Embeddings of Complete Binary Trees into Star Graphs ... - CiteSeerX
Proceedings of the 28th Annual Hawaii Inrernarional Conference on System Sciences - 1995
Embeddings of Complete Binary Trees into Star Graphs with Congestion 1* M.-C. Heydemann, J. Opatmyi, D. Sotteau LRI. U.4 410 CNRS. bit 490, Universitd de Paris-Sud 91405 Orsay, France iDepartment of Computer Science, Concordia University Montrial. Canada
Abstract
u as an int.erna.1 vertex. The vertex-congestion of f is the masimum of the congestions of the vertices of H. The cougestion of an. edge (u, V) of H in the embedding f is the number of paths in {P,(e) : e E E(G)} containing (u, v) as an edge. The edge-congestion off is the maximum of the congestions of the edges of H. The siar graphs were proposed in [l] as a topology for int.erconnecting processors in large scale parallel computers. These graphs belong to the family of Cayley graphs [4], a family of graphs obtained from representations of groups, and they have many interesting properties [ 11. Let n be a positive integer, n 1 2. The star graph S, of dimension 11is a graph whose vertex set consists of all permut,ations of { 1,2, . . , n}. The ith coordinatc of any vertex ~1x1.. . I, of a star graph denotes the ith position from the left in ~1x2 . . . t,. A vertex ;u = tlz?. . . z,, is adjacent. to the vertices Xix?. . . Xi-lXlXi+l . . .X,, for 2 < i 5 n, i.e., vertices obtained by a transposition of the symbol in the first coordinate with the symbol in the ith coordinate of the vertex for 2 5 i < n. For 2 _< i 5 n, we define the generator 9; to be a permutation which transposes the symbol in the 1st coordinate with the symbol in the ith coordinate. Thus a vertex I is adjacent to 93(x), 93(x), . . . , g,,(x). The star graph of dimension n has n! vertices and each of its vertices is adjacent to n - 1 other vertices. The diameter of the star graph is [+(n - 1)J (see [l]). The following property of star graphs will be used in this paper. For any coordinate greater than 1, the subgraph of S, induced by all the vertices having the same symbol in a given coordinate is isomorphic to a star graph S,-1 of dimension n - 1. In particular, the subgraph induced by all the vertices of S,, having the same symbols in the last k coordinates, k < n - 1, is isomorphic to a st.a.r graph of dimension n - k. For any nonnegakive integer h. the complete binary
we giae a conslruc2io1~ of embed1 and dilaiiou 4 of complete binary trees into star graph,s. The heighl of the trees embedded into th.e n-dimeasloaal star graph S, L’OgznJ+l + 1, which tmprorcs th.e is(7?+1)~log~nJ-2 previous result from. [5],[6] by more thnzt n/2 - 1. Il’e then consiruct em.beddings of zjerter-coagestion 1. dilation at mosi an + 2, of complete bzuary trees tato ihe n-dim.ensional star graph, wh.osf heighi differs from the theoretical upper bound of log, n! by less then 3 [log, nl . Our results show that the star netzrorks call eficiedy simulate algorithms thai arc lnfended for a binary tree architecture. In
this paper,
dings of vertex-congestion
1
Introduction
In the field of interconnection net.works, the study of graph embeddings is motivated in particular by the problem of efficient simulation of an interconnection network or a parallel algorithm on some other interconnection network. Let G and H denote two simple, undirected graphs. An embedding of the graph G into the graph H is a mapping f of the vertices of G into the vertices of H together with a mapping P, which assigns to ea.ch edge (u, u) of G a path between f(u) and f(u) in H. In this paper, we will consider only injective mappings. The dilation. of the embedding f is t.he ma.ximum of the lengths of the paths in (P,(e) : e E E(G)}. The congestion of a vertez u of H in the embedding f is the number of paths in {P!(e) : e E E(G)} containing *The work was supported partially by NSERC of Canada and bv PRC C3 of France and was done while the 3rd author was vking McGill Universit.y.
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Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS'95) 1060-3425/95 $10.00 © 1995 IEEE
Proceedings of the 28th Annual Hawaii International Conference on System Sciences - 1995
lree of height. h. denot.ed Tj,. is the binary tree where each internal vertex has esactly two children aud all the leaves are at distance h from the root of the tree. For a comp1et.e bina,ry t,ree Th. t.he le?:~I i. 0 < i 5 h. is defined as the set of all vertices of T,, a.t distance i from the root. of the tree. The t,ree Th has h + 1 levels and level i, 0 5 i 5 h, conta.ins 2’ vertices. Thus Th has 2”+’ - 1 vertices. The problem of embedding a network into anot.her one has been considered in t,he last. t.en years for net.works such as grids, hypercubes, and t,rees (see [lo]). The problem of embedding a graph int.o st.ar graphs has been already studied for some families of graphs. Nigam et, al. [11] showed that. t.he star graph S,, contains a Hamiltonian cycle for every n, n > 2, and presented an embedding of hypercubes ilit. t,he st,ar graphs (see also [3], [9]). Jwo et, al. [S] considered embeddings of cycles and grids into star graphs. In [5],[6], we have considered embeddings of complete binary trees into the st.ar and pancake graphs with dilations 1, and 26, 26 + 1 for 6 2 1. Our aim was to optimize the height. of the embedded trees. irrespective of the congestion. Let G be a net,work that is t.o be simulat.ed on a. network H. and f be an embedding of G int.o H. If, in H. communications use the st.ore and forward mode of communication, then t.he efficient simulat.ion of G will require that, the dilation of f is small. However, if transmitt.ed messages are not. st.ored in t,he intermediate nodes, then it can be more important t,o minimize the vertes-congestion or t,he edge-congest.ion of the embeddings. The embeddings of complete binary trees int.o star graphs from [5],[6] 1iave vertex-congestion 1 only if their dilation is 1. Alt.hough t,he height. of the complete binary tree embedded into the n-din1ensiona.l st,ar graph S, with dilation 1 is asympt.ot.ically optimal, it differs from the theoretical upper bound of log,(n!) by a factor Q(n). In this paper. we improve the height of complete binary trees that can be embedded into S-, with vertex-congestion 1 by considering embeddings with dilation greater than 1. In Section 2, we give a construction of embeddings of vert,ex-congestion 1 and dilation 4 of comp1et.e binary trees into star graphs. The height. of the trees embedded into the star graph S,, is (11+ 1) [log? nJ 2L’OgznJ+l + 1, which improves the previous result by more than n/2 - 1. Section 3 describes the construction of embedclings of vertex-congestion 1, dilation at most jn+2, of complete binary trees into the n-dimensional star graph. The height. of the embedded t,rees differs from the bhe-
oretical upper bound by less then 3 [log, nl . We also give there a table that summarizes our results for small values of n. Our results show that the star networks can efficiently simulate algorithms that are intended for a binary tree architecture.
2
Embeddings
of constant
dilation
Let us recall the following result from [5],[6] which gives an embedding with dilation 1 and therefore vertex-congestion 1. 2.1 For n = 5 or 6 there exists a dilation 1 em.beddisg of th.e complete binary tree of height 2n - 5 into the star graph S,. For n 2 7, there exists a dilatioa I embedding of a complete binary tree of height at leasi equal to (1/2+c(n.))n log:! n into th.e star graph S,, where e(n) tends to 0 when n tends to infinity. Theorem
In [5],[6], 1.t .IS s1lown that the height of complete binary trees embedded into S, with dilation 2 excedes the height, of the trees from the above theorem by at. least n/2. Thus the difference between the theoret,ical upper bound of [log*(n! + 1)J - 1 and the height of the trees embedded into S,, according to the above theorem is Q(n). We will now show how to improve t.he a.bove result on the embeddings with vertexcongestion 1 by considering embeddings with higher, constant dilation. From the proof of Theorem 2.1 given in [5] we can derive t.he following weaker result that will be sufficient, in our constructions. Corollary
2.2 For n 2 5, there exists a dilation 1 em.bedding of a com.plete binary tree of height at least equal fo 2[logz(n + 3)1 - 1 into the star graph S,.
