Hamilton Circuits in the Directed Butterfly Network

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(1)Hamilton Circuits in the Directed Butterfly Network Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Stéphane Pérennes. To cite this version: Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Stéphane Pérennes. Hamilton Circuits in the Directed Butterfly Network. RR-2925, INRIA. 1996. �inria-00073773�. HAL Id: inria-00073773 https://hal.inria.fr/inria-00073773 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Hamilton circuits in the directed Butterfly network Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Ste´phane Perennes. N˚ 2925 Juillet 1996 THE`ME 1. ISSN 0249-6399. apport de recherche.

(3)

(4) Hamilton circuits in the directed Butterfly network Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Ste´phane Perennes The`me 1 — Re´seaux et syste`mes Projet SLOOP Rapport de recherche n˚2925 — Juillet 1996 — 24 pages.

(5)

(6)

(7)

(8)

(9) !

(10) "# $ %'&($)& *,+-+.+.

(11) . . of degree and Abstract: in this paper, we prove that the wrapped Butterfly digraph dimension contains at least arc-disjoint Hamilton circuits, answering a conjecture of D. Barth. We also conjecture that can be decomposed into Hamilton circuits, except for , and . We show that it suffices to prove the conjecture for prime and . Then, we give such a Hamilton decomposition for all primes less than 12000 by a clever computer search, and so, as corollary, we have a Hamilton decomposition of for any divisible by a number , with .. .

(12) . . . Key-words: Butterfly graph, graph theory, Hamiltonism, Hamilton decomposition, Hamilton cycle, Hamilton circuit, perfect matching.. (Re´sume´ : tsvp). This work has been supported by the CEFIPRA (French-Indian collaboration) and the European project HCM MAP. Version of July 2, 1996 submitted to D ISCRETE A PPLIED M ATHEMATICS. Email : {bermond, darrot, delmas, sp}@unice.fr. Unite´ de recherche INRIA Sophia-Antipolis 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex (France) Te´le´phone : (33) 93 65 77 77 – Te´lećopie : (33) 93 65 77 65.

(13) Circuits Hamiltoniens dans le re´seau Butterfly oriente´.

(14) . .

(15).

(16) (

(17)

(18)

(19)

(20)

(21) %)& $ & *,+.+-+. Re´sume´ : dans cet article, nous prouvons que le graphe Butterfly reboucle´ (dans sa version oriente´) de degre´ et de dimension contient au moins circuits Hamiltoniens arc-disjoints, peut eˆtre dećomre´pondant a` une conjecture de D. Barth. Nous conjecturons aussi que pose´ en circuits Hamiltoniens, sauf pour , et . Nous montrons qu’il suffit de prouver cette conjecture pour premier et . Puis nous donnons une telle dećomposition pour tous les premiers jusqu’a` 12000 graˆce a` une subtilite´ de programmation, induisant comme corollaire que admet une dećomposition Hamiltonienne pour tout divisible par un nombre tel que .. $. . Mots-cle´ : graphe Butterfly, theórie des graphes, Hamiltonisme, dećomposition Hamiltonienne, cycle Hamiltonien, circuit Hamiltonien, couplage parfait..

(22) Hamilton circuits in the directed Butterfly network. 3. 1 Introduction and notations 1.1 Butterfly networks Many interconnection networks have been proposed as suitable topologies for parallel computers. Among them the Butterfly networks have received particular attention, due to their interesting structure. First, we have to warn the reader that under the name Butterfly and with the same notation, different networks are described. Indeed, if some authors consider the Butterfly networks as multistage networks used to route permutations, others consider them as point-to-point networks. In what follows, we will call the multistage version Butterfly and we will use Leigthon’s terminology [13], namely wrapped Butterfly, for the point-to-point version. Furthermore, these networks can be considered either as undirected or directed. To be complete, we recall that some authors consider only binary Butterfly networks that is the restricted class of networks obtained when the out-degree is 2 (directed case) or 4 (undirected case). will denote the set of integers In this article, we will use the following definitions and notation. modulo . (For definitions not given here see [15]).. $. . .

(23).

(24) . Definition 1.1 The Butterfly digraph of degree and dimension , denoted

(25)

(26) has as vertices the couples , where is an element of that is a word where the and ( is called the level). For , a vertex

(27)

(28) letters belong to

(29)

(30)

(31) is joined by an arc to the vertices where is any element of .. +& &

(32) .

(33) - .. .

