Embedding Hypercubes into Pancake, Cycle

Proceedings of the 28th Annual Hawaii International Conference on System Sciences -

1995

Embedding Hypercubes into Pancake, Cycle Prefix and Substring Reversal Networks Linda Gardner‘, Zevi Mille?, Dan Pritikin2, I. Hal Sudborough’ ’ Computer Science Program, University of Texas at Dallas, Richardson, Texas 75083 2 Dept. of Mathematics and Statistics, Miami University, Oxford, Ohio 45056

Abstract

Permutations Rev[i j] are called substrin~ reversals, permutations Rev[l,j] are called prefix reversals, and permutations CyCi or (CyCi)-l are called prefix cvcles. The Qylev graph determined by Cn and a set of generators Tn, denoted by G(Cn,Tn), is the graph whose vertex set is Cn, with two permutations/vertices a$ connected by an edge from a to p if and only if p = a 0 t for some generator t E T . Each of A,, Bn, Cn, Dn, F happens to be closed under the inverse operator, so th: Cayley graphs listed below are regarded as undirected graphs. G(Zn,An) is called the lransoosition network of

Among Cayley graphs on the symmetric group, some that have noticeably low diameter relative to the number of generators involved are the pancake network, burnt pancake network, and cycle prefix network [1,6,7,8]. Another such Cayley graph is the star network, and constructions have been given for one-to-one and one-tomany low dilation embeddings of hypercubes into star networks [11,12,15]. For each of dilations 1 through 4, in this paper we construct embeddings of hypercube (and hypercube-like) networks into pancake, burnt pancake, cycle prefix, and substring reversal networks. Embeddings of hypercubes into pancake networks is our focus, the other results being closely related.

dimension n, G(C,,Bn) is called the &V network of dimension n, G(C,,C,) is called the substring reversal network of dimension n, denoted by SSR,, G(Cn,D,) is called the pancake network of dimension n, denoted by Pn, G(Zn,Fn) is called the cvcle prefix network of dimension n, denoted by CP,. The first, second and fourth of these networks can be found in [ 11, and CPn can

1. Introduction Much recent attention has been given to n-dimensional interconnection networks with n! many processors, one for each permutation on n symbols [1,4,6,7,9,11,12,15]. Such networks, usually defined as Cayley graphs [l] on the symmetric group of order n, often have smaller diameter and fewer connections per processor than correspondingly large hypercubes [lo]. We consider the ability of some of these networks to efficiently simulate hypercubes. That is, we show how to embed hypercubes into them with low dilation. We assume familiarity with basic concepts. Details are found in the literature [ 10,133. Let Cn denote the group of all permutations on n symbols (with the operation of functional composition). Let (ij) denote the permutation which transposes symbols i and j. Here are some sets of generators for Xn, relevant to our study: An= [(ij): llicjln) Bn = ((lj) : j 2 2) Cn = (Rev[i j] : 1 I i < j I n], where Rev[i j] = 12...

i-l

i

i+l

( 1 2 . . . i-1 j j-1 Dn = (Rev[l j] : j 2 2) En = ( Cyci cyci = ( Fn = (CyCi

:i2 1 2 i 1 :i2

j-l

j

j+l

n

i+l

i

j+l

n

be found in [7]. The directed Cayley graph G(C,,E,)

vertices FDCBAGE and CDFBAGE are adjacent (via the generator Rev[ 1,3]). In SSR7 on the ordered symbol set ABCDEFG there

2), where i i+l . . . n 3 . . . i-l 2 . . . i-2 i-l i+l . . . n 1 2) U ((CyCi)-’ : i 2 2)

exists a path of length 2 from EGBDFCA EGFDBCA (applying the generator Rev[3,5]) ACBDFGE (applying Rev[ 1,73).