Proof. The corollary is true for 5 5 n 5 16 by considering the values in Table 1 of [5]. By Theorem 2 of [5], for (p + 1)2P-’ < n 5 (p + 2)2P and p 2 3 (which implies n > IS), there exists a dilation 1 embedding of a complete binary tree of height at least p(n - 2P) + 3 into S, . So it remains to verify that, for the considered values of n, we have p(n -2P)+3 2 2[logZ(n+3)1 - 1. cl The constructions of embeddings of complete binary trees of vertex congestion 1 and dilation 4 will be based on the following proposition. Proposition
2.3 Let n be an integer greater than or equal to 6. If n. is n.ot a power of 2 and there exists an embeddang of a complete biaay tree of height
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Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS'95) 1060-3425/95 $10.00 © 1995 IEEE
Proceedings of the 28th Annual Hawaii Inrernarional Conference on System Sciences - 1995
h tat0 a (IJ - 1)-&men.sionnl si.nr groplt S,,-, ~if.h vetiet-congestion 1 and dilaitotl 4. then then tx~sfs an embedding into a n-dimensional star graph S,, of a com.plete bin.ary tree of heighl h + [log, I?]. ofsn with vertex-congestion 1 and dilalzoi, 4. If n is a power of 2 and there errs13 (III embedding of a com.plete binary tree of hezghf h’ into 5’,,-r, with vertex-congestion I and dilation 4. their tberc et&s an embedding into S,, of a complete bttaary IIW of heighf h’ + 2 log, n. - l! also with llerie~-colrgcsfiott 1 and dilation 4.
Proof : Let p = [log? !I!. Decompose the comp1et.e binary tree Th+p (or Tht+zl,-l ) int.o vert.es-disjoint, complete binary trees T”. T’. . . . T?’ where T” is the subtree induced by t.he first. 11 levels of T,,+* (or Tht+2p-1) and T’, T?. . . . . T?’ are t.he subt.rees of T,,+p (or Thl+zP-l) of height. h (or h’+y- 1) having root,s a.t level p. Decompose the st.ar graph S,, int.0 n disjoint substars S’ ( S? . . , 5’” of dimension VJ - 1 where S’ 1 5 i 5 n. is the substar of S,, induced by the vertiiei ending with i. M’e consbruct. an embedding of Th+,, (or T~J+~~-I ) into S,, by embedding T” into 9. T’ into Si , for 1 5 i 5 2&‘( and by specifying vert,ex-disjoint. paths on which will be mapped t,he edges bet.neen the leaves of T” and the roots of T’ . T’. . T?’ (see Figure 1). The details of the proof are divided in two cases. Case 1 : n is not. a power of 2. By hypothesis. for 1 5 i 5 21’. t.here esist.s an embedding of T’. which is a complet,e binary t.ree of height. h, into s’, a star gra.ph of dimension I) - 1. By the vertex transitivity of star graphs. t,he root of T’ can be mapped on any vertex of S’. Now we define the embedding of T” and of t,he edges of Th+p incident with the leaves of T”. Consider Sn-‘)“‘, the substar of S” of dimension n - 2 induced by all the vertices ending with (n - 1)n. B, Corollary 2.2, for n 2 T, there esists a.n embedding of dilation 1 of a complete binary t.ree of height, 2 [logz( n+ 1)1- 1 into a star graph of dimension n - 2. Since 2[log2(n. + l)l - 1 2 [log, nJ = y. t,liere exists an embedding f of dilation 1 of a complete binary tree R of height p into S(n-l)n. Such an embedding also exists for n = 6 since T2 can be easily embedded into Sd with dilation 1. Obviously, the vertex-congestion of such an embedding is 0. The embedding of the levels 0 up to p - 1 of R serves as the embedding of T” into 9’. The embedding of the levels y- 1 and y of R is used to const.ruct vertex-disjoint paths on1.o which we map the edges of Th+P bet.ween t.he leases of T” and the roots of the T”s. Let. xj, 1 < j 5 2!‘-‘. be the jth leaf of Tu and lj . 1 5 j 5 2”. hr the jt,h 1ea.f
of R from the left.. For any j, 1 _< j _< 2*“, the edge of T bet.ween xj and the root. of T’j-l is mapped on the following path: f(tj), f(l?j-1) = C.fltl?...(n - l)n, (n - l)az...aln, (2j - l)az. ..aln, if al # 2j - 1, and f(Xj). f(/,j-1) = (2j- l)az...(nl)n, na.? . . . ( TJ - l)(Zj - 1) otherwise. Similarly, the edge of T between xj and the root of T2j is mapped on the following path: f(tj), j(lzj) = blb?...(n - l)n, (n - l)ba...bln, (2j)bz . . ,bln, nb:!. . .b1(2j) if bl # 2j and f(xj), f(l2j) = (2j)ba..-(nl)n, nbz...(n1)(2j) otherwise. Note that in this case, if 2j = n - 1, which may ha.ppen if n. = 2P + 1, then we simply omit the loop in the given path. For each j, 1 5 j 5 2P-l, the last vertices of the above paths will be the images of the roots of T2jW1 and T’j in the embeddings of these trees into S2jm1 and S’j Any vertex of S” is used at, most once either as an image of a vertex of T”, or as an internal vertex of a path on which is mapped an edge connecting 2’” to T’, 1 5 i 5 2F. Indeed, since the first internal vertices of the a.bove paths are obtained from the embedding of the leaves of R, they are pairwise different and distinct from the images of the vertices of 2”‘. Since all the second internal vertices (if they are needed) are obtained from the first ones using the same generator y,-l, they must. be also all distinct and they are in S” - 9”-l)“. The third internal vertices are obtained from t,he second ones, bringing 2j - 1 or 2j in the first positions. and are therefore also all distinct and different from the previous ones. The last vertices of the paths are in different, substars s’, 1 5 i 5 2P, and these vertices are used as the vertices on which are mapped the roots of T’. Clearly, the dilation of the embedding is at most 4. Case 2 : ii is a power of 2, i.e., n = 2P, p > 3. Notice that, in that, case, the last subtree T” rooted at level p is embedded into the same substar S” of dimension I) - 1 as the tree T” . Thus its embedding will involve more work to make it of vertex-congestion 1, and will be treated separately. For 1 5 i 5 2P - 1, T’ is a tree of height h’ + p - 1. Since n - 1 is not a power of 2, and Llog2( n - l)J = p - 1, by hypothesis and the result, of Case 1, there exists an embedding of r’ into s’, a star graph of dimension n - 1. By the vertex transitivity of star graphs, the root of r’ can be mapped on a.ny vertex of 5”. We will use such an embedding for 1 2 i 5 n - 1. Define the embedding of T”, of T”, and of the edges of T,,t+zl,- 1 incident to the leaves of T” ( into s” as fol-
Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS'95) 1060-3425/95 $10.00 © 1995 IEEE
Proceedings of the 28th Annual Hawaii International Conference on System Sciences - 1995
lows. Consider S’n-3’(“-1 In. a star graph of dimension n - 3 induced by all vertices of S, endiug with (n - 2)(n - 1)n. By Corollary 2.2. r.here exist.s an embedding f’ of dilation 1 of a complete binar!. tree R’ of height 2p - 1 = 21og, n - 1 int.o S”‘-L’)“‘-l)‘l. a substar of dimension n - 3. Obviously. t,he vert.escongestion of such an embedding is 1. The embedding of the levels 0 up to p - 1 of R’ serves as the embedding of T” into 9’. The embedding of the first 2t’- 1 vertices of level p of R’ is used for the first internal vertices of the vertex-disjoint. paths on which the edges of Thi+zp-l between the lea.ves of T” and the roots of the T”s, 1 5 i < 2P - 1. will be mapped. These paths are construct.ed esactly as in Case 1 above. The second internal vert.ex of each pat,11is mapped on the vertes obtained by int,erchanging 7) - 1 and t.he symbol in the first, co0rdinat.e of t,he vert.es of level p. etc. Notice tha.t all internal vert,ices of these pa.ths are either from the embedding of level 1’of R’ (and therefore in CJn-2)(n-l)n)q or ill CJnm2byT_ S(tz-l bn. \\~]lere Sn-‘)*” is the substar of S” induced by the vert.ices having n - 2 and R a.t coordinates II - 2 and n. Also. all the paths a.re vertex-disjoinr. The last vert,ices of the paths are in different, substars s’. 1 2 i 5 2” - l( and these vertices are used as the vert.ices on which are mapped the roots of T’.
