 6 1 62 7 < AJ L1 6 1 62 7 < AJ L

π β β

π β β

, ,

~, , .

For b ¶ {0, 1}l-k:

k d l k k− ≤ − − ≤ −2 1 1π β1 62 7, .

PROOF OF CLAIM B4. The proof of 1) and 2) is exactly like theproof of Claim A4. For 3), define

d l k d l kπ β π β− − = − −1 11 62 7 1 62 7,~

,

for all b ¶ {0, 1}l-k. Then, the claimed properties in 3)follow immediately from the corresponding proper-ties of

~d . o

5 CONCLUSION

In this paper, we have presented a new technique for theembedding of large cube-connected cycles networks intosmaller ones. Using the new embedding strategy, weshowed:

Let k, l ¶ N, k � 8, such that 53 2+ < ≤ck

lk , ck

kk= +

⋅4 3

3 22 3 .

Then, there is a dilation 1 embedding of CCC(l) into CCC(k)

with load lk

l k⋅ −2 .

This is optimal, and improves the results from [16]. In thecase that l

kl k⋅ ∈−2 N , the embedding technique can be

adapted to yield an even stronger result:


Let k, l ¶ N, such that 53 2 2< ≤ ⋅ ∈−l

klk

l k, N . Then, there

is a dilation 1 embedding of CCC(l) into CCC(k) with loadlk

l k⋅ −2 .

The embedding technique can also be applied in the case32

53< ≤ +l

k kc yielding:

1)�Let k, l ¶ N, k � 8, such that 53

53< ≤ +l

k kc , ckk

k= +⋅4 3

3 22 3 .

Then, there is a dilation 1 embedding of CCC(l) into

CCC(k) with load 53 2+ ⋅ −ck

l k3 8 .

2)�Let k, l ¶ N such that 32

53< <l

k . Let p ¶ {1, 2, ¤} such

that 5 43 2

5 13 1

pp

lk

pp

−−

++< ≤ . Then, there is a dilation 1 embedding

of CCC(l) into CCC(k) with load 5 13 1 2p

pl k+

+−⋅ .

This also improves results from [16].Unfortunately, the new embedding technique does not

lead to any improvement in the case 1 32< ≤l

k . Hence, it isstill of interest to improve the load of the nonoptimal dila-tion 1 embeddings when 1 5

3< ≤ +lk kc (or to prove their

optimality). Finally, a further study should also consider thecongestion of the embeddings.

ACKNOWLEDGMENTS

The author would like to thank the anonymous referees fortheir constructive comments which helped to improve thepresentation of the paper. This work was partially sup-ported by EU ESPRIT Long Term Research Project ALCOM-IT under contract no. 20244.

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Ralf Klasing studied computer science at theUniversity of Paderborn from 1984 to 1990,where he received the MSc. degree in 1990.From 1990 to 1995, he was a member of theresearch group of Prof. Monien at the Universityof Paderborn, where he received the PhD de-gree in 1995. From 1995 to 1997, he was a sci-entific assistant in the research group of Prof.Hromkovic at the University of Kiel. At present,he is a research fellow on algorithms and com-plexity in the research group of Prof. Paterson at

the University of Warwick.Dr. Klasing has published a number of scientific papers in refereed

international periodicals. His research interests include the theory ofgraph embeddings and the design of algorithms for communication in par-allel architectures. Recently, his interests also include the study of approxi-mative and randomized methods for hard combinatorial problems.