537 1060-3425/95

is

called the dicvcle prefix network of dimension n, denoted by DCP . As isnstandard, we typically regard Cn as the set of all n! arrangements of a set of specified and ordered symbols i-th (where symbol listed in an the arrangement/permutation is regarded as the functional image of the i-th symbol of the standard, specified order), and we regard the generators as acting upon these symbol strings by appropriately shuffling around the n symbols. For example: In P7 on the ordered symbol set ABCDEFG the

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Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS'95) 1060-3425/95 $10.00 © 1995 IEEE

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the 28th Annual Hawaii International Conference on System Sciences - 1995

In CP7 on the ordered symbol set ABCDEFG there exists a path of length 2 from EGBDFCA to GBDFECA to AGBDFEC (applying generators (Cyc# and Cyc+.

a substack is flipped, each of the i pancakes flipped changes orientation. Pn has the similar interpretation, except that the pancakes have no orientation, i.e. turning a pancake over renders it indistinguishable from its previous state. In a manner entirely analogous to how BPn was defined, the n-dimensional burnt substring reversal network, denoted BSSRn, has generator set (Rev[2i-1,2j]: 1 I i I j 5 n) = (rev[ij]: 1 I i I j 5 n).

Henceforth we specify merely a symbol string such as ABCDEFG or X1X2...Xn to inform the reader of the set of symbols being permuted, and will not bother reminding the reader that the order of the symbols in that string is relevant concerning which strings correspond to which group elements as functions, although that order is in fact irrelevant in virtue of the symmetry properties of Cayley graphs. We let “;” denote concatenation of strings. For a symbol string S = X1X2...Xm let S’ denote its “reverse,”

Let 2’ denote the power set of X. Let dis$-,(x,y) denote the distance between x and y in G. Given two networks G = (V,E) and H = (V’,E’) and an integer d, a function f : V-+2” - (0 ) is called a one-to-manv dilation d embedding of G into H if for every pair of vertices x,y in V, 1) f(x) n f(y) = 0, and 2) If x and y are adjacent in G, then for each x’ E f(x) there corresponds at least one vertex y’ I f(y) for which distB(x’,y’) I d.

i.e. S’ = X,X,-l...X1. In this notation, SSR, is the network whose vertex set is Cn with two symbol strings X,Y adjacent if and only if X,Y are expressible in the forms X = A;B;C and Y = A;B’;C where substring B must have at least 2 symbols, substrings A,C possibly empty. We also consider so-called “burnt” networks related to those described above. The n-dimensional burnt pancake network [6,8,9], denoted by BP,, is defined as follows. Given Z2n on a symbol string X1,X2,...,XZn, the vertex

We write G 2 H to indicate that there exists a one-tomany dilation d embedding of G into H. We write G 5 H to indicate that there exists a one-to-one dilation d embedding of G into H, i.e. the traditional sort of embedding. We write G C H to indicate that there exists a one-to-one dilation 1 embedding of G into H, i.e. that G is isomorphic to a subgraph of H. A one-to-many dilation d embedding of G = (V,E) into H = (V’,E’) is essentially a particular kind of relation ‘dt from V to V’, where xaty if and only if y E f(x). Requirement 1) above stipulates that each element of V’ is related to exactly one or zero elements of V, so that the inverse relation of 3$., when restricted to the range of 33,) is in fact a function. Therefore it is typically convenient to specify a one-to-many dilation d embedding of G into H by specifying the range R r V’ of 33, and specifying the function g: R+V corresponding to the inverse of 33,, thereby specifying the ‘actual’ embedding implicitly by f(x) = [y E V’ : g(y) = x), where R is the part of H actually used in embedding G within it. A co e bedding of dilation d of G = (V,E) into H = (V’,E’) is? pair (R,g) where 0 f R r V’ and onto function g : R+V such that for every y E R and neighbor x’ of g(y) in G there exists some y’ E R such that g(y) = x’ and distB(y,y’) 5 d. Since R is the domain of g we