and the last, vertex is obtained from the previous one by transposing the first and (n - 2)nd coordinate. The last vertex of the path is then used as the image of the root in the vertex-congestion 1 and dilation 4 embedding of T’j, of height h’, into Sj*“, a star graph of dimension n - 2, an embedding which exists by the hypothesis of the proposition. Notice that all the internal vertices of these paths are either from the embedding of level 2p - 1 of R’ into S(n-2)(n-1)n, or in _ s(n-1)” s n-2f*n where S(“-2)‘” is the substar of S” of dimension n, - 2 induced by the vertices having n - 2 and n at, coordinates n - 2 and n. Therefore, all the paths are vertex-disjoint. Clearly this embedding of T” 1 as well as the embedding of Th!+zp-l, is 0 of vertex-congestion 1 and dilation 4.
Theorem
2.4 For euery integer n > 5, there is a verler-congestion. 1 and dilation 4 embedding of a complete bin.ary tree of height at least equal to (II + l)[log, n.J - 2L’06znJ+1 + 1 into the star graph S,.
Proof. Let h(n) denote the height of a binary tree tha.t we ca.n embed into S,. We will show by induction on n that h(n) 1 (n + l)[log, nj - 2L106znJ+1+ 1. There exists a dilation 1, and therefore congestion 1, embedding of T5 into Ss (result from [5],[6]). Thus the lower bound for h(n) is true for n = 5. Assume that the lower bound is true for L, 5 5 k 5 n - 1. Let n be an int,eger grea.ter than 5, and let p = Llog, nJ. If n is not a power of 2 then [log, 7?J= [log,(n - l)J = p. By Proposition 2.3, h(n) > p+ h(n. - 1). Thus, by the induction hypothesis, h.(n.) 1 p+ n[log:,(n - l)J - 2Ll”g~(~-~)J+~ + 1, and we get the lower bound for n. If n. is a power of 2, i.e. n = 2P,p 1 3, then [log, nJ = p = 1 + [log,(n - 2)J. Since h(n - 2) 2 (11 - l)(p - 1) - 2’ + 1 = (n - 1)p - 2n + 2 by the induction hypothesis, and h(n.) 2 h(n - 2) + 2p - 1 by Proposition 2.3, we get h(n.)>(n+l)p-2n+l=(n+l)p-2P+‘+l. Cl
Denot,e by 1’the last. vertex of level p of R’. In the embedding of T” into S” , the root, of T” is mapped on the image of r. Since n - 1 is not. a. power of 2. and since by hypothesis there esist.s an embedding of a t,ree of height h’ int,o a star of dimension n - 2. then. by t,he result of the first case of this proposition. there exist.s an embedding of II”‘, a tree of height h’+ [log?( n - I)], into S” , a star of dimension n - 1. The coiistruction of such an embedding is similar t,o that of Case 1: T” is decomposed into T’“, T’l~ T”. . . . T@‘-’ The embedding of the first p - 1 levels of t.he subtree of R’ rooted at 7’(levels p up to 2p- 2 of R’) serves for the ‘” , and the T’j’s a.re embedding of T’u into .Sn-“(“embedded into substars Sj** of S”, 1 < j 5 2P-‘. where Sj*” is the substar of dimension n - 2 induced by the vertices of S having j in coordina.te n - 2. and n in the last coordinate. The embedding of the last level of the subtree of R’ rooted at 1’(level 2p - 1 of R’) is used for the first internal vertices of the vertexdisjoint paths onto which are mapped the edges between T”’ and the subtrees T’j’s. The second internal vertex of any such path is obtained from the first. one by a transposition of n - 1 in coordinate n - 1 wit.11the first symbol (if necessary). The third internal vertex is obtained from the second one by t.he transposition of the coordinat,e containing j wit,11 the first coordinat,e.
The values for the heights of the trees we can embed with vertex-congestion 1 for n up to 16, based on the results of this section, are included in the table summarizing all results at the end of the next section. Notice that. the height of the complete binary tree embedded with dilation 2 and vertex-congestion great,er than 1 into S, from [5],[6] is at least (n + l)[log, 17J - 2i’06,nJ+1 + 2, which is one more that t,he height of the complete binary tree embedded with vertex-congestion 1 and dilation 4 into S,, in the above 549
Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS'95) 1060-3425/95 $10.00 © 1995 IEEE
Proceedings of&e 28th Annual Hawaii International Conference on System Sciences - 1995
theorem. Since the difference between the heights of complete binary trees that can be embedded with dilation 2 and 1 was shown in [5],[6] to be at. least g for n. > 12. we obtain that the difference between the heights of the trees from Theorem 2.1 and Theorem 2.4 is more than 5 - 1 for n 1 12. Recall that an upper bound on the height of the complete binary tree that can be embedded into S,, irrespective of any constraints is llog,( n! + 1) - 11. In [5],[6], there is a construction of an embedding of dilation 4, vertex-congestion > 1, of a complete binary tree into S, . The height of the tree is shown to excede the bound on the embeddings of trees with dilation 2 by more than 9 for n 2 12. Thus, the height of complete binary trees that. can be embedded into S, with vertex-congestion 1 a.nd dilation 4 according to Theorem 2.4, differs from the theoretical upper bound by more than 0 for n. 2 12. We will show in the next section that the height, of the trees embedded with congestion 1 can be increased by not. putting a.ny restriction on the dilation, so that the height. will differ from the theoretical upper bound by at most 3 [log? n.1.