(34) has

(35) ! , .

(36). ". . . . vertices. Each vertex in level has out-degree . This digraph is not strongly connected. It is mainly used as a multistage interconnection network (the levels corres inputs (nodes at level 0) to ponding to the stages) in order to route some one-to-one mapping of outputs (nodes at level ). The underlying undirected graph obtained by forgetting the orientation will be denoted .. . .

(37). . . The orientation on

(38) being ob-. .

(39) . Figure (1) shows simultaneously and tained by directing the edges from left to right..

(40) .

(41) . Note that is often represented example in [13, 15]) in an opposite way to our dra

(42) (for

(43) . We have chosen the representation which emwing as the authors denote the nodes phasises the most on the recursive decomposition of and provides us an easy representation of our inductive construction (see section (3))..

(44)

(45) + .

(46) . Definition 1.2 The wrapped Butterfly digraph by identi is obtained from fying the vertices of the last and first level namely with . In other words the vertices are

(47) #

(48) the couples where is an element of that is a word where the letters $ belong to and ( is called the level). For any , a vertex

(49)

(50) is joined. %

(51)

(52) &

(53) '( by an arc to the vertices where is any element of (and ) where has to be taken modulo ).. . RR n˚2925. . .

(54). . -. ,. . - .

(55) 4. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. +.

(56).

(57) ,

(58) .. Usually to represent the wrapped Butterfly (di)graph we use the representation of by repeating at the end level 0. Hence the reader has to remind that levels and are identified for vertices; its diameter is . is a is -regular digraph with . The underlying wrapped Butterfly network will be denoted ; it is regular of degree , with diameter ..

(59)

(60) . .

(61)

(62) . Levels 0. 1. 2 (or duplicated level 0). 00 01. Lines or words. 02. Node (10,2) in BF(3,2) 10. or Node (10,0) in WBF(3,2). 11 12. 20 21 22. +. . . Figure 1: The graphs (multistage version) with 3 levels, or (point-to-point version) by duplicating level . For digraphs or the edges must be directed into arcs from left to right .. 1.2 Other definitions and general results .

(63). will denote the complete graph on. . vertices,. . will denote the complete bipartite graph where each set of the bipartition has size ,. will denote the symmetric digraph obtained from the graph by replacing each edge by. two opposite arcs. In particular

(64) (resp.

(65) ) will denote the complete symmetric (resp. bipartite) digraph on (resp. ) vertices,. . . . . . will denote the complete symmetric digraph with a loop on each vertex,. a circuit or directed cycle of length.

(66). will be denoted. . . , and a dipath of length.

(67) . . .. INRIA.

(68) Hamilton circuits in the directed Butterfly network. . 5. . . will denote the directed digraph obtained from part to the right part.. by orienting each edge from the left. . Definition 1.3 (see [15]) Let be a directed graph. The line digraph of is the directed graph whose vertices are the arcs of and whose arcs are defined as follows: there is an arc from a vertex to a vertex in if and only if, in , the initial vertex of is the end vertex of .. . . is nothing else than and that . is Note that is the line digraph of

(69) . (corollary (4.1)) that . . . . . . We will see in section (4). Definition 1.4 A 1-difactor of a digraph is a spanning subgraph of with in and out-degree 1. It corresponds to a partition of the vertices of into circuits. Definition 1.5 A Hamilton cycle (resp. circuit) of a graph (resp. digraph) is a cycle (resp. circuit) which contains every vertex exactly once. Definition 1.6 We will say that a graph (resp. digraph) has a Hamilton decomposition or can be decomposed into Hamilton cycles (resp. circuit) if its edges (resp. arcs) can be partitioned into Hamilton cycles (resp. circuits). Remark 1 A Hamilton circuit is a connected 1-difactor. The existence of one and if possible many edge(arc)-disjoint Hamilton cycles (circuits) in a network can provide advantage for algorithms that make use of a ring structure. Furthermore, the existence of a Hamilton decomposition allows also the message traffic to be evenly distributed across the network. Various results have been obtained on the existence of Hamilton cycles in classical networks (see for example the survey [2, 11]). For example it is well-known that any Cayley graph on an abelian group is hamiltonian. Furthermore it has been conjectured by Alspach [1] that: Conjecture 1 (Alspach) Every connected Cayley graph on an abelian group has a Hamilton decomposition.. . . This conjecture has been verified for all connected 4-regular graphs on abelian groups in [9]. That includes in particular the toroidal meshes (grids). For the hypercube it is also known that can. be decomposed into Hamilton cycles (see [3, 2]). Concerning line digraphs it has been shown in [12] that -regular line digraphs always admit Hamilton circuits. In the case of de Bruijn or Kautz digraphs which are the simplest line digraphs, partial results have been obtained successively in [6, 14], and quasi optimal results have been obtained for undirected de Bruijn and Kautz graphs [4].. . . . 1.3 Results for the Butterfly networks The wrapped Butterfly (di)graph is actually a Cayley graph (on a non abelian group) and a line digraph. So the decomposition into Hamilton cycles (resp. circuits) of this graph (resp. digraph) has received some attention. First it is well-known that is hamiltonian (see [13, page 465] for a proof in the case ). In [7], Barth and Raspaud proved that has a Hamilton decomposition answering a conjecture of J. Rowley and D. Sotteau (private communication).. . RR n˚2925.