set of BPn is the subgroup of 2” n! arrangements of those 2n symbols so that each symbol X2i-l appears consecutive to its paired symbol X2i. The generator set for BPn is (Rev[l,2i]: i > 1 ), thereby specifying the edges of the Cayley graph BP,. For our purposes it is convenient to regard the vertex set of BPn as being all arrangements

of

n

of

the

2n

symbols

P1’P2’...pn,p1r,p2r,... Pnr in which exactly one symbol appears from each of the symbol pairs [ Pi,Pir j , where P.1 denotes the string of two symbols X2i-1X2i and Pi’ denotes its reverse. In this alternative notation we think of there as being n symbols Pi arranged, each of which is also assigned an orientation, i.e. is in either ordinary or reversed mode. When vertices of BPn are regarded as arrangements of n “bumt”/oriented symbols instead of 2n non-oriented symbols we denote the generator Rev[ 1,2i] by rev[ 1 ,i] (with a lower-case “r”) in keeping with how that generator acts to reverse and change the orientation of i (as opposed to 2i) symbols. Thus each vertex of BP n can be thought of as a stack of n pancakes of various sizes, each burnt on one side, and the edges of BPn are present to indicate the option of grabbing the top i pancakes and turning the substack of i upside-down, replacing them back on the rest of the stack. When such

usually refer to g as being the co-embedding, R understood implicitly. By the comments above we see that there exists a one-to-many dilation d embedding of G into H if and only if there exists a dilation d coembedding of G into H. Given a dilation d co-embedding g of a hypercube Q, into H, to &ggle bit i of a vertex y E R is to specify a choice of y’ E R such that the bit strings g(y) and g(y) differ in exactly the i-th bit, with distB(y,y’) I d. Throughout this paper we routinely and

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implicitly specify one-to-many dilation d embeddings of Q,, into H by specifying a co-embedding (R,g) along with a demonstration of how to toggle each of the n bits for eachyE R. The one-to-many embedding thereby specified will be a one-to-one embedding if and only if g is one-to-one. Some of these notions are illustrated in the following proof. Let 89 denote the bitwise modulo-2 sum operation for bit strings.

Proof: In each case the embedding f is simply the identity map. Thus to verify that each is a dilation d embedding it suffices to show that each generator in the “guest” network is a composition of d or fewer generators in the “host” network. For a), Cyc. = Rev[ l,i] 0 Rev[l,i1 11 and (Cyci)-‘= Rev[l,i-11 e Rev[l,i]. For b), Rev[ij] = Rev[l j] * (Rev[l j-i] 0 Rev[l j] ). For c), rev[i,j] = rev[l j] * (rev11j-i] 0 rev[l j] ). I

-1:

Our motivation is to find efficient embeddings of hypercubes into pancake networks. Hence in the rest of the paper we construct low dilation maps of hypercubes into pancake networks and their relatives.

For any m,n,d 2 1,

a) If Q,,, 5 Pn then Q,n+l 2 P2n. If Q, 2 BPn then Qm+l

b)

2

BP2n-

If Qm 5 Pn then Qm+t 3 P2,. If Q, 3 BPn then

Qm+l

$

2.

Proof: Let embedding permutations X n+lXn+2”’

(R’,g’) [resp. (R”,g”)] be a dilation d coof Q, into Pn, where Cn is viewed as of the symbol string X1X2...X, [resp. X2J.

With C2n being permutations

Dilation

1 embeddings

We begin with a theorem concerning embeddings of hypercubes and other Cartesian products of cliques into dicycle prefix networks.