3
Embeddings of stricted dilation
trees
with
graph S,, into the substar S” of dimension n - 1 and (n - l)(n - 2) . . . (n - k) disjoint substars S’I~~“‘~* of dimension n - k where s’~~a.“~k is induced by the vertices ending with ili2 . . . ik for different sequences of k symbols, where, for any j, ij is different from n. We construct, an embedding of Th+p into S,, by embedding Tu into S”, and each T’ into a different substar S ili2”‘ik of dimension n - k for 1 s i _< 2P (this is pos**-(n-k)), and by specsible since 2P 5 (n- l)(n-2) ifying vertex-disjoint paths on which will be mapped the edges between the leaves of T” and the roots of T’,Tz ,..., l-2’. By the hypothesis, for 1 5 i 2 2P, there exists an embedding of r’, which is a complete binary tree of height h, into pliz”‘iL, a star graph of dimension n - 6. By the vertex transitivity of star graphs, the root. of T’ can be mapped on any vertex of sili2...ik. Now we define the embedding of T“ and of the edges of Th+p incident with the leaves of T” . Consider S(“-L)“‘(“-l)“, the substar of 5’” of dimension n - k - 1 induced by all the vertices ending - 1)n. By hypothesis there exists with (n - k)...(n a. congestion 1, dilation d2 embedding f of a complete binary tree R of height [log,(n - l)(n - 2). . -(n - k)J into S(n-k)..,(n-l)n. The embedding of the first p levels of R serves as the embedding of T” into 9’. The embedding of levels p - 1 and p of R is used to construct. vertex-disjoint paths on which we map the edges of Th+&, connecting the leaves of T” and the roots of the T”s. Let xj, 0 < j < 2P-1 - 1, be the jth leaf of T” and lj, 0 5 j-5 2; - 1, be the jth leaf of R from the left. For any j, 1 2 j 5 2P-‘, the edge of T between xj and the root of T2jW1 is mapped on the following pa.th of length at most 2k + 2:
unre-
The constructions of embeddings of complete binary trees of congestion 1 and unrest,rict,ed dilation will be based on the following proposition which is a generalisation of Proposition 2.3. Proposition
f(Xj
3.1 Let n an.d k be integers such th.al
)rf(l?j-1)
=
ala2 . ..a.,~-l(n.-k)(n-k+l).~.(n-l)n, (71.- k)a2 * * .a,_r.-lal(n-k+l)...(n-l)n, (il)U? * * .a,-)-lal(n - k + 1). f .(n - l)n, (n-k+l)a2...a,-r-lalil(nk+2)... (n - l)n, . . . , *ik-z(n - l)n, &-la?. . ~an-k-lc41ili2~~ (n - 1)~. . .an-t-lalili2 . ..it-ln. &a?. . .an,t-lalili2 .-.i~-ln, . .
n>4andl log,(2’)! - 3; + 3. By applying Lemma 3.2 with 7. = 2’. we get t,hat, h(2’+‘) > log&2’)! - 3i + 3 + + log,(2’+‘(2’+’ - 1)‘. .(2’ + l)] - 3 > logq(zi+l)! 3(i + 1) + 3 and therefore the result is true for n = 2’+’ So t,he result is true for powers of 2. Assume that n is not a power of 2. i.e.. 2’ < 1) < 2’+’ for some integer i. We just proved t,hat h(2’) > log,(2’)! -3i+3. Then, using the result of Lemma 3.2 we obtain that h(n) >log2(2’)!-3i-t3+log,[n(n-1)...(2’+1)]-3 0 = log, n! - 3(i + 1) + 3. It follows from Lemma. 3.2 t,hat. t.he dilation of the embedding in Theorem 3.3 is bounded by 5 . 2L’“g2nJ-? $2 _< $n $2 (which. indeed. is not so small compared to the diamet.er [$(n - l)] of S,, ). In the above theorem, the lower bound on the height. of trees is not sharp for large values of n. The act.ual height. of the trees that, we can embed int,o S,, . for a given n. is better. For esample, by recursively applying t,he construction from Proposition 3.1. we obtain embeddings of trees into star graph of dimension n whose height differs from the theoretical upper bound by less than 1.5 log, n for all values of n up to 1000.
4
Since our constructions of embeddings of vertexcongest,ion 1 of complete binary trees into star graphs are based on a decomposition of a star graph of dimension n into star graphs of smaller dimensions by fixing the content of the last coordinates of vertices of S,,? these constructions will be essentially valid for any Cayley graph that allows a similar decomposition, for exa.mple the pancake graphs [2], or more generally the recursively decomposable Cayley graphs [7].
Conclusions
In the previous sections. we have presemed two results on embeddings of vertex-congestion 1 of complete binary trees into the star graph S,,. In the first. result, the embeddings are of dilation 4 while they are of dilation at most in + 2 in the second one. However, embeddings of other dilations can be a.lso obtained from the presented results. Clearly, a repeat.ed applica.tion of Proposition 3.1 with a fixed value of P gives constructions of embeddings of vertes-congestion 1 and dilation 2k+2 for any value of R, 2 5 k 5 5.2L’“gznl-3. In all these constructions, there is a. trade-off between the height of the embedded trees and the dilat.ion. Note that the best expansion of our embeddings (i.e. the ratio of the number of vertices of the host. gra.ph to the number of vertices of bhe guest graph) is in 0( n1.5).
There are many problems dealing with embeddings of trees into star graphs that remain open, e.g., embeddings of arbitrary trek and dynamic embeddings. Since the height of complete binary trees that can be embedded into star graphs with vertex or edgecongestion 1 approaches closely the theoretical upper bound. we can conclude that star graphs are very suitable for simulation of complete binary trees on parallel computers having the star graph as interconnection network. if small vertes or edge-congestion is the main fa.ctor 552
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Proceedings of the 28th Annual Hmvaii International Conference on System Sciences- 19%
Table 1: Vertex-congest,ion 1 embeddings of complete binary trees into S, n 3
4 5 6 7 8 9 10 11 l? 13 14 15 16 n
order of S, = n.! 6 24 la0 720 5040 40320 362880 z 3.6 lo6 z4 10’ z 4.8 lo8 z 6.2 10’ cz 8.710” z 1.3 1o12 x 2.1 1ol3
U!
-
I
upper bound on beighi of T 1 3 5 8 11 14 li ‘0 24 Pi 31 35 39 43 [log2(n.! + l)J - I
i
9 11 13 15 li 19 21 “3 25 ‘i
star graph : An attractive alternative to the nConference cube. Proceedings of the International in Parallel Processing, pages 393-400. 198i.
[8] J. Jwo, S. Lakshmivarahan, and S. Dhall. Embedding of cycles and grids in star graphs. Proceedings of the 2nd IEEE Parallel and Distributed Processing Symposium, Dallas, Texas, 1990.
[2] S. Akers and B. Krishna.murthy. A group theoretic model for interconnection networks. IEEE 38:555-5(X.
[9] Z. Miller, D. Pritikin, and I. Sudborough. Small dilation embeddings of hypercubes into star networks. Research Report TX 75083, University of
1989.
[3] S. Bettayeb, B. Gong, M. Girou, and I. Sudborough. Simulating permutation networks on hypercubes. Proceedings of th.e Is1 Latin American Symposium on Theoretical Informatics, Notes in Computer Science, 583:61-70, [4] N. Biggs.
Algebraic graph theory.
28 32 35
[i] C. GowriSankaran. Broadcasting on recursively decomposable Cayley graphs. Discrete Applied Mathematics, to appear, 1994.
[l] S. Akers. D. Hare], and B. I~rishnal~~urt~hy. The
on Com.puters.
12 15 18 21 24 27 29 32
[lo;;n!, -3pog, n1 + 3
References
Transactions
n with higher dilation
height of a t L embedded into dilation 4 1 3 2 5
dilation 1
Texas at Dallas.
[lo] B. Monien and H. Sudborough. Embedding one interconnection network in another. Computing Supplement, 7:257-282, 1990.
Lecture
1992.
[ll]
Cambridge
University Prws, 1974.
M. Nigam, S. Sahni, and B. Krishnamurthy. Em.bedding hamiltonians and hypercubes in the star interconnection graphs. Proceedings of the International
[5] A. Bouabdallah, M.-C. Heydemann, J. Opatrny, and D. Sotteau. Embedding complete binary trees into star and pancake graphs. Submitted to Mathematical Systems Theory, 1993.
Conference
in Parallel
111.340-111.343, 1990.