(70) .

(71) .

(72) 6. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. Theorem 1.1 (Barth, Raspaud).

(73) can be decomposed into 2 Hamilton cycles.

(74) ,

(75) can be decomposed into . They also gave the following conjecture. Conjecture 2 (Barth, Raspaud) For. Hamilton cycles..

(76) ,

(77) contains ( arc-disjoint Hamilton circuits.

(78) . is nothing else than which itself is the arc-disjoint sum of Recall that for

(79) and loops. So conjecture (3) can be seen as an extension of a theorem of Tillson [17]. Hamilton. Theorem 1.2 (Tillson) The complete symmetric digraph

(80) can be decomposed into circuits except for % and 6.

(81) . In this paper we focus mainly on the decomposition of the wrapped Butterfly digraph

(82) Our main result implies

(83) that the number of arc-disjoint Hamilton circuits contained in can only increase when increases.

(84) contains arc-disjoint Hamilton circuits, then

(85) contains Proposition 1.1 If

(86) (

(87) . also at least arc-disjoint Hamilton circuits, for any % and 6, implies conjecThis proposition with Tillson’s theorem and a special study for ture (3).

(88) ,

(89) contains arc-disjoint Hamilton circuits. Theorem 1.3 For

(90) can be Furthermore it appears that, except for three cases, for all small values of , decomposed into Hamilton circuits, so we conjecture that:

(91)

(92) Conjecture 4 For ,

(93) # or 3) and () ,,

(94) # ). can be decomposed into Hamilton circuits, except for (

(95) " . Using results of section (4) By the proposition (1.1) it suffices to prove the conjecture for In his thesis [5] , Barth also stated the following conjecture for the directed case:. Conjecture 3 (Barth) For. on conjunction of graphs, we have been able to reduce the study to prime degrees. So conjecture (4) would follow from conjecture (5). Conjecture 5 For any prime number cuits.. . ,. can be decomposed into . Hamilton cir-. With a clever computer search, we have been able to prove conjecture (5) for any prime less than 12000, leading to the following statement:.

(96) . . $ %& $ & *,+-+.+. Theorem 1.4 If is divisible by any number , has a Hamilton decomposition.. then. and consequently. Finally the methods used in this paper can be applied with other ideas to the undirected case and lead us to prove conjecture (2) in a forthcoming paper [8]. Theorem 1.5 For.

(97) ,

(98) can be decomposed into . Hamilton cycles.. INRIA.

(99) Hamilton circuits in the directed Butterfly network. 7. 2 Circuits and Permutations.

(100)

(101)

(102) . 2.1 More definitions. First, we will show that the existence of arc-disjoint Hamilton circuits in , is equivalent . to the ability to route compatible cyclic realisable permutations between levels 0 and in For this purpose we need some specific definitions.. .

(103)

(104) . + . . In the whole paper, will always denote a permutation of which associates to , % . It follows from the definition of that there exists a unique dipath connecting a vertex to a . So we can associate to a permutation of a set of dipaths in connecting vertex % vertex to vertex for any in ..

(105) + .

(106) . . . of two . permutations and is the permutation which associate to the

(107) or equivalently

(108) realizes the perDefinition 2.1 A permutation is realisable in associated dipaths from the inputs to the outputs are vertex-disjoint. mutation if the + . Definition 2.2 A permutation is cyclic if all the elements are distinct for & ,.

(109) is compatible if the Definition 2.3 A set of permutations in dipaths from % + to

(110) for in

(111) # and +

(112) &realisable are arc-disjoint. We will also & . . .