BP2xl*

Theorem: a) Forallm21, Q,,.,=!,DCP2m. b) For ml, m2 ,..., mn 22, Km1 x Km2 x . . . x K,

of

X1X2...X2n, let R = (a;P’ : (a E R’and p E R”) or (a E R” and p E R’)). Let h : R’uR”+Qm be defined by h(a)

n

A DCPs where s = ml+m2+...+mn . Proof: For any permutation rc in R = C2m of the 2m

= g’(a) if a E R’ and h(a) = g”(a) if a E R”. Then define g : R+Qm+l by g(a;p’) = (h(a)@h(P));B, where B=OifaER’andB=lifaER”. Weshowthatgisa dilation d co-embedding. To toggle the (m+l)-th bit of a;P’ we apply Rev[l,2n] to obtain p;a’, noting that g(a$‘) and g(p;a’> differ in exactly the (m+l)-th bit. To toggle the i-th bit of a;P’ for 1 I i I m when a E R [resp. a E R”] we apply the same sequence of d or fewer generators that serve to toggle the i-th bit of a relative to the co-embedding g’ [resp. g”]. Having shown that any bit of a;pr can be toggled via a path of length d or less in P2n, we have proven the first part of a). The proof of the

symbols XI,OXI,IX~.OX~,~...X,,OX,,I, let gt@ = blb2...bm. where bi = 0 or 1 depending upon which appears first (leftmost) in rc among the symbols Xi,0 and To toggle the i-th bit, apply the appropriate cyclic xi,l* shift to bring to the front whichever of Xi,0 and Xi.1 occurs after the other. This reverses their order and leaves every other pair Xk,o and Xk,l in the same relative order, so g is a dilation 1 co-embedding, proving a). The proof of b) is similar, involving the s symbols X~,OX~,I...XI,~~-~...X~,OX~,~...X~,~~-I. where bi is determined by the leftmost second-subscript k appearing in a symbol Xi,k I

second part of a) is essentially the same. As for b), the further supposition that co-embeddings g’and g” are oneto-one is the same as supposing the conditions in the premise of b), and since the resulting g constructed is then one-to-one we have also proven b). I

A corollary of Theorem 2b) is that the n dimensional r-ary hypercube (Kr)n has a one-to-many dilation 1 embedding into DCPnr .

As for how the some of the Cayley graph networks defined above relate to one another, we have the following.

Lemma: a) If Qm c BSSRk then Qm+2 C BSSRk+l.

Q,ml!Z BSSR, for each positive integer m, via an embedding f’ such that no two vertices f’(x), f”(y) are the same m-symbol string when regarded as strings in SSR,, i.e. with orientations suppressed. c) Qmel G SSR,.

b)

Theorem: a) CP, 3 Pn . b) SSR, 5 P,, . a) BSSR, 5 BPn .

539



f(x;x1;x2;...;xm) and f(y;xl;x2;...;xm) differ by a substring reversal which reverses the concatenation of several substrings made up of Yi’s and/or (Yir)‘s [i.e. one

Proof: Suppose f : Q,+ BSSRk embeds Q, as a subgraph of BSSRk, with Ck being on the n burnt symbols X1X2... Xk. Define f’ : Qm+2+ BSSRk+l as follows, with BSSRk+l being on the burnt symbols x1x2... X,A. For x E R, let f’(x;OO) = f(x);A, let f

can always toggle any one of the bits corresponding to xl, and if xi,y. in Qni then are adjacent f(x;xl;x2;...;x.:.l.;x ) and f(x;x1;x2;...;yi;...;x ) differ by a substring’rever% internal to the substringmfi(xi) or

‘(x;Ol) = f(x);A’, let f’(x;lO) = A’;[f(x)]‘, and let f’(x;ll) = A;[f(x)]‘. It is straightforward to verify that f’ embeds Qm+2 as a subgraph of BSSRk+l, proving a). For b) we proceed by induction on m, noting that Q, has such an embedding in BSSR1, serving as a basis step. Suppose f;l; is such an embedding of Q,-, in BSSRm, with Cm on the m burnt symbols XlX2...X . Define f;+l : G+ BSSRm+l as follows, with ZmTl on the burnt symbols X1X2...X,,,A.

