EMBEDDING VARIANTS OF HYPERCUBES WITH DILATION 2

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Journal of Interconnection Networks Vol. 13, Nos. 1 & 2 (2012) 1250004 (16 pages) c World Scientific Publishing Company

DOI: 10.1142/S0219265912500041

EMBEDDING VARIANTS OF HYPERCUBES WITH DILATION 2

PAUL MANUEL

J. Inter. Net. 2012.13. Downloaded from www.worldscientific.com by KUWAIT UNIVERSITY on 01/27/14. For personal use only.

Department of Information Science, Kuwait University, Safat, Kuwait, 13060 [email protected] INDRA RAJASINGH School of Advanced Sciences, VIT University, Chennai, India, 600 127 [email protected] R. SUNDARA RAJAN Department of Mathematics, Tagore Engineering College, Chennai, India, 600 127 [email protected] Received 4 July 2012 Revised 30 December 2012 Graph embedding has been known as a powerful tool for implementation of parallel algorithms and simulation of interconnection networks. In this paper, we introduce a technique to obtain a lower bound for the dilation of an embedding. Moreover, we give algorithms for embedding variants of hypercubes with dilation 2 proving that the lower bound obtained is sharp. Further, we compute the exact wirelength of embedding folded hypercubes and augmented cubes into hypercubes. Keywords: Embedding; dilation; wirelength; hypercubes; folded hypercubes; augmented cubes; crossed cubes.

1. Introduction Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. An interconnection network is often represented as a graph, where nodes and edges correspond to processors and communication links between processors, respectively. In the design and analysis of an interconnection network, its graph embedding ability is a major concern. An ideal interconnection network (host graph) is expected to possess excellent graph embedding ability which helps efficiently execute parallel algorithms with regular task graphs (guest graphs) on this network.1,2 The quality of an embedding can be measured by certain cost criteria. One of these criteria which is considered very often is the dilation. The dilation of an 1250004-1

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P. Manuel, I. Rajasingh & R. S. Rajan

embedding is defined as the maximum distance between a pair of vertices of H that are images of adjacent vertices of G. It is a measure for the communication time needed when simulating one network on another.2 Another important cost criteria is the wirelength. The wirelength of a graph embedding arises from VLSI designs, data structures and data representations, networks for parallel computer systems, biological models that deal with cloning and visual stimuli, parallel architecture, structural engineering and so on.3,4 Even though there are numerous results and discussions on the wirelength problem, most of them deal with only approximate results and the estimation of lower bounds.5,6 Graph embeddings have been well studied for binary trees into paths,4 binary trees into hypercubes,2,7 complete binary trees into hypercubes,8 incomplete hypercube in books,9 tori and grids into twisted cubes,10 meshes into locally twisted cubes,11 meshes into faulty crossed cubes,1 meshes into crossed cubes,12 generalized ladders into hypercubes,13 grids into grids,14 binary trees into grids,15 hypercubes into cycles,6,16 star graph into path,17 snarks into torus,18 generalized wheels into arbitrary trees,19 hypercubes into grids,20 m-sequencial k -ary trees into hypercubes,21 meshes into m¨obius cubes,22 ternary trees into hypercubes,23 enhanced and augmented hypercubes into complete binary trees,24 circulant into arbitrary trees, cycles, certain multicyclic graphs and ladders,25 hypercubes into cylinders, snakes and caterpillars,26 hypercubes into necklace, windmill and snake graphs,27 embedding of special classes of circulant networks hypercubes and generalized Petersen graphs.28 In recent years, among many interconnection networks, the hypercube has been the focus of many researchers due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance.29 Hypercubes are known to simulate other structures such as grids and binary trees.7,20 The hypercube has many desirable and attractive properties. However, the hypercube has its own intrinsic drawbacks, such as its large diameter. As a result, several enhancements of the hypercube have been proposed to improve some properties such as diameter.3 There are many variants proposed, such as the folded hypercube F Qr ,3 the augmented hypercube AQr ,30 the crossed cubes CQr ,12 the twisted cube T Qr ,10 the M¨ obius cube M Qr22 and so forth. In this paper, we focus r our discussion on F Q , AQr and CQr . Our objective is to embed these variants of hypercubes with dilation 2. Also, we estimate the exact wirelength of embedding folded hypercubes and augmented cubes into hypercubes. The outline of the paper is given in Fig. 1, where the numbers on the directed lines from graph A to graph B denote the dilation of embedding the guest graph A into the host graph B.