(113) say that realises compatible permutations.. The composition element the element . . . . . . . . . . . . . . .

(114) . . . 2.2 Hamilton circuits and permutations.

(115) .

(116)

(117) contains arc-disjoint Hamilton circuits if and only if

(118) realizes Lemma 2.1 k compatible cyclic permutations.

(119) and let be the unique dipath joirealisable in Proof. Let be a cyclic % +

(120) % permutation

(121)

(122) obtained from by identi

(123) % to

(124) % . Let ning be the dipath of

(125) . + fying with . The concatenation of the dipaths forms a Hamilton circuit .

(126) of because, being cyclic, the span the set of vertices of level 0. Conversely,

(127) , let % + % +

(128) + be the vertices we if is a Hamilton circuit of meet 0 by following the cycle . Let us consider the permutation defined by % successively

(129) . As onislevel a Hamilton circuit, all the ’s are distinct and all the inside dipaths are vertex

(130) . Finally by the equivalence above, arcdisjoint, so is a cyclic realisable permutation in We are now ready to prove that there is an immediate connection between the existence of compatible cyclic realisable permutations in and that of Hamilton circuits in .. . . . . . . . . . . . . . . . . . . . . . disjoint Hamilton circuits correspond to compatible cyclic realisable permutations (see Figure (2) for an example). . RR n˚2925.

(131) J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. 1. 00. lev (o el 2 rd up lic. (00,0) (00,1). lev el. lev el. 0. ate. dl. ev el. 0). 8. 00. 01 (01,0) (01,1) 01 (10,0) (10,1). 10. 10. 11. 11 (11,0) (11,1). figure a. figure b. figure c. (figure a) or equivalently the associated permutation Figure 2: A Hamilton circuit of (figure b) and the cyclic permutation which is used (figure c). realisable in 3 Recursive construction 3.1 Recursive decomposition of.

(132) ,.

(133) . .

(134).

(135).

(136) . . The permutation network has a simple recursive property: the first levels of form vertex-disjoint subgraphs isomorphic to . We shall call them left But

(137) $ terflies. If the vectors of are denoted , then each left Butterfly connects . In the the set of vertices having the same left part . So we will label a left Butterfly by disjoint subgraphs isomorphic to same way, the two last levels of are built with that we shall call right Butterflies; each right Butterfly connects all the vertices having the same right part and we will label it by . We can summarize the situation as follows:. '

(138) ,. , . . . .

(139) . are denoted , $ the left Butterfly labeled by is formed by the vertices , will be denoted. . vertices of. . . . . . the right Butterfly with label be denoted .. . $ '. is formed by the vertices. . . .

(140) first levels. It of the last levels. It will. of the. INRIA.

(141) Hamilton circuits in the directed Butterfly network.

(142) ! ,. 9.

(143) . Remark 2 In , vertices of level are shared by the left and right Butterflies, the outputs of the left Butterflies being considered as the inputs of the right Butterflies. Moreover all the subgraphs defined above are arc-disjoint. Figure (3) displays such a recursive decomposition. Left Butterflies. Right Butterflies. 0 0. 0 0. 0 1. 0 1. 0 2. 0 2 B Left (0). 1 0. 1 0. 1 1. 1 1. 1 2. K d,d(0). K d,d(1). 1 2 K d,d(2). B Left (1). 2 0. 2 0. 2 1. 2 1. 2 2. 2 2 B Left (2). . ". . Figure 3: The recursive decomposition of . To obtain the directed version the $

(144) edges must be directed into arcs from left to right. The vertices are denoted . In the 2 first levels form 3 vertex-disjoint subgraphs, each one isomorphic to . . In the same way, the 2 last levels of These 3 subgraphs are labeled are built with %

(145) disjoint subgraphs isomorphic to labeled .. . . . ,. . 3.2 Iterative Construction.

(146) ,. .

(147) . We will now give a simple construction which enables us to construct compatible cyclic realisable permutations in from compatible cyclic realisable permutations in . will denote a permutation realisable in the right Butterfly . To is In what follows associated a perfect matching in ; the arcs in the perfect matching join to . . . RR n˚2925. . .