For x

E

Q,-,

[fi(xi)lr

Theorem: a) If Qn L SSRk then Qn+l L SSRk+l. b) If Q, C SSRk then Qmn+m-l L SSRmk. c SSR2n. d Q2n+2n-2-1 Proof: Claim a) follows from Lemma 3 via the parameters m=2, nl=O, n2=n, k,=l, %=k. Claim b) follows from Lemma 3 via the parameters m, n1=n2=...=nm=n, kl=k2=...=km=k. Claim c) is proven

let

by induction on n. The basis case n=2 is handled by the embedding illustrated in Figure 1. Assuming that Q2n+2n-2-1 C SSR2n’ then from b) with m=2 we have

Q 2(2n+2n.2-1)+2-1

Lemma lb) fuels the proof of the following Lemma, showing how one can combine various dilation 1 embeddings of hypercubes into SSR networks to produce an embedding of a larger hypercube into a larger SSR network, the dimension of the large hypercube now embedded being a bit larger than might otherwise be anticipated. If Qni L SSRki for i=l,2,...,KI

SSR2n+l,

i.e.

Q2n+~+2n-~-1

c

I

Since Q, has 4-cycles and Pn does not, it is never possible to embed Q, into Pn for m>l via a one-to-one dilation 1 embedding. Hence we consider one-to-many embeddings. Toward embedding hypercubes into pancake networks we first take up the more general problem of embedding Cartesian products of cliques into pancake and burnt pancake networks.

then

Theorem 4: For nl, n2 ,..., nm 22, let m

L = -nm + 2 c

tn,-1)

.

i=l

Then our dilation 1 embedding

a) K,1 x Kn2 x . . . ’

f:Q,-l+~ni-+ SSR+ is defined as follows. For x E Qm-l and xi E Qni let f(x;xl;x2;...;xm) be the string resulting from f’(x) by replacing each Yi in f”(x) by

Knm

b> Kn 1 X Kn2 X ... X Knm I?rxd

Let

AIA2...A

fi(xi),

each Yir in f”(x) by [fi(xi)lr. In other words, f(x;x1;x2;..., ‘xm) is the concatenation of substrings

f(x,),f(x,),...,f(x,)

C

completing the induction.

SSR2n+1>

Qm-l+~ni’ SSR~ki . F&&: Let f’ : Qmml -+ BSSR, be the embedding from Lemma 2b), relative to the m burnt symbols Y 1Y2...Y . Let fi : Qni+ SSRki be dilation 1 embeddings ts assumed, where each SSRki is on the symbols Xi,lXi,J...Xi,ki.

xm) [i.e. one can always

of f(x;x1;x2;...;x+..;

toggle any one of the bits corresponding to xi]. I

f;+l(x;O) = f,“(x);A and let f;+l(x;l) = A’;[f;(x)]‘. It is straightforward to verify that fr;l+l is an embedding satisfying b), completing the induction. Suppressing the orientations of the burnt symbols in each f’(x), we see that f’ induces a dilation 1 embedding of en-, in SSR,. I

-3:

1995

sting in symbols symbols good if

formed by arranging them in the order

determined by f’(x), reversing those f(xi)‘s for which Yi appears reversed in f”(x). To see that f is a dilation 1 embedding, observe that if x,y are adjacent in Qmel then

A ‘m(2L+2) A

BPm(2~+l

)

P,(,L+~) he on permutations of the symbols

2mL+IB1B*‘“Bm-1DlD2’.‘D m’ A symbol PrnczL+2)is called go& if between each pair of Dh and Di is some Bk and between each pair of Bh and Bi is some Dk. That is, the string is upon deletion of the Ai symbols the string

alternates between D and B symbols, starting and ending

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Procredirqs of the 28th Annual Hawaii International Conference on System Sciences -

with a D symbol. For example when m=2 and nI=2 and n2=3 we have L=3, and concerning the strings y = ADAA AABAAA DA AAA A and 4 2 2 10 6 5 1 7 9 11 1 13 8 3 12 1 AADAAAAA BA A A ’ =A1A2D2% 4 5 1 6 7 8 9 10 1 11 12 13 in P16 we have that y is good while z is not. In a good string the D symbols partition the set of A and B symbols into exactly m+l blocks, some of them possibly empty. If symbols Bi and Ah are in the same block then we say that Ah is in the i-th block; notice that if Bh is to the left of Bi then the h-th block is to the left of the i-th block, even if ich. Symbol Ah is said to be