2. Preliminaries In this section we give the basic definitions and preliminaries related to embedding problems. 1250004-2

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Embedding Variants of Hypercubes with Dilation 2 2

2 1

1

2 2

1 2

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2 2

2


Fig. 1.

The outline of the paper using diagram.

Definition 2.1.5 Let G and H be finite graphs. An embedding f of G into H is defined as follows: (1) f is a one-to-one map from V (G) → V (H) (2) Pf is a one-to-one map from E(G) to {Pf (u, v) : Pf (u, v) is a path in H between f (u) and f (v) for (u, v) ∈ E(G)}. Definition 2.2.5 If e = (u, v) ∈ E(G), the length of Pf (u, v) in H is called the dilation of the edge e. The maximal dilation over all edges of G is called the dilation of the embedding f . The dilation of embedding G into H denoted by d(G, H) is the minimum dilation taken over all embeddings f of G into H. The dilation problem 5,15,19,20 of a graph G into H is to find an embedding of G into H that induces the minimum dilation d(G, H). The edge congestion of an embedding f of G into H is the maximum number of edges of the graph G that are embedded on any single edge of H. Let ECf (e) denote the number of edges (u, v) of G such that e is in the path Pf (u, v) between f (u) and f (v) in H. In other words, ECf (e) = |{(u, v) ∈ E(G) : e ∈ Pf (u, v)}| where Pf (u, v) denotes the path between f (u) and f (v) in H with respect to f . If we think of G as representing the wiring diagram of an electronic circuit, with the vertices representing components and the edges representing wires connecting them, then the edge congestion EC(G, H) is the minimum, over all embeddings f : V (G) → V (H), of the maximum number of wires that cross any edge of H.31 Definition 2.3.20 The wirelength of an embedding f of G into H is given by X X W Lf (G, H) = dH (f (u), f (v)) = ECf (e) (u,v)∈E(G)

e∈E(H)

where dH (f (u), f (v)) denotes the length of the path Pf (u, v) in H. 1250004-3

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The wirelength of G into H is defined as W L(G, H) = min W Lf (G, H) where the minimum is taken over all embeddings f of G into H.