[6] A. Bouabdallah, M.-C. Heydemann, J. Opatrny, Embedding complete binary and D. Sotteau. trees into star graphs. Proceedings of the conference on the Mathematical Foundation of Computer Science, Lecture Motes ia Computer Science, 841~266-275, 1994. 553
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Processing, pages
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Case
1
Case
2
Figure 1: Construction
of an embedding of a complete binary tree with congestion 1 and dilation
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4
Embeddings of Complete Binary Trees into Star Graphs with Congestion 1* M.-C. Heydemann, J. Opatmyi, D. Sotteau LRI. U.4 410 CNRS. bit 490, Universitd de Paris-Sud 91405 Orsay, France iDepartment of Computer Science, Concordia University Montrial. Canada
Abstract
u as an int.erna.1 vertex. The vertex-congestion of f is the masimum of the congestions of the vertices of H. The cougestion of an. edge (u, V) of H in the embedding f is the number of paths in {P,(e) : e E E(G)} containing (u, v) as an edge. The edge-congestion off is the maximum of the congestions of the edges of H. The siar graphs were proposed in [l] as a topology for int.erconnecting processors in large scale parallel computers. These graphs belong to the family of Cayley graphs [4], a family of graphs obtained from representations of groups, and they have many interesting properties [ 11. Let n be a positive integer, n 1 2. The star graph S, of dimension 11is a graph whose vertex set consists of all permut,ations of { 1,2, . . , n}. The ith coordinatc of any vertex ~1x1.. . I, of a star graph denotes the ith position from the left in ~1x2 . . . t,. A vertex ;u = tlz?. . . z,, is adjacent. to the vertices Xix?. . . Xi-lXlXi+l . . .X,, for 2 < i 5 n, i.e., vertices obtained by a transposition of the symbol in the first coordinate with the symbol in the ith coordinate of the vertex for 2 5 i < n. For 2 _< i 5 n, we define the generator 9; to be a permutation which transposes the symbol in the 1st coordinate with the symbol in the ith coordinate. Thus a vertex I is adjacent to 93(x), 93(x), . . . , g,,(x). The star graph of dimension n has n! vertices and each of its vertices is adjacent to n - 1 other vertices. The diameter of the star graph is [+(n - 1)J (see [l]). The following property of star graphs will be used in this paper. For any coordinate greater than 1, the subgraph of S, induced by all the vertices having the same symbol in a given coordinate is isomorphic to a star graph S,-1 of dimension n - 1. In particular, the subgraph induced by all the vertices of S,, having the same symbols in the last k coordinates, k < n - 1, is isomorphic to a st.a.r graph of dimension n - k. For any nonnegakive integer h. the complete binary
we giae a conslruc2io1~ of embed1 and dilaiiou 4 of complete binary trees into star graph,s. The heighl of the trees embedded into th.e n-dimeasloaal star graph S, L’OgznJ+l + 1, which tmprorcs th.e is(7?+1)~log~nJ-2 previous result from. [5],[6] by more thnzt n/2 - 1. Il’e then consiruct em.beddings of zjerter-coagestion 1. dilation at mosi an + 2, of complete bzuary trees tato ihe n-dim.ensional star graph, wh.osf heighi differs from the theoretical upper bound of log, n! by less then 3 [log, nl . Our results show that the star netzrorks call eficiedy simulate algorithms thai arc lnfended for a binary tree architecture. In
this paper,
dings of vertex-congestion
1
Introduction
In the field of interconnection net.works, the study of graph embeddings is motivated in particular by the problem of efficient simulation of an interconnection network or a parallel algorithm on some other interconnection network. Let G and H denote two simple, undirected graphs. An embedding of the graph G into the graph H is a mapping f of the vertices of G into the vertices of H together with a mapping P, which assigns to ea.ch edge (u, u) of G a path between f(u) and f(u) in H. In this paper, we will consider only injective mappings. The dilation. of the embedding f is t.he ma.ximum of the lengths of the paths in (P,(e) : e E E(G)}. The congestion of a vertez u of H in the embedding f is the number of paths in {P!(e) : e E E(G)} containing *The work was supported partially by NSERC of Canada and bv PRC C3 of France and was done while the 3rd author was vking McGill Universit.y.
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Proceedings of the 28th Annual Hawaii International Conference on System Sciences - 1995
lree of height. h. denot.ed Tj,. is the binary tree where each internal vertex has esactly two children aud all the leaves are at distance h from the root of the tree. For a comp1et.e bina,ry t,ree Th. t.he le?:~I i. 0 < i 5 h. is defined as the set of all vertices of T,, a.t distance i from the root. of the tree. The t,ree Th has h + 1 levels and level i, 0 5 i 5 h, conta.ins 2’ vertices. Thus Th has 2”+’ - 1 vertices. The problem of embedding a network into anot.her one has been considered in t,he last. t.en years for net.works such as grids, hypercubes, and t,rees (see [lo]). The problem of embedding a graph int.o st.ar graphs has been already studied for some families of graphs. Nigam et, al. [11] showed that. t.he star graph S,, contains a Hamiltonian cycle for every n, n > 2, and presented an embedding of hypercubes ilit. t,he st,ar graphs (see also [3], [9]). Jwo et, al. [S] considered embeddings of cycles and grids into star graphs. In [5],[6], we have considered embeddings of complete binary trees into the st.ar and pancake graphs with dilations 1, and 26, 26 + 1 for 6 2 1. Our aim was to optimize the height. of the embedded trees. irrespective of the congestion. Let G be a net,work that is t.o be simulat.ed on a. network H. and f be an embedding of G int.o H. If, in H. communications use the st.ore and forward mode of communication, then t.he efficient simulat.ion of G will require that, the dilation of f is small. However, if transmitt.ed messages are not. st.ored in t,he intermediate nodes, then it can be more important t,o minimize the vertes-congestion or t,he edge-congest.ion of the embeddings. The embeddings of complete binary trees int.o star graphs from [5],[6] 1iave vertex-congestion 1 only if their dilation is 1. Alt.hough t,he height. of the complete binary tree embedded into the n-din1ensiona.l st,ar graph S, with dilation 1 is asympt.ot.ically optimal, it differs from the theoretical upper bound of log,(n!) by a factor Q(n). In this paper. we improve the height of complete binary trees that can be embedded into S-, with vertex-congestion 1 by considering embeddings with dilation greater than 1. In Section 2, we give a construction of embeddings of vert,ex-congestion 1 and dilation 4 of comp1et.e binary trees into star graphs. The height. of the trees embedded into the star graph S,, is (11+ 1) [log? nJ 2L’OgznJ+l + 1, which improves the previous result by more than n/2 - 1. Section 3 describes the construction of embedclings of vertex-congestion 1, dilation at most jn+2, of complete binary trees into the n-dimensional star graph. The height. of the embedded t,rees differs from the bhe-
oretical upper bound by less then 3 [log, nl . We also give there a table that summarizes our results for small values of n. Our results show that the star networks can efficiently simulate algorithms that are intended for a binary tree architecture.
2
Embeddings
of constant
dilation
Let us recall the following result from [5],[6] which gives an embedding with dilation 1 and therefore vertex-congestion 1. 2.1 For n = 5 or 6 there exists a dilation 1 em.beddisg of th.e complete binary tree of height 2n - 5 into the star graph S,. For n 2 7, there exists a dilatioa I embedding of a complete binary tree of height at leasi equal to (1/2+c(n.))n log:! n into th.e star graph S,, where e(n) tends to 0 when n tends to infinity. Theorem
In [5],[6], 1.t .IS s1lown that the height of complete binary trees embedded into S, with dilation 2 excedes the height, of the trees from the above theorem by at. least n/2. Thus the difference between the theoret,ical upper bound of [log*(n! + 1)J - 1 and the height of the trees embedded into S,, according to the above theorem is Q(n). We will now show how to improve t.he a.bove result on the embeddings with vertexcongestion 1 by considering embeddings with higher, constant dilation. From the proof of Theorem 2.1 given in [5] we can derive t.he following weaker result that will be sufficient, in our constructions. Corollary
2.2 For n 2 5, there exists a dilation 1 em.bedding of a com.plete binary tree of height at least equal fo 2[logz(n + 3)1 - 1 into the star graph S,.