1995

block, where we are free to exchange those x0 many A’s from the 0-th block with any number of the xii many A’s on the left side of the i-th block. Since the range of allowable values L < xii 5 L+ni-1 consists of ni many consecutive integers, exactly one of them, say wli, such that w’~+x”~ mod ni = zi (and similarly if i=m). Thus we are done if there is a prefix reversal which renders things so that the left side of the i-th block has wli many As. Some thought shows that this is always possible due to the constraints involved, in such a way that the number of A’s in the new 0-th block is still between 0 and L+l, so that the resulting string is in R. Thus a) is proved. The proof of b) is nearly identical. One difference is that in defining the domain R of g we require that each A symbol in the i-th block appears reversed if either it is to the left of Bi in its block or if i=m. Thus the m many D symbols which previously served to delimit the extent of the various blocks are no longer required; upon deletion of the D symbols one can still determine the extent of each block by observing the breaks between consecutively reversed A’s and ordinary As. Therefore without the D’s there are only m(2L+l) symbols are now involved, although they are now burnt. I

in the Q-th block [resp. m-th block1 if it appears to the left [resp. right] of all D symbols. For a good string x let xi denote the number of A symbols in the i-th block and for i=1,2,...,m-1 let xti [resp. x”~] denote the number of symbols in the i-th block that are to the left [resp. right] of Bi. We are finally in position to specify our coembedding (R,g). Let R be the set of good strings x satisfying 0 5 x0 I L+l and L < xm 5 L+nm-1 and L 5 x’. I L+ni-1 and L I x’li I L+ni-1 for each i=li , ,..., m-l. For x E R let g(x) = (x1,x2 ,..., xm) where we regard the i-th coordinate xi by its remainder modulo ni so that it is one of the integers O,l,...,ni-1, thereby representing one of the ni vertices of K,.. For our 1 example good string y above, 0 5 yo=l I L+l = 4 and 3 = L I ytl = 4 2 L+nl-1 = 4 and 3 = L < y’ll = 3 < L+nl-1 = 4 and 3 = L I y2 = 5 5 L+n2-1 = 5, so y E R. We see that g(y) = (4+3 mod 2,5 mod 3) = (1,2), regarded as a vertex in K2 x K3 [whose vertex set is ((a,b): a E

Theorem: @ Q,

if n is even, and

A p(3n+3)(n-2)

Qn

& P(3n.2)(n-1)

b>

Q,

:

Qn

&

BP(3n_2)(n_l)

if n is o&i

BP (3n+3)(n-2)

- (n-2)/2

_ (n-1),i

if n is even, and

if n is odd.

m For n even, apply Theorem 4a) with parameters m = (n-2)/2 and n1=n2=...= n,_l=4 and nm=16 to obtain

(O,l),b~ (U2))l.

that K4 X K4 X . . . X K4 X K16 A P(3n+3)(n-2).

To show that g is a dilation 1 co-embedding we now show for each good string x E R how to “toggle” the i-th coordinate xi of g(x) to any choice of zi. That is, given x E R with g(x) = (x1,x2 ,..., xm) and some zi E (Ql,..., ni-

Q, C K4 x K, x . . . x K4 x K16, we have proven a) when n is even. For n odd apply Theorem 4a) with parameters m = (n-1)/2 and n1=n2=...= nrnl=4 and n,=8, completing the proof of a) in analogous fashion. The proof of b) follows the exact same choices of parameters as above, except as applied in Theorem 4b). I