The wirelength problem 5,6,15,19,20,31 of a graph G into H is to find an embedding of G into H that induces the minimum wirelength W L(G, H). Definition 2.4. 3 For r ≥ 1, let Qr denote the graph of r-dimensional hypercube. The vertex set of Qr is formed by the collection of all r-dimensional binary strings. Two vertices x, y ∈ V (Qr ) are adjacent if and only if the corresponding binary strings differ exactly in one bit. The decimal representation of the vertices is given by the elements of the set {0, 1, 2, . . . , 2r − 1}. For convenience we use the symbol x + 1 instead of x and therefore the set of labels of the vertices is {1, 2, . . . , 2r }. Next, mathematically we define the variants of hypercube network, such as the folded hypercube F Qr , the augmented hypercube AQr and the crossed cubes CQr . Definition 2.5. 3 For two vertices x = x1 x2 · · · xr and y = y1 y2 · · · yr of Qr , (x, y) is a complementary edge if and only if the bits of x and y are complements of each other, that is, yi = xi for each i = 1, 2, . . . , r. The r-dimensional folded hypercube, denoted by F Qr is an undirected graph obtained from Qr by adding all complementary edges. Definition 2.6. 30 Let r ≥ 1 be an integer. The augmented cube AQr of dimension r has 2r vertices, each labeled by an r-bit binary string a1 a2 · · · ar . We define AQ1 = K2 . For r ≥ 2, AQr is obtained by taking two copies of the augmented cube AQr−1 , r−1 edges between the two as follows: and AQr−1 denoted by AQr−1 0 1 , and adding 2×2 r−1 Let V (AQ0 ) = {0a2 a3 · · · ar : ai = 0 or 1} and V (AQr−1 1 ) = {1b2 b3 · · · br : bi = r−1 0 or 1}. A vertex A = 0a2 a3 · · · ar of AQ0 is joined to a vertex B = 0b2 b3 · · · br of AQr−1 iff for every i, 2 ≤ i ≤ r, either 1 1. ai = bi ; in this case, AB is called a hypercube edge, or 2. ai = bi ; in this case, AB is called a complement edge. Definition 2.7. 32 Two 2-digit binary strings x = x1 x0 and y = y1 y0 are pairrelated, denoted by x ∼ y, if and only if (x,y) ∈ {(00, 00), (10, 10), (01, 11), (11, 01)}. Definition 2.8. 32 The r-dimensional crossed cube CQr , is a graph CQr = (V, E) that is recursively constructed as follows: CQ1 is a complete graph with two vertices labeled by 0 and 1. CQr consists of two identical (r − 1)-dimension crossed cubes, r−1 CQr−1 and CQr−1 0 1 . The vertex u = 0un−2 un−3 · · · u0 ∈ V (CQ0 ) and vertex v = r−1 r 1vn−2 vn−3 · · · v0 ∈ V (CQ1 ) are adjacent in CQ if and only if 1. ur−2 = vr−2 , if r is even; and 2. 0 ≤ i < ⌊ r−1 2 ⌋, u2i+1 u2i ∼ v2i+1 v2i 1250004-4

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Embedding Variants of Hypercubes with Dilation 2

The binary reflected Gray code is well known and is widely used in digital systems because it is cyclic and the binary number system can easily be translated to the Gray code. For the hypercube embedding problem, the Gray code has a few useful properties that can be used to provide upper and lower bounds on the embedding distance.7 Definition 2.9.7 The k-bit Gray code sequence Gk is defined recursively as: G1 = (0, 1) Gk = (0Gk−1 , 1GR k−1 ) where GR k−1 is the reverse of Gk−1 .


In this paper, we use the decimal equivalent of the binary Gray code sequence. Definition 2.10. For r ≥ 1, label the vertex x1 x2 · · · xr of hypercube Qr as r P xi 2r−i which is equivalent to the lexicographic labeling.5 i=1

Definition 2.11.28 A Hamiltonian cycle is a cycle that visits each node of the graph exactly once. By convention, the trivial graph on a single node is considered to possess a Hamiltonian cycle. A graph possessing a Hamiltonian cycle is said to be a Hamiltonian graph. Lemma 2.1. 28 Let G be an r-dimensional hypercube Qr and the vertices of Qr be labeled using Gray code labeling. (i) The Gray code labeling of Qr determines a hamiltonian cycle Cr = (1, 2, . . . , 2r , 1) in Qr , r ≥ 2. (ii) Let x and y be the Gray code labeling of vertices u and v in Qr . If x − y = 2m for some m, 1 ≤ m ≤ r, then d(u, v) = 2 in Qr . (iii) Let x and y be the Gray code labeling of vertices u and v in Qr . If y − x = 2m − 2x + 1 for some m, 1 ≤ m ≤ r, then d(u, v) = 1 in Qr . (iv) Let 1 and y be the Gray code labeling of vertices u and v in Qr . If y−1 = 2m+i −2m for some m and i, 1 ≤ m,i ≤ r, then d(u, v) = 2 in Qr . In addition, we have the following result. Lemma 2.2. Let x and y be the lexicographic labeling of vertices u and v in Qr . If x − y = 2m − 2 for some m, 1 ≤ m ≤ r, then d(u, v) = 2 in Qr . 3. Main Results In Sec. 3.1, we have obtained the exact wirelengths of embedding F Qr into Qr and AQr into Qr . It is interesting to note that the method used in finding the exact wirelength of F Qr into Qr fails for finding the same in embedding AQr into Qr and vice-versa. In Sec. 3.2, we obtain a lower bound for the dilation of embedding 1250004-5

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of graphs using minimum wirelength. We prove that the lower bound is sharp in embedding variants of hypercubes. From the definitions, we know that Qr ⊂ F Qr ⊂ AQr . Hence we have the following result: Lemma 3.1. Let r ≥ 1 be an integer. Then d(Qr , F Qr ) = d(F Qr , AQr ) = d(Qr , AQr ) = 1.