Proof. The corollary is true for 5 5 n 5 16 by considering the values in Table 1 of [5]. By Theorem 2 of [5], for (p + 1)2P-’ < n 5 (p + 2)2P and p 2 3 (which implies n > IS), there exists a dilation 1 embedding of a complete binary tree of height at least p(n - 2P) + 3 into S, . So it remains to verify that, for the considered values of n, we have p(n -2P)+3 2 2[logZ(n+3)1 - 1. cl The constructions of embeddings of complete binary trees of vertex congestion 1 and dilation 4 will be based on the following proposition. Proposition
2.3 Let n be an integer greater than or equal to 6. If n. is n.ot a power of 2 and there exists an embeddang of a complete biaay tree of height
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Proceedings of the 28th Annual Hawaii Inrernarional Conference on System Sciences - 1995
h tat0 a (IJ - 1)-&men.sionnl si.nr groplt S,,-, ~if.h vetiet-congestion 1 and dilaitotl 4. then then tx~sfs an embedding into a n-dimensional star graph S,, of a com.plete bin.ary tree of heighl h + [log, I?]. ofsn with vertex-congestion 1 and dilalzoi, 4. If n is a power of 2 and there errs13 (III embedding of a com.plete binary tree of hezghf h’ into 5’,,-r, with vertex-congestion I and dilation 4. their tberc et&s an embedding into S,, of a complete bttaary IIW of heighf h’ + 2 log, n. - l! also with llerie~-colrgcsfiott 1 and dilation 4.
Proof : Let p = [log? !I!. Decompose the comp1et.e binary tree Th+p (or Tht+zl,-l ) int.o vert.es-disjoint, complete binary trees T”. T’. . . . T?’ where T” is the subtree induced by t.he first. 11 levels of T,,+* (or Tht+2p-1) and T’, T?. . . . . T?’ are t.he subt.rees of T,,+p (or Thl+zP-l) of height. h (or h’+y- 1) having root,s a.t level p. Decompose the st.ar graph S,, int.0 n disjoint substars S’ ( S? . . , 5’” of dimension VJ - 1 where S’ 1 5 i 5 n. is the substar of S,, induced by the vertiiei ending with i. M’e consbruct. an embedding of Th+,, (or T~J+~~-I ) into S,, by embedding T” into 9. T’ into Si , for 1 5 i 5 2&‘( and by specifying vert,ex-disjoint. paths on which will be mapped t,he edges bet.neen the leaves of T” and the roots of T’ . T’. . T?’ (see Figure 1). The details of the proof are divided in two cases. Case 1 : n is not. a power of 2. By hypothesis. for 1 5 i 5 21’. t.here esist.s an embedding of T’. which is a complet,e binary t.ree of height. h, into s’, a star gra.ph of dimension I) - 1. By the vertex transitivity of star graphs. t,he root of T’ can be mapped on any vertex of S’. Now we define the embedding of T” and of t,he edges of Th+p incident with the leaves of T”. Consider Sn-‘)“‘, the substar of S” of dimension n - 2 induced by all the vertices ending with (n - 1)n. B, Corollary 2.2, for n 2 T, there esists a.n embedding of dilation 1 of a complete binary t.ree of height, 2 [logz( n+ 1)1- 1 into a star graph of dimension n - 2. Since 2[log2(n. + l)l - 1 2 [log, nJ = y. t,liere exists an embedding f of dilation 1 of a complete binary tree R of height p into S(n-l)n. Such an embedding also exists for n = 6 since T2 can be easily embedded into Sd with dilation 1. Obviously, the vertex-congestion of such an embedding is 0. The embedding of the levels 0 up to p - 1 of R serves as the embedding of T” into 9’. The embedding of the levels y- 1 and y of R is used to const.ruct vertex-disjoint paths on1.o which we map the edges of Th+P bet.ween t.he leases of T” and the roots of the T”s. Let. xj, 1 < j 5 2!‘-‘. be the jth leaf of Tu and lj . 1 5 j 5 2”. hr the jt,h 1ea.f
of R from the left.. For any j, 1 _< j _< 2*“, the edge of T bet.ween xj and the root. of T’j-l is mapped on the following path: f(tj), f(l?j-1) = C.fltl?...(n - l)n, (n - l)az...aln, (2j - l)az. ..aln, if al # 2j - 1, and f(Xj). f(/,j-1) = (2j- l)az...(nl)n, na.? . . . ( TJ - l)(Zj - 1) otherwise. Similarly, the edge of T between xj and the root of T2j is mapped on the following path: f(tj), j(lzj) = blb?...(n - l)n, (n - l)ba...bln, (2j)bz . . ,bln, nb:!. . .b1(2j) if bl # 2j and f(xj), f(l2j) = (2j)ba..-(nl)n, nbz...(n1)(2j) otherwise. Note that in this case, if 2j = n - 1, which may ha.ppen if n. = 2P + 1, then we simply omit the loop in the given path. For each j, 1 5 j 5 2P-l, the last vertices of the above paths will be the images of the roots of T2jW1 and T’j in the embeddings of these trees into S2jm1 and S’j Any vertex of S” is used at, most once either as an image of a vertex of T”, or as an internal vertex of a path on which is mapped an edge connecting 2’” to T’, 1 5 i 5 2F. Indeed, since the first internal vertices of the a.bove paths are obtained from the embedding of the leaves of R, they are pairwise different and distinct from the images of the vertices of 2”‘. Since all the second internal vertices (if they are needed) are obtained from the first ones using the same generator y,-l, they must. be also all distinct and they are in S” - 9”-l)“. The third internal vertices are obtained from t,he second ones, bringing 2j - 1 or 2j in the first positions. and are therefore also all distinct and different from the previous ones. The last vertices of the paths are in different, substars s’, 1 5 i 5 2P, and these vertices are used as the vertices on which are mapped the roots of T’. Clearly, the dilation of the embedding is at most 4. Case 2 : ii is a power of 2, i.e., n = 2P, p > 3. Notice that, in that, case, the last subtree T” rooted at level p is embedded into the same substar S” of dimension I) - 1 as the tree T” . Thus its embedding will involve more work to make it of vertex-congestion 1, and will be treated separately. For 1 5 i 5 2P - 1, T’ is a tree of height h’ + p - 1. Since n - 1 is not a power of 2, and Llog2( n - l)J = p - 1, by hypothesis and the result, of Case 1, there exists an embedding of r’ into s’, a star graph of dimension n - 1. By the vertex transitivity of star graphs, the root of r’ can be mapped on a.ny vertex of 5”. We will use such an embedding for 1 2 i 5 n - 1. Define the embedding of T”, of T”, and of the edges of T,,t+zl,- 1 incident to the leaves of T” ( into s” as fol-
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Proceedings of the 28th Annual Hawaii International Conference on System Sciences - 1995
lows. Consider S’n-3’(“-1 In. a star graph of dimension n - 3 induced by all vertices of S, endiug with (n - 2)(n - 1)n. By Corollary 2.2. r.here exist.s an embedding f’ of dilation 1 of a complete binar!. tree R’ of height 2p - 1 = 21og, n - 1 int.o S”‘-L’)“‘-l)‘l. a substar of dimension n - 3. Obviously. t,he vert.escongestion of such an embedding is 1. The embedding of the levels 0 up to p - 1 of R’ serves as the embedding of T” into 9’. The embedding of the first 2t’- 1 vertices of level p of R’ is used for the first internal vertices of the vertex-disjoint. paths on which the edges of Thi+zp-l between the lea.ves of T” and the roots of the T”s, 1 5 i < 2P - 1. will be mapped. These paths are construct.ed esactly as in Case 1 above. The second internal vert.ex of each pat,11is mapped on the vertes obtained by int,erchanging 7) - 1 and t.he symbol in the first, co0rdinat.e of t,he vert.es of level p. etc. Notice tha.t all internal vert,ices of these pa.ths are either from the embedding of level 1’of R’ (and therefore in CJn-2)(n-l)n)q or ill CJnm2byT_ S(tz-l bn. \\~]lere Sn-‘)*” is the substar of S” induced by the vert.ices having n - 2 and R a.t coordinates II - 2 and n. Also. all the paths a.re vertex-disjoinr. The last vert,ices of the paths are in different, substars s’. 1 2 i 5 2” - l( and these vertices are used as the vert.ices on which are mapped the roots of T’.