(148) 10. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. . & . + & '& . . $. + & . Definition 3.1 Let . , be families of permutations realisable in $ , . The families are compatible, if for every the set of permutations. . % , is compatible (in other words they correspond to arc-disjoint matchings of ).. Definition 3.2 A family of permutations satisfies the cyclic property if the composition of the permutations of the family is a cyclic permutation for any order of the composition.. . . + & & of permutahas the cyclic property. . be the permutation of defined by # + and Proof. Let if is realisable in % . Furthermore for , the matchings associated to and. Clearly, are arc-disjoint; are compatible. Finally for so for a given the permutations a given the composition of the is the permutation which . taken inany order which permutations associates to the element is clearly cyclic. . $ . Lemma 3.1 There exist compatible families . $ , such that for each , the family tions realisable in. . . . . . . ),. . .

(149) . . . . .

(150) . . Lemma 3.2 (Inductive lemma) Let be a cyclic permutation realisable in and let $ be a family of realisable permutations satisfying the cyclic property. Then the permu

(151) tation of defined by where , is a cyclic permutation realisable in ..

(152) . . . . of the dipath from from + to %

(153) , consists Proof. + to The dipath

(154) inassociated toassociated (which to the permutation of is isomorphic

(155) ) and then of the arc joining %

(156) to

(157) , in defined to . . We claim that the dipaths joining two distinct inputs by the+ matching , that is + to their outputs are vertex-disjoint and so is realisable. Indeed, if their and and and the last arcs are disjoint as either first parts are in two different % . , being realisable the first dipaths are or and is realisable in . If vertex-disjoint and as the last arcs belong to two different . it remains to

(158) show ( . . Finally & that is cyclic. To prove that it suffices to show that for & Suppose . So , which implies that . But for , . Byis thedefinition image of by the composition of the of in some where is aelements , and order as has the cyclic property for cyclic permutation. So for & ( . So for any & &

(159) , .

(160) , then there exist Corollary 3.1 If there exist compatible cyclic realisable permutations in .

(161) . compatible cyclic realisable permutations in . +

(162) and Proof. Let , & & cyclic realisable permutations of $ + , be compatible. let , & & , be compatible families of realisable permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INRIA.

(163) Hamilton circuits in the directed Butterfly network. 11.

(164) & . . . . each satisfying the cyclic property. Such families exist by lemma (3.1) as . By the above proposition for each the are cyclic realisable permutations of and they are compatible as both the

(165) and the are compatible (all the dipaths associated are arc-disjoint). . Now, we are ready to prove our main proposition stated in the introduction..

(166) . .

(167)

(168). contains arc-disjoint Hamilton circuits, then Proposition 3.1 (Main) If contains also at least arc-disjoint Hamilton circuits, for any .. .

(169)

(170) .

(171) . Proof. The result for follows from corollary (3.1) and lemma (2.1). A recursive application of this property gives the above proposition. For an example of the construction see Figure (4). 0. 2. C. 0. 1. 2. figure a. C. C. C. C1. 0. 1 figure b. C1. 0. C. 0. 1. C1. 0. figure a’. figure b’. are obtained from two Hamilton cirFigure 4: Two Hamilton circuits (figures and ) of . by the construction of lemma (3.2). Figures ( ) and ( ) show two arccuits of and

(172)

(173) ). We disjoint Hamilton circuits of , used the families (figure ) and

(174) (figure ) defined in the proof of lemma (3.1). . . . RR n˚2925.

(175) 12. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. 3.3 Consequences.

(176) can be decomposed into two Hamilton circuits as soon as

(177) % . For &

(178) &

(179) admits only one Hamilton circuit. % into two arc-disjoint HaProof. A computer search has given a decomposition of .

(180) milton circuits. Therefore by proposition (3.1) Hamilton decomposition, for any

(181) % . For &

(182) & , an exhaustive computer search showshasthata there cannot exist two arc-disjoint Corollary 3.2 ,. Hamilton circuits.. .

(183) can be decomposed into three Hamilton circuits as soon as

(184) # . For &

(185) &

(186) admits only two arc-disjoint Hamilton circuits. obtained Proof. That follows from the existence of a Hamilton decomposition of. Corollary 3.3 ,. by computer (figures of decompositions available on demand). . Now we are able to prove Barth’s conjecture (3)..

(187) ,

(188) contains arc-disjoint Hamilton circuits. % , and , . contains Proof. By Tillson’s decomposition ((1.2)), for arc-disjoint Hamilton circuits. So by proposition (3.1), for % and ,

(189) contains at least arc-disjoint Hamilton circuits. For % % (resp. (resp.6),wehave found ). Sobybycomputer search 4 (resp. 6) arc-disjoint Hamilton circuits in %

(190) (resp.