3.1. Exact Wirelength of Embedding Variants of Hypercube into Hypercube In this section, we obtain the exact wirelength of embedding folded hypercube and augmented cube into hypercube. The following two versions of the edge isoperimetric problem of a graph G(V, E) have been considered in the literature,33 and are N P complete.34 Problem 1: Find a subset of vertices of a given graph, such that the edge cut separating this subset from its complement has minimal size among all subsets of the same cardinality. Mathematically, for a given m, if θG (m) = min |θG (A)| A⊆V , |A|=m

where θG (A) = {(u, v) ∈ E : u ∈ A, v ∈ / A}, then the problem is to find A ⊆ V such that |A| = m and θG (m) = |θG (A)|. Problem 2: Find a subset of vertices of a given graph, such that the number of edges in the subgraph induced by this subset is maximal among all induced subgraphs with the same number of vertices. Mathematically, for a given m, if IG (m) = max |IG (A)| where IG (A) = {(u, v) ∈ E : u, v ∈ A}, then the A⊆V , |A|=m

problem is to find A ⊆ V such that |A| = m and IG (m) = |IG (A)|. For a given m, where m = 1, 2, . . . , n, we consider the problem of finding a subset A of vertices of G such that |A| = m and |θG (A)| = θG (m). Such subsets are called optimal. We say that optimal subsets are nested if there exists a total order O on the set V such that for any m = 1, 2, . . . , n, the first m vertices in this order is an optimal subset. In this case we call the order O an optimal order.33,37 This implies n P that W L(G, Pn ) = θG (m). m=0

Further, if a subset of vertices is optimal with respect to Problem 1, then its complement is also an optimal set. But, it is not true for Problem 2 in general. However for regular graphs a subset of vertices S is optimal with respect to Problem 1 if and only if S is optimal for Problem 2.33 In the literature, Problem 2 is defined as the maximum subgraph problem.34

Lemma 3.2. (Congestion Lemma)20 Let G be an r-regular graph and f be an embedding of G into H. Let S be an edge cut of H such that the removal of edges of S leaves H into 2 components H1 and H2 and let G1 = f −1 (H1 ) and G2 = f −1 (H2 ). Also S satisfies the following conditions. 1250004-6

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(i) For every edge (a, b) ∈ Gi , i = 1, 2, Pf (a, b) has no edges in S. (ii) For every edge (a, b) in G with a ∈ G1 and b ∈ G2 , Pf (a, b) has exactly one edge in S. (iii) G1 is an optimal set. P Then ECf (S) is minimum and ECf (S) = ECf (e) = r |V (G1 )| − 2 |E(G1 )|. e∈S

Lemma 3.3. (Partition Lemma)20 Let f : G → H be an embedding. Let {S1 , S2 , . . . , Sp } be a partition of E(H) such that each Si is an edge cut of H. Then W Lf (G, H) =

p X ECf (Si ).