and the last, vertex is obtained from the previous one by transposing the first and (n - 2)nd coordinate. The last vertex of the path is then used as the image of the root in the vertex-congestion 1 and dilation 4 embedding of T’j, of height h’, into Sj*“, a star graph of dimension n - 2, an embedding which exists by the hypothesis of the proposition. Notice that all the internal vertices of these paths are either from the embedding of level 2p - 1 of R’ into S(n-2)(n-1)n, or in _ s(n-1)” s n-2f*n where S(“-2)‘” is the substar of S” of dimension n, - 2 induced by the vertices having n - 2 and n at, coordinates n - 2 and n. Therefore, all the paths are vertex-disjoint. Clearly this embedding of T” 1 as well as the embedding of Th!+zp-l, is 0 of vertex-congestion 1 and dilation 4.
Theorem
2.4 For euery integer n > 5, there is a verler-congestion. 1 and dilation 4 embedding of a complete bin.ary tree of height at least equal to (II + l)[log, n.J - 2L’06znJ+1 + 1 into the star graph S,.
Proof. Let h(n) denote the height of a binary tree tha.t we ca.n embed into S,. We will show by induction on n that h(n) 1 (n + l)[log, nj - 2L106znJ+1+ 1. There exists a dilation 1, and therefore congestion 1, embedding of T5 into Ss (result from [5],[6]). Thus the lower bound for h(n) is true for n = 5. Assume that the lower bound is true for L, 5 5 k 5 n - 1. Let n be an int,eger grea.ter than 5, and let p = Llog, nJ. If n is not a power of 2 then [log, 7?J= [log,(n - l)J = p. By Proposition 2.3, h(n) > p+ h(n. - 1). Thus, by the induction hypothesis, h.(n.) 1 p+ n[log:,(n - l)J - 2Ll”g~(~-~)J+~ + 1, and we get the lower bound for n. If n. is a power of 2, i.e. n = 2P,p 1 3, then [log, nJ = p = 1 + [log,(n - 2)J. Since h(n - 2) 2 (11 - l)(p - 1) - 2’ + 1 = (n - 1)p - 2n + 2 by the induction hypothesis, and h(n.) 2 h(n - 2) + 2p - 1 by Proposition 2.3, we get h(n.)>(n+l)p-2n+l=(n+l)p-2P+‘+l. Cl
Denot,e by 1’the last. vertex of level p of R’. In the embedding of T” into S” , the root, of T” is mapped on the image of r. Since n - 1 is not. a. power of 2. and since by hypothesis there esist.s an embedding of a t,ree of height h’ int,o a star of dimension n - 2. then. by t,he result of the first case of this proposition. there exist.s an embedding of II”‘, a tree of height h’+ [log?( n - I)], into S” , a star of dimension n - 1. The coiistruction of such an embedding is similar t,o that of Case 1: T” is decomposed into T’“, T’l~ T”. . . . T@‘-’ The embedding of the first p - 1 levels of t.he subtree of R’ rooted at 7’(levels p up to 2p- 2 of R’) serves for the ‘” , and the T’j’s a.re embedding of T’u into .Sn-“(“embedded into substars Sj** of S”, 1 < j 5 2P-‘. where Sj*” is the substar of dimension n - 2 induced by the vertices of S having j in coordina.te n - 2. and n in the last coordinate. The embedding of the last level of the subtree of R’ rooted at 1’(level 2p - 1 of R’) is used for the first internal vertices of the vertexdisjoint paths onto which are mapped the edges between T”’ and the subtrees T’j’s. The second internal vertex of any such path is obtained from the first. one by a transposition of n - 1 in coordinate n - 1 wit.11the first symbol (if necessary). The third internal vertex is obtained from the second one by t.he transposition of the coordinat,e containing j wit,11 the first coordinat,e.
The values for the heights of the trees we can embed with vertex-congestion 1 for n up to 16, based on the results of this section, are included in the table summarizing all results at the end of the next section. Notice that. the height of the complete binary tree embedded with dilation 2 and vertex-congestion great,er than 1 into S, from [5],[6] is at least (n + l)[log, 17J - 2i’06,nJ+1 + 2, which is one more that t,he height of the complete binary tree embedded with vertex-congestion 1 and dilation 4 into S,, in the above 549
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Proceedings of&e 28th Annual Hawaii International Conference on System Sciences - 1995
theorem. Since the difference between the heights of complete binary trees that can be embedded with dilation 2 and 1 was shown in [5],[6] to be at. least g for n. > 12. we obtain that the difference between the heights of the trees from Theorem 2.1 and Theorem 2.4 is more than 5 - 1 for n 1 12. Recall that an upper bound on the height of the complete binary tree that can be embedded into S,, irrespective of any constraints is llog,( n! + 1) - 11. In [5],[6], there is a construction of an embedding of dilation 4, vertex-congestion > 1, of a complete binary tree into S, . The height of the tree is shown to excede the bound on the embeddings of trees with dilation 2 by more than 9 for n 2 12. Thus, the height of complete binary trees that. can be embedded into S, with vertex-congestion 1 a.nd dilation 4 according to Theorem 2.4, differs from the theoretical upper bound by more than 0 for n. 2 12. We will show in the next section that the height, of the trees embedded with congestion 1 can be increased by not. putting a.ny restriction on the dilation, so that the height. will differ from the theoretical upper bound by at most 3 [log? n.1.
3
Embeddings of stricted dilation
trees
with
graph S,, into the substar S” of dimension n - 1 and (n - l)(n - 2) . . . (n - k) disjoint substars S’I~~“‘~* of dimension n - k where s’~~a.“~k is induced by the vertices ending with ili2 . . . ik for different sequences of k symbols, where, for any j, ij is different from n. We construct, an embedding of Th+p into S,, by embedding Tu into S”, and each T’ into a different substar S ili2”‘ik of dimension n - k for 1 s i _< 2P (this is pos**-(n-k)), and by specsible since 2P 5 (n- l)(n-2) ifying vertex-disjoint paths on which will be mapped the edges between the leaves of T” and the roots of T’,Tz ,..., l-2’. By the hypothesis, for 1 5 i 2 2P, there exists an embedding of r’, which is a complete binary tree of height h, into pliz”‘iL, a star graph of dimension n - 6. By the vertex transitivity of star graphs, the root. of T’ can be mapped on any vertex of sili2...ik. Now we define the embedding of T“ and of the edges of Th+p incident with the leaves of T” . Consider S(“-L)“‘(“-l)“, the substar of 5’” of dimension n - k - 1 induced by all the vertices ending - 1)n. By hypothesis there exists with (n - k)...(n a. congestion 1, dilation d2 embedding f of a complete binary tree R of height [log,(n - l)(n - 2). . -(n - k)J into S(n-k)..,(n-l)n. The embedding of the first p levels of R serves as the embedding of T” into 9’. The embedding of levels p - 1 and p of R is used to construct. vertex-disjoint paths on which we map the edges of Th+&, connecting the leaves of T” and the roots of the T”s. Let xj, 0 < j < 2P-1 - 1, be the jth leaf of T” and lj, 0 5 j-5 2; - 1, be the jth leaf of R from the left. For any j, 1 2 j 5 2P-‘, the edge of T between xj and the root of T2jW1 is mapped on the following pa.th of length at most 2k + 2:
unre-
The constructions of embeddings of complete binary trees of congestion 1 and unrest,rict,ed dilation will be based on the following proposition which is a generalisation of Proposition 2.3. Proposition
f(Xj
3.1 Let n an.d k be integers such th.al
)rf(l?j-1)
=
ala2 . ..a.,~-l(n.-k)(n-k+l).~.(n-l)n, (71.- k)a2 * * .a,_r.-lal(n-k+l)...(n-l)n, (il)U? * * .a,-)-lal(n - k + 1). f .(n - l)n, (n-k+l)a2...a,-r-lalil(nk+2)... (n - l)n, . . . , *ik-z(n - l)n, &-la?. . ~an-k-lc41ili2~~ (n - 1)~. . .an-t-lalili2 . ..it-ln. &a?. . .an,t-lalili2 .-.i~-ln, . .