(191) ) contains 4 (resp. 6) arc-disjoint Hamilton circuits.proposition (3.1), % , there exists for

(192) a Hamilton decomposition of the proof above for Asseen

(193) .inThese results and those of the next section lead us to propose the following conjecture,

(194) . which would completely close the study of the Hamilton decomposition of % and

(195) ,

(196) can be decomposed into Hamilton circuits. Conjecture 6 For

(197) "# or equivalently, as (3.1) it suffices to prove the conjecture for By

(198) proposition (see corollary . . (4.1)) that

(199) admits compatible eulerian tours (see [12]).. Theorem 3.3 For. . 4. .

(200) Decomposition of. into Hamilton circuits. 4.1 Line digraphs and conjunction We need some more definitions and results concerning conjunction, line digraphs and de Bruijn digraphs.. INRIA.

(201) Hamilton circuits in the directed Butterfly network. Definition 4.1 (see [2]). 13. is the digraph

(202) to

(203)

(204) and. if and only if there is an arc.

(205) of two digraphs

(206) 1. The conjunction and an arc joining

(207) with vertex-set

(208) . .

(209)

(210)

(211) joining to in and an arc joining to. . . . . . . in. 2. The edge-disjoint union (sum) of two digraphs ,

(212) .. . . . on the same set of vertices is denoted.

(213). 3. will denote the digraph made of disjoint copies of .. and % % . Hamilton circuits of For example

(214). Properties 4.1. . . . . where and

(215).

(216). . . ) !! ) ) . .

(217). . . . . . . . .

(218). are the two arc-disjoint.

(219) . .

(220) . . . .

(221). Proof. These results are clear from the definitions. . There is a very strong connection between the de Bruijn digraph and the wrapped Butterfly digraph. We recall the definition of the de Bruijn digraph:.

(222). .

(223) . .

(224). Definition 4.2 The de Bruijn digraph of out-degree and diameter has as vertices the &

(225) words of length on an alphabet of letters. Vertex is joined by an arc to the vertices

(226) &

(227) where is any letter from the alphabet. Proposition 4.2. RR n˚2925.

(228)

(229)

(230)

(231)

(232)

(233)

(234)

(235)

(236)

(237)

(238)

(239)

(240)

(241)

(242)

(243)

(244) . (1) (2) (3) (4) (5).

(245) 14. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. Proof. Equality (1) is well known, and even sometimes considered as the proper definition of de Bruijn digraphs (see [10, 15]).

(246)

(247) Result (2) can be found in [16] and can be proved as follows:

(248)

(249)

(250) from (1), but . By properties (4.1),

(251) . which is indeed Result (3) is implicit in different papers. It can be obtained by considering the following isomor to with

(252)

(253) , phism from . To the vertex in $

(254) , we associate the vertex in , where and % are the verand . By definitions (4.1-1) and (4.2), the out-neighbors of in

(255)

(256) such that

(257) for tices with , and

(258) , being any are letter from the alphabet. The associated vertices in where .

(259) . For , or equivalently , ,

(260) # and and for , . So by definition (1.2), the vertices are exactly the out-neighbors of %

(261) , hence in . The second part of the equality is due to the fact that

(262)

(263)

(264)

(265) . An example is displayed in Figure (5). Result (4) is equivalent to (3). The last equality follows directly from (2) and (3).. .

(266)

(267)

(268)

(269) ! ". ,

(270) . Corollary 4.1.

(271) . .

(272) . . . . .

(273)

(274)

(275)

(276) , ,

(277) ,. . . .

(278) .. Proof. Follows from proposition (4.2) equality (3) for.

(279) # . . $ " . " " is a regular digraph with in and out-degree Proof. is the union of circuits. Star 1. So itwhere ting from a vertex we find at distance the vertex (resp. ) has to be taken modulo $

(280) (resp. $ ). So the length of any circuit is the smallest common multiple of

(281) $ and $ , that is $

(282) as and are relatively prime. As the number of vertices in the digraph is $

(283) there are $ such cycles.

(284)

(285)

(286)

(287)

(288)

(289) Proposition 4.3

(290)

(291) Proof. Let. (3) of proposition (4.2), we have.

(292)

(293)

(294)

(295) . As .

(296) By property

(297) (from lemma (4.1) with $ and ), then we obtain:.