i=1

Definition 3.1. 36 An incomplete hypercube on i vertices of Qr is the subcube induced by {0, 1, . . . , i − 1} and is denoted by Li , 1 ≤ i ≤ 2r . Theorem 3.1. 37–39 Let Qr be an r-dimensional hypercube. For 1 ≤ i ≤ 2r , Li is an optimal set on i vertices. Definition 3.2. A reverse incomplete hypercube on i vertices of Qr is the subcube induced by {2r − 1, 2r − 2, ..., 2r − i} and is denoted by RLi , 1 ≤ i ≤ 2r . It is easy to see that any r-dimensional folded hypercube F Qr can be viewed as G(0Qr−1 , 1Qr−1 ; C + C) where 0Qr−1 and 1Qr−1 are two (r − 1)-dimensional hypercubes with the prefix 0 and 1 of each vertex, respectively, and C = {(0u, 1u) : 0u ∈ V (0Qr−1 ) and 1u ∈ V (1Qr−1 )}, C = {(0u, 1u) : 0u ∈ V (0Qr−1 ) and 1u ∈ V (1Qr−1 )}.40 Lemma 3.4. 41 In the folded hypercube F Qr , Li is isomorphic to RLi for 1 ≤ i ≤ 2r . In the next result, we use the Partition technique to find the exact wirelength of embedding folded hypercube into hypercube. Wirelength Algorithm A Input: The r-dimensional folded hypercube F Qr and the r-dimensional hypercube Qr , r ≥ 1. Algorithm: Label the vertices of F Qr and Qr either by lexicographic labeling or Gray code labeling from 1 to 2r and map f (x) = x. Output: An embedding f of F Qr into Qr given by f (x) = x with minimum wirelength. Proof of correctness: We know that Qr has r edge cuts, say Si , 1 ≤ i ≤ r, which partition E(Qr ) such that |Si | = 2r−1 and each Si satisfies conditions (i),(ii) and (iii) of the Congestion Lemma. Therefore ECf (Si ) minimum and ECf (Si ) = 1250004-7

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(r + 1)2r−1 − 2(r − 1)2r−2 = 2r for all i. Partition Lemma implies that the wirelength is minimum and is given by W L(F Qr , Qr ) =

r X

2r

i=1


= r 2r .

The Partition technique fails in computing the exact wirelength of embedding augmented hypercube into hypercube. In what follows, we obtain a different method to compute the wirelength of networks. Let G and H be a graph on (V1 , E1 ) and (V2 , E2 ) vertices and edges respectively, where V1 ⊇ V2 and E1 ⊇ E2 . For an embedding of G into H, if |E2 | number of edges in E1 have dilation 1 and the remaining edges in E1 have dilation 2, then the wirelength is minimum and is given by |E2 | + 2(|E1 | − |E2 |). For proving the main result, we need the following Lemma. Lemma 3.5. Let G and H represent two copies of the r-dimensional hypercube Qr . Let f : G → H be an embedding, where G is labeled using lexicographic labeling and H is labeled using Gray code labeling and f (x) = x. Then df (G, H) = 2. Proof. We prove the result by induction on r. Since the diameter Qr is ≤ 2, the result is obvious for r = 1, 2. Assume that the result is true for hypercube of dimension r − 1, r ≥ 3. Consider the graph G ∼ = Qr . By definition, Qr contains two copies of Qr−1 , and by induction hypothesis, the dilation of embedding f of Qr−1 in G into Qr−1 in H is 2. Let e = (u, v) ∈ G\Qr−1 . Then f (x) − f (y) = 2r−1 , where x and y are the lexicographic labeling of u and v in G. By Lemma 2.1(ii), d(u, v) = 2 in H. Hence d(G, H) = 2. Wirelength Algorithm B Input: The r-dimensional augmented hypercube AQr and the r-dimensional hypercube Qr , r ≥ 1. Algorithm: Label the vertices of AQr by lexicographic labeling from 1 to 2r and label the vertices of Qr by Gray code labeling from 1 to 2r and map f (x) = x. Output: An embedding f of AQr into Qr given by f (x) = x with minimum wirelength. Proof of correctness: By Lemma 3.5, each hypercube edge in AQr is mapped into a path of length 2 in Qr . Thus 2r edges in AQr have dilation 2 in Qr and by Lemma 2.1(i), the remaining (r − 2)2r−1 edges have dilation 1. 1250004-8

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By Embedding Algorithm B, each complement edge in AQr is mapped into an edge in Qr . Thus, 2r number of complement edges in AQr have dilation 1 in Qr . Hence, r2r−1 number of edges in AQr have dilation 1 in Qr and the remaining 2r edges have dilation 2. Since (r − 2)2r−1 + 2r = r(2r−1 ) = |E(Qr )|, the wirelength is minimum and is given by r2r−1 + 2(2r ) = (r + 4)2r−1 .