1) with zi f xi we must show that there exists z E R for which g(z) = (x1 ,..., x~_~,z~,x~+~ ,..., xm). For our example string y, to toggle the 2-nd coordinate x2 from its current value of 2 to the new value z,=O we apply the prefix reversal Rev[l,l4] to obtain the string z = AAA DA AABAAA ADAA A 3 8 13 1 11 9 7 1 5 6 10 2 2 4 12 1 which is in R and satisfies g(z) = (1 ,O), there now being a multiple of 3 many A’s in the 2-nd block. While a prefix reversal may reverse some entire blocks, notice that applying a prefix reversal can only change the number of A symbols in the 0-th block and in one other block, say the i-th block. Such a prefix reversal which affects x. will bring all elements of the 0th block into the i-t;

Since

We claim that Theorem 5a) is the best result of its kind that can be derived from direct application of Theorem 4a), although we omit the optimization proof for brevity. While on an asymptotic basis Theorem 5 yields our best known results for dilation 1 embeddings (one-to-many or otherwise) of hypercubes into pancake and burnt pancake networks (as measured by how large the dimension is of the hypercube being embedded), the following result gives our best results when the dimension of the target pancake network is small.

541



Thmrem:

is a magnitude better than each of our previous embeddings in which each Cayley graph symbol accounted for at most a constant number of hypercube bits.

Qk ‘Z p2k and Q, c BP2k.

EcQQf: The result is proved by induction. The basis case k=O is easily checked, and the induction step follows from Lemma lb). I

3.

Dilation

1995

Thenrem: a) For n 2 m, (Km)n-m+l ?, CPn , where

2 embeddings

We next give dilation 2 embeddings of hypercubes into pancake networks. The one given in Theorem 7a) is one-to-one, but otherwise inferior to the one-to-many embedding given in Theorem 7b).

(Km)n-m+l =KmxKmx...xK k

m’ 2) CPn .

2 9 Qk(n-2k+l) Let Zk be on the symbols A1,A2,...,A,Let R be the set of those m+l,D1,D2....,Dm~1. permutations of the form ao;Dl;al;D2;a2;D3;...;am~ 2;Dm-l;am-l in which within each substring ai it is

b) For

n 2

Proof:

Theorem: a) For n 2 4, Qn ?, P2”-2 .

b) Q, *2 P2n’ && Figure 2 shows a dilation 2 embedding of Q4 into P4, proving a) when n=4. Straightforward iteration of Lemma lb) completes the proof of a). For b), let f’ be the one-to-many dilation 1 embedding of Qn into DCP2, as given in Theorem 2a). Network DCP211 embeds as a subdigraph into CP2n via the identity embedding function id, and in turn CP2, has a dilation 2 embedding f’ into P2n, as shown in Theorem

the case that the A’s appear in lexicographic order, i.e. Ae appearing to the left of Ad within substring ai implies that c < d. For an arbitrary string S = in R define the coao;D1;a1;D2;...;am~2;Dm~1;am~l embedding g by g(S) = t(l);t(2);...;t(n-m+l) where symbol Ai appears within the substring at(i) of S. To toggle t(i) we can apply a single prefix cycle to bring symbol Ai to the front of the string and then another

la). Therefore the composition f’ 0 id 0 f’ is a one-tomany dilation 2 embedding, proving b). I

prefix cycle to move symbol Ai into any ah of our choosing, making sure that it is filed within ah

As should be no surprise, we can do considerably better embedding into burnt pancake networks.

lexicographically. Thus t(i) is rendered equal to any integer from 0 to m-l without affecting any other coordinates t(i). proving a). Claim b) follows from a) via the choice m = 2k, since Q, C K2k. 1

-em

8: Q$ BP,. &&I Let C2n be on the symbols X1X2...Xn. Let R = C2n, and for a string S E X2,, define the co-embedding g

4.

by g(S) = blb2...bn where bi = 1 if Xi appears in reversed orientation, bi = 0 if Xi appears in ordinary orientation. To toggle the i-th bit of S when burnt symbol Xi or its reverse appears as the j-th symbol of S where j>l, apply the generator rev[ 1 j-l] and then rev[ 1 j]. This reverses the orientation of Xi without altering the orientations of any other burnt symbols. rev[l,ll. I