(298)

(299)

(300)

(301)

(302)

(303)

(304)

(305)

(306)

(307)

(308) Lemma 4.1 When and are relatively prime,. . . ". ". . . . . . . . . . . INRIA.

(309) Hamilton circuits in the directed Butterfly network. 15. (o. l2 ve le. 000. 000. 000. 100. 001. 010. 100. 010. 100. 001. 010. 110. 101. 011. 110. 001. 010. 100. 001. 101. 011. 110. 101. 011. 110. 101. 011. 111. 111. 111. 111. ). l1 ve le. l0 l3 ve ve le le ed at lic up. rd. l0 ve le 000. Figure 5: The graph. . as a conjunction of with . . . ..

(310)

(311) (resp.

(312) ) admit

(313)

(314) admit

(315) arc-disjoint Hamilton cir-. Corollary 4.2 If

(316) and are relatively prime, and if

(317) (resp. ) arc-disjoint Hamilton circuits, then cuits.. . . .

(318) ). From

(319)

(320)

(321)

(322) has

(323)

(324) vertices, the 1-difactor consists of

(325) circuits each As

(326)

(327) one being a Hamilton circuit of a connected component of to

(328)

(329) . So, the conjunction of one Hamilton circuit of

(330)

(331)

(332) with oneisomorphic Hamilton

(333) provides one Hamilton circuit in

(334)

(335) . Applying this results circuit of to the

(336) different couples of circuits, provides

(337) arc-disjoint Hamilton circuits in

(338)

(339) .

(340) with prime, corollary (4.2) enGiven a Hamilton decomposition of the digraphs

(341) , so it is enough to prove conjecture (6) ables to construct a Hamilton decomposition of for being a power of a prime. . . .

(342) . .

(343) Proof. Let (resp. ) be a Hamilton circuit in (resp. is a set of circuits of length

(344) . lemma (4.1), the conjunction . . . . . . . RR n˚2925. .

(345) 16. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. 4.2 Reduction to the case where. is prime.

(346) # . has a Hamilton decomposition. That appears We would like to prove that has a Hamilton decomposition. , quite difficult. However, we will prove that for Such a decomposition will then be sufficient to reduce the problem to case of prime degrees. Lemma 4.2 For any number circuits.. . .

(347) . . and any prime ,. . % . . . can be decomposed into Hamilton. $. $. $. be labeled . with , and Proof. Let the nodes of The digraph is similar to the wrapped butterfly digraph, so we can also define a multistage network obtained by duplicating the level into a level . Formaly this multistage network is where $ is a directed with vertices); its vertices will be path of $ $ length (.i.e. . labeled with , and

(348) . +.

(349). + ..

(350) .

(351). .

(352) . + & & . +

(353) ,. +

(354) . except that now Like in section (2), we can define a notion of realisable permutation in %

(355)

(356) realizes there is more than one dipath connecting to . We will say that

(357)

(358) %

(359) compatible permutations of if there exist dipaths , , . $ %

(360) satisfying the following pro, where connects in dipaths to are perties. For a given the vertex-disjoint (realisability’s property) and all the dipaths %

(361) are arc-disjoint (compatibility’s property).. . . . . . . . realizes compatible cyclic permutations; more We will show by induction that

(362)

(363) . First we give for exactly that if %the

(364) " % weandwill theprove property is true for , it is also true for dipaths associated to compatible cyclic permutations. , with a vertex is followed by a vertex % In all. the dipaths that we consider, . , where and depend on the level . % . For any , the dipaths are vertex-disjoint if and only if the functions # + . are for given and one to one mappings. As is prime that is satisfied if and only if % . The dipaths are arc-disjoint if and only if the functions are for a given + one to one mappings. That is satisfied if and only if .. can be decomposed Using the same argument than in lemma 2.1, we can state that realizes compatible cyclic permutations. into Hamilton circuits if and only if. .

(365) . . . . . . . . . . .

(366) . . .

(367) . . . . In order to simplify the notations we omit in the labels of the vertices the values of the levels; as ) a vertex of level is always followed by a vertex of level .. INRIA.

(368) Hamilton circuits in the directed Butterfly network. 4.2.1 Initial constructions. . Let. . denote the function of. 17. + :. . +. into.

(369) " . %

(370) .

(371) " % % .

(372) . % . %. . %. . +. if if. . . , . . . . . . , . into circuits. To produce a clearer figure, verFigure (6) shows one decomposition of tices on and those on even levels in the following order: ifodd levels or are ranked and lexicographically .