3.2. Computing Dilation From Minimum Wirelength The dilation problem and the wirelength problem are different in the sense that an embedding that gives minimum dilation need not give minimum wirelength and vice-versa. However in this section we obtain a relation between the two embedding parameters, namely, the dilation and the wirelength whcih turns out to be a powerful technique to obtain minimum dilation using minimum wirelength. In the literature there is no efficient method to compute exact dilation of graph embeddings.7,8,23 The following result leads to exact dilation. Lemma 3.6. (IPS Lemma) Let G and H be graphs on same number of vertices. Let δ and W L be the dilation and wirelength of embedding graph G into graph H and let f : G → H be an embedding realizing δ. If dij number of edges in G are of dilation ij with respect to f , 1 ≤ j ≤ k, then WL −

k P

ij dij

j=1

δ≥

|E| −

k P

dij

j=1

Proof. Let W L′ be the wirelength of the embedding f . We have W L′ ≤ δ| + δ +{z· · · + δ}

(3.2)

|E| times

If dij number of edges are of dilation ij with respect to f , then dij terms on the R.H.S of equation 3.2 replaced by i1 does not affect the inequality in Eq. 3.2. Repeating this step for all j, 1 ≤ j ≤ k, we get W L′ ≤ |E|δ − (di1 + di2 + · · · + dik )δ + (i1 di1 + i2 di2 + · · · + ik dik ) = δ(|E| −

k k X X dij ) + ij dij j=1

But W L ≤

W L′ .

j=1

Therefore, W L ≤ δ(|E| −

k k X X dij ) + ij dij j=1

WL − (i.e)

δ≥

k P

ij dij

j=1

|E| −

j=1

k P

. Hence the Lemma. dij

j=1

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Remark 3.1. Let δ and W L be the dilation and wirelength of embedding graph WL G into graph H. Then δ ≥ |E(G)| . Embedding Algorithm A Input: The r-dimensional folded hypercube F Qr and the r-dimensional hypercube Qr , r ≥ 1. Algorithm: Label the vertices of F Qr by lexicographic labeling from 1 to 2r and label the vertices of Qr by Gray code labeling from 1 to 2r and map f (x) = x.


Output: An embedding f of F Qr into Qr with dilation 2. Proof of correctness: We show that 2r edges of F Qr are mapped to edges in Qr giving dilation 1 as follows: Case 1: If e = (u, v) is a complement edge, then f (y)− f (x) = 2r − 2x+ 1, where x and y are the lexicographic labeling of u and v respectively. By Lemma 2.1(iii), d(u, v) = 1 in Qr . Case 2: If e = (u, v) is a hypercube edge with the labels of u, v as i and i + 1 respectively, then (f (i), f (i + 1)) is an edge in Qr . Thus, d(u, v) = 1 in Qr . Thus we have 2r−1 + 2(2r−2 ) = 2r edges having dilation 1 from F Qr to Qr . By IPS Lemma, we get W L − 2r |E(G)| − 2r r 2r − 2r = (r + 1)2r−1 − 2r = 2.

δ≥

Next we prove that the bound obtained is sharp. Dilation Algorithm A Input: The r-dimensional folded hypercube F Qr and the r-dimensional hypercube Qr , r ≥ 1. Algorithm: Label the vertices of F Qr by lexicographic labeling from 1 to 2r and label the vertices of Qr by Gray code labeling from 1 to 2r and map f (x) = x. See Fig. 2. Output: An embedding f of F Qr into Qr with dilation 2. Proof of correctness: Since F Qr contains Qr , by Lemma 3.5, the dilation of embedding Qr ⊂ F Qr into Qr is 2.

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Let e = (u, v) ∈ F Qr \Qr , then f (y) − f (x) = 2r − 2x + 1, where x and y are the lexicographic labeling of u and v respectively. By Lemma 2.1(iii), d(u, v) = 1 in Qr . Hence d(F Qr , Qr ) = 2.