Dilation

3 embeddings

Continuing with the theme of using divider symbols, we present our best dilation 3 result. As notation, given a set of strings ~~,s~+~,...,sd over some alphabet, we let Con I i=c&$ denote the string s = s&;...sd obtained by concatenating the Si in the order of increasing i proceeding along s from left to right. Theorem: For any m I [log2(k) i-1, there is aone to many dilation

In the case j=l simply apply

3 map EQ,+PQ, where u=m(k-2m+1). &J& We view V(PQ) as the set of permutations on the set AUBUD, where B = (B(l),B(2) ,..., B(2m)), D = (D(l),D(2) ,..., D(2m)). and A = (A(l),A(2) ,..., A(k2m+')}. We will define a set RC P(k) and a coembedding g:R+Q,. For convenience, in the following description we will omit the indices of the D-symbols, and just write each of them as D. We then let R be the set of all strings in P(k) of the form

In one instance we know of a dilation 2 embedding of an (n+l)-dimensional hypercube into an n-dimensional burnt pancake network, an improvement over Theorem 8. This is in the case n=2, where BP2 happens to be isomorphic to an 8-cycle, and there exists a dilation 2 embedding of Q, into BP2 To embed hypercubes into cycle prefix networks we again make use of designated “divider” symbols. In the Theorem 9b) below, each non-divider symbol accounts for k of the bits of the hypercube being embedded. Thus this 542


Proceedings of the 28th Annual Hawaii International Conference on System Sciences - 1995

the integer is such that A(r) is a symbol in the substring a(s) when i,>O, and be the all O’s string of length k-l when A(r) is a symbol in the substring a(0). Proceeding inductively suppose that for each i, Oliltck-2, we have constructed a string S(‘)EP(~~-’ -1) together with with a binary string b(r,S(i)) of length k-i-l for each llr >* (*I where d = 2m, and where the a(i) are strings over the A’s whose letter sets partition the set A. For a given SER expressed as in (*). define g(S) as follows. For each r, 15 r I k-2m+1, we let P(r,S) be the length m binary string which is the binary equivalent of the integer it such that the symbol A(r) is in the substring a(t)B($)a’(t). We then let g(S) = Con(r=l,k2m+1;b(r,S)). To see how to toggle an arbitrary bit, let this bit be found in the substring b=P(r,S) of g(S), and let b’ be the string obtained from b by flipping this bit, Let it be the integer equivalent of b’, and write x for the substring a(t)B(i&‘(t) of S. We can then write S = DaDxDPA(r)G for suitable substrings a, /3, and 6 of S, where a similar argument can be given when x follows A(r) in the string S. Now let S’= DaRDA(r)xDPG, so that P(r,S’)=b’ and pci.s’)=pci,s) f or any j#r, and hence S’ is a string obtained from S by toggling the given bit. Finally we observe that dist(S,S’) I 3 by via the pancake steps S-$RDxRDaRDA(r)G+A(r)DaDxD~&-+S’, thereby completing the proof of the theorem. I

5. Dilation

$9 = a(‘)(O)B(‘)(i,)a(‘)(

1)B(i)(i2)a(i)(2)...B(i)(im)a(i)(m),

where m = 2k-i-1 -1, and where the a(‘)‘s are strings over the A(i)‘s. We call the substring B(i)(it)a(i)(t) of S(‘) a block of S(i) with index it, and also a block level i. Now let S(i+l) ~P(2~~~~l-l) be the string from S(i) by replacing block B(i)(Cd)a(i)(d) by B(i+l)(r) where 15r12 k-i-2 -1 and cd=2r, and

(of S) at obtained a symbol replacing

such a block by the symbol A(i+l)(r) where 1 + QT(n>v Now suppose n is an arbitrary integer different than a power of 2 less one, say 2k -l