n>4andl log,(2’)! - 3; + 3. By applying Lemma 3.2 with 7. = 2’. we get t,hat, h(2’+‘) > log&2’)! - 3i + 3 + + log,(2’+‘(2’+’ - 1)‘. .(2’ + l)] - 3 > logq(zi+l)! 3(i + 1) + 3 and therefore the result is true for n = 2’+’ So t,he result is true for powers of 2. Assume that n is not a power of 2. i.e.. 2’ < 1) < 2’+’ for some integer i. We just proved t,hat h(2’) > log,(2’)! -3i+3. Then, using the result of Lemma 3.2 we obtain that h(n) >log2(2’)!-3i-t3+log,[n(n-1)...(2’+1)]-3 0 = log, n! - 3(i + 1) + 3. It follows from Lemma. 3.2 t,hat. t.he dilation of the embedding in Theorem 3.3 is bounded by 5 . 2L’“g2nJ-? $2 _< $n $2 (which. indeed. is not so small compared to the diamet.er [$(n - l)] of S,, ). In the above theorem, the lower bound on the height. of trees is not sharp for large values of n. The act.ual height. of the trees that, we can embed int,o S,, . for a given n. is better. For esample, by recursively applying t,he construction from Proposition 3.1. we obtain embeddings of trees into star graph of dimension n whose height differs from the theoretical upper bound by less than 1.5 log, n for all values of n up to 1000.
4
Since our constructions of embeddings of vertexcongest,ion 1 of complete binary trees into star graphs are based on a decomposition of a star graph of dimension n into star graphs of smaller dimensions by fixing the content of the last coordinates of vertices of S,,? these constructions will be essentially valid for any Cayley graph that allows a similar decomposition, for exa.mple the pancake graphs [2], or more generally the recursively decomposable Cayley graphs [7].
Conclusions
In the previous sections. we have presemed two results on embeddings of vertex-congestion 1 of complete binary trees into the star graph S,,. In the first. result, the embeddings are of dilation 4 while they are of dilation at most in + 2 in the second one. However, embeddings of other dilations can be a.lso obtained from the presented results. Clearly, a repeat.ed applica.tion of Proposition 3.1 with a fixed value of P gives constructions of embeddings of vertes-congestion 1 and dilation 2k+2 for any value of R, 2 5 k 5 5.2L’“gznl-3. In all these constructions, there is a. trade-off between the height of the embedded trees and the dilat.ion. Note that the best expansion of our embeddings (i.e. the ratio of the number of vertices of the host. gra.ph to the number of vertices of bhe guest graph) is in 0( n1.5).
There are many problems dealing with embeddings of trees into star graphs that remain open, e.g., embeddings of arbitrary trek and dynamic embeddings. Since the height of complete binary trees that can be embedded into star graphs with vertex or edgecongestion 1 approaches closely the theoretical upper bound. we can conclude that star graphs are very suitable for simulation of complete binary trees on parallel computers having the star graph as interconnection network. if small vertes or edge-congestion is the main fa.ctor 552
Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS'95) 1060-3425/95 $10.00 © 1995 IEEE
Proceedings of the 28th Annual Hmvaii International Conference on System Sciences- 19%
Table 1: Vertex-congest,ion 1 embeddings of complete binary trees into S, n 3
4 5 6 7 8 9 10 11 l? 13 14 15 16 n
order of S, = n.! 6 24 la0 720 5040 40320 362880 z 3.6 lo6 z4 10’ z 4.8 lo8 z 6.2 10’ cz 8.710” z 1.3 1o12 x 2.1 1ol3
U!
-
I
upper bound on beighi of T 1 3 5 8 11 14 li ‘0 24 Pi 31 35 39 43 [log2(n.! + l)J - I
i
9 11 13 15 li 19 21 “3 25 ‘i
star graph : An attractive alternative to the nConference cube. Proceedings of the International in Parallel Processing, pages 393-400. 198i.
[8] J. Jwo, S. Lakshmivarahan, and S. Dhall. Embedding of cycles and grids in star graphs. Proceedings of the 2nd IEEE Parallel and Distributed Processing Symposium, Dallas, Texas, 1990.
[2] S. Akers and B. Krishna.murthy. A group theoretic model for interconnection networks. IEEE 38:555-5(X.
[9] Z. Miller, D. Pritikin, and I. Sudborough. Small dilation embeddings of hypercubes into star networks. Research Report TX 75083, University of
1989.
[3] S. Bettayeb, B. Gong, M. Girou, and I. Sudborough. Simulating permutation networks on hypercubes. Proceedings of th.e Is1 Latin American Symposium on Theoretical Informatics, Notes in Computer Science, 583:61-70, [4] N. Biggs.
Algebraic graph theory.
28 32 35
[i] C. GowriSankaran. Broadcasting on recursively decomposable Cayley graphs. Discrete Applied Mathematics, to appear, 1994.
[l] S. Akers. D. Hare], and B. I~rishnal~~urt~hy. The
on Com.puters.
12 15 18 21 24 27 29 32
[lo;;n!, -3pog, n1 + 3
References
Transactions
n with higher dilation
height of a t L embedded into dilation 4 1 3 2 5
dilation 1
Texas at Dallas.
[lo] B. Monien and H. Sudborough. Embedding one interconnection network in another. Computing Supplement, 7:257-282, 1990.
Lecture
1992.
[ll]
Cambridge
University Prws, 1974.
M. Nigam, S. Sahni, and B. Krishnamurthy. Em.bedding hamiltonians and hypercubes in the star interconnection graphs. Proceedings of the International
[5] A. Bouabdallah, M.-C. Heydemann, J. Opatrny, and D. Sotteau. Embedding complete binary trees into star and pancake graphs. Submitted to Mathematical Systems Theory, 1993.
Conference
in Parallel
111.340-111.343, 1990.
[6] A. Bouabdallah, M.-C. Heydemann, J. Opatrny, Embedding complete binary and D. Sotteau. trees into star graphs. Proceedings of the conference on the Mathematical Foundation of Computer Science, Lecture Motes ia Computer Science, 841~266-275, 1994. 553
Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS'95) 1060-3425/95 $10.00 © 1995 IEEE
Processing, pages
Proceedings of the 28th Annual Hawaii International Conference on System Sciences - 1995
Case
1
Case
2
Figure 1: Construction
of an embedding of a complete binary tree with congestion 1 and dilation
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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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