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In the next result we embed AQr into Qr with dilation 2 using minimum wirelength. Embedding Algorithm B Input: The r-dimensional augmented hypercube AQr and the r-dimensional hypercube Qr , r ≥ 1. Algorithm: Label the vertices of AQr by lexicographic labeling from 1 to 2r and label the vertices of Qr by Gray code labeling from 1 to 2r and map f (x) = x. Output: An embedding f of AQr into Qr with dilation 2. Proof of correctness: By lexicographic labeling, AQr contains 2r−1 edges having dilation 1 in Qr . By IPS Lemma, we get δ≥ =

W L − r 2r−1 |E(G)| − r 2r−1 (r + 4)2r−1 − r 2r−1 r 2r−1 + 2r − r 2r−1

= 2. Next we prove that the bound obtained is sharp.

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Dilation Algorithm B Input: The r-dimensional augmented cube AQr and the r-dimensional hypercube Qr , r ≥ 1. Algorithm: Label the vertices of AQr by lexicographic labeling from 1 to 2r and label the vertices of Qr by Gray code labeling from 1 to 2r and map f (x) = x.


Output: An embedding f of AQr into Qr with dilation 2. See Fig. 3(a). Proof of correctness: We prove the result by induction on r. Since the diameter Qr is ≤ 2, the result is obvious for r = 1, 2. Assume that the result is true for dimension r − 1, r ≥ 3. That is, d(AQr−1 , Qr−1 ) = 2. Consider AQr . By definition, AQr contains two copies of AQr−1 , and by induction hypothesis, the dilation of embedding AQr−1 into Qr−1 is 2. Let e = (u, v) ∈ AQr \AQr−1 , then we have the following two cases. Case 1: If e is a hypercube edge, then f (x) − f (y) = 2r−1 , where x and y are the lexicographic labeling of u and v in AQr . By Lemma 2.1(ii), d(u, v) = 2 in Qr . Case 2: If e is a complement edge, then f (y) − f (x) = 2r − 2x + 1, where x and y are the lexicographic labeling of u and v respectively. By Lemma 2.1(iii), d(u, v) = 1 in Qr . Hence d(AQr , Qr ) = 2.

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(a) Embedding of AQ3 into Q3 with dilation 2 (b) Embedding of CQ3 into Q3 with dilation 2.

We further compute dilation for embedding certain other invariants of Qr . Dilation Algorithm C Input: The r-dimensional augmented cube AQr and the r-dimensional folded hypercube F Qr , r ≥ 1. Algorithm: Label the vertices of AQr by lexicographic labeling from 1 to 2r and label the vertices of F Qr by Gray code labeling from 1 to 2r and map f (x) = x. 1250004-12

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Output: An embedding f of AQr into F Qr with dilation 2. Proof of correctness: We prove the result by induction on r. Since the diameter of F Qr is 1, for r = 1 and 2, the result is obvious. Assume that the result is true for dimension r − 1, r ≥ 3. That is, d(AQr−1 , F Qr−1 ) = 2. Consider AQr . By definition, AQr contains two copies of AQr−1 , and by induction hypothesis, the dilation of embedding AQr−1 into F Qr−1 is 2. Let e = (u, v) ∈ AQr \AQr−1 , then we have the following two cases. Case 1: If e is a hypercube edge, then f (x) − f (y) = 2r−1 , where x and y are the lexicographic labeling of u and v in AQr . By Lemma 2.1(ii), d(u, v) = 2 in F Qr . Case 2: If e is a complement edge, then f (y) − f (x) = 2r − 2x + 1, where x and y are the lexicographic labeling of u and v respectively. Then by Lemma 2.1(iii), d(u, v) = 1 in F Qr . Hence d(AQr , F Qr ) = 2. Dilation Algorithm D Input: The r-dimensional crossed cube CQr and the r-dimensional hypercube Qr , r ≥ 1. Algorithm: Label the vertices of CQr and Qr by lexicographic labeling from 1 to 2r and map f (x) = x. Output: An embedding f of CQr into Qr with dilation 2. See Fig. 3(b). Proof of correctness: We prove the result by induction on r. Since the diameter Qr is ≤ 2, for r = 1 and 2, the result is obvious for r = 1, 2. Assume that the result is true for dimension r − 1. That is, d(CQr−1 , Qr−1 ) = 2. Consider CQr . By definition, CQr contains two copies of CQr−1 , and by induction hypothesis, the dilation of embedding CQr−1 into Qr−1 is 2. Let e = (u, v) ∈ CQr \CQr−1 , then we have the following two cases. Case 1: If e is a hypercube edge, then by definition d(u, v) = 1 in Qr . Case 2: If e is not a hypercube edge, then f (x) − f (y) = 2r−1 − 2, where x and y are the lexicographic labeling of u and v in CQr . Then by Lemma 2.2, d(u, v) = 2 in Qr . Hence d(CQr , Qr ) = 2. Proceeding along the same lines, we prove the following result. Theorem 3.2. Let r be an integer ≥ 1. 1. Let G and H be the r-dimensional hypercube Qr and the r-dimensional crossed cube CQr respectively. Then the dilation of embedding G into H is 2.

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2. Let G and H be the r-dimensional augmented cube AQr and the r-dimensional crossed cube CQr respectively. Then the dilation of embedding G into H, and that of H into G are both 2. 3. Let G and H be the r-dimensional crossed cube CQr and the r-dimensional folded hypercube F Qr respectively. Then the dilation of embedding G into H, and that of H into G are both 2. 4. Conclusion


In this paper, we have considered embedding variants of hypercubes with minimum dilation. Also, we have obtained a lower bound for computing dilation of embeddings and proved that the bound is sharp. The strategies used in the paper fail to compute exact wirelength of CQr into Qr and the problem is under investigation. Acknowledgment We are greatly indebted to the referees whose valuable suggestions led us to make changes in the paper. References 1. X. Yang, Q. Dong and Y.Y. Tan, Embedding meshes/tori in faulty crossed cubes, Information Processing Letters, Vol. 110, no. 14-15, 559 - 564, 2010. 2. T. Dvoˆr´ak, Dense sets and embedding binary trees into hypercubes, Discrete Applied Mathematics, Vol. 155, no. 4, 506 - 514, 2007. 3. J.M. Xu, Topological Structure and Analysis of Interconnection Networks, Kluwer Academic Publishers, Netherlands, 2001. 4. Y.L. Lai and K. Williams, A survey of solved problems and applications on bandwidth, edgesum, and profile of graphs, J. Graph Theory, Vol. 31, 75 - 94, 1999. 5. S.L. Bezrukov, J.D. Chavez, L.H. Harper, M. R¨ ottger and U.P. Schroeder, Embedding of hypercubes into grids, Mortar Fine Control System, 693 - 701, 1998. 6. J.D. Chavez and R. Trapp, The cyclic cutwidth of trees, Discrete Applied Mathematics, Vol. 87, 25 - 32, 1998. 7. W.K. Chen and M.F.M. Stallmann, On embedding binary trees into hypercubes, Journal on Parallel and Distributed Computing, Vol. 24, 132 - 138, 1995. 8. S.L. Bezrukov, Embedding complete trees into the hypercube, Discrete Applied Mathematics, Vol. 110, no. 2-3, 101 - 119, 2001. 9. J.-F. Fang and K.-C. Lai, Embedding the incomplete hypercube in books, Information Processing Letters, Vol. 96, 1 - 6, 2005. 10. P.-L. Lai and C.-H. Tsai, Embedding of tori and grids into twisted cubes, Theoretical Computer Science, Vol. 411, no. 40-42, 3763 - 3773, 2010. 11. Y. Han, J. Fan, S. Zhang, J. Yang and P. Qian, Embedding meshes into locally twisted cubes, Information Sciences, Vol. 180, no. 19, 3794 - 3805, 2010. 12. J. Fan and X. Jia, Embedding meshes into crossed cubes, Information Sciences, Vol. 177, no. 15, 3151 - 3160, 2007. 1250004-14

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EMBEDDING VARIANTS OF HYPERCUBES WITH DILATION 2

EMBEDDING VARIANTS OF HYPERCUBES WITH DILATION 2

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