(373) .

(374) . %. . + . %. . . . . Figure 6: A decomposition of. . . .

(375) "

(376) " . . RR n˚2925. #. % # and % and.

(377)

(378) . . . . . Level 0. Level 3. Level 2. Level 1. Level 0. Level 0. Level 3. Level 2. , presented with a special ranking of the vertices.. %. Level 1. 20 21 22. Level 0. 02 12 22. . Level 0. 10 11 12. Level 3. 01 11 21. Level 2. 00 01 02. Level 1. 00 10 20. Level 0. . Odd levels. . Even levels. . , . . ( % . . .. . . . . . % . , . . . . . . . . , . . ..

(379) 18. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. of the form

(380) "# , thearefunctions implicitely defined. " %

(381) & & .

(382) . . & , % .. , . To complete the proof it remains to note that in the three first cases the permutation induced by % . . .

(383) , the construction is cyclic, and that in the case % % , is also cyclic as .. +. %. . # +. . In all the cases one can easily verify that the functions with and . For example, in the construction for

(384) are: . . . . . . .

(385) . . . . . . . . .

(386) . . . . .

(387) . . . . . . . . . . . . . 4.2.2 Induction step. + & & . . . realizes The induction step follows from two facts. First, it can be easily seen that compatible permutations , , if and only if there exist two sets of permutations . and , , such that:. + & '& ( + , , for & & + ( , realizes the compatible permutations , & '& . + realizes the compatible permutations , & & + Secondly, realizes compatible permutations , & & , such that each. . . . . is the identity permutation. Indeed, let us consider the dipaths:. . . ' . % .

(388) . +. # +.

(389) . . . . . . %. . . . %. . . . %.

(390) . . . . . . Once again of level is joined to a vertex of level with # a vertex with and . realizes compatible permutations then realizes the same So if compatible permutations. can be decomposed into Hamilton circuits So we can conclude by induction that for any number . . . .

(391) . . . . Theorem 4.3 If a digraph with at least vertices, contains contains arc-disjoint Hamilton circuits.. arc-disjoint Hamilton circuits, then. . Proof. First we prove the result for prime. by hypothesis the number of vertices of is greater than . Hence. $. . &% '. ".

(392). . . ' %. ". .

(393). ! #". .

(394). . . . , where being. . INRIA.

(395) Hamilton circuits in the directed Butterfly network. . From lemma (4.2) which gives. $. . ! . . '.

(396). .

(397) .. . 19. . . So,. . . '. . . . ".

(398) . '. .

(399). . . . . Suppose now that the result holds for all integers strictly less than . If is prime we just proved

(400) where is a prime and

(401) . By proposition (4.2-2), the result. Otherwise . . By induction.

(402) . As a consequence

(403)

(404) contains at least

(405) arc-disjoint Hamilton circuits. Moreover being prime. will contain

(406) arc-disjoint Hamilton circuits. . . When can be decomposed into Hamilton circuit the above theorem can be restated as: Theorem 4.4 If with more than vertices can be decomposed into Hamilton circuits, then. can also be decomposed into Hamilton circuits.. . $ . . $. . with can be decomposed into Hamilton circuits, then Corollary 4.3 If can also be decomposed into Hamilton circuits for any integer .. . . $ . . Proof. Just apply theorem (4.4) to which is by proposition (4.2) Note that has only vertices so the corollary cannot be applied for. % $. . . $ . . %-$ . . . Example 1 As has a Hamilton decomposition then has a Hamilton decom position for any integer . In particular has a Hamilton decomposition for . Corollary 4.4 To prove conjecture (6) it suffices to prove that position for any prime .. . . . has a Hamilton decom-. % % . Proof. Let be a non prime number. If has a prime factor different from 2 or 3, by corollary (4.3), it suffices to prove the conjecture for . If has only prime factors equal to 2. or 3, then with . A computer search shows that , and . have a Hamilton decomposition. So, according to corollary (4.3), with have a Hamilton decomposition. . . . . . Remark 3 Although it is not the purpose of this article, proposition (4.3) can be used to improve results on the decomposition of de Bruijn digraphs into Hamilton circuits, Proposition 4.4. . . . . If is the greatest prime dividing , then. $ contains , $. RR n˚2925. . . contains

(407) . Hamilton circuits.. Hamilton circuits..

(408) 20. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. . . . Proof. The result holds for , now for , by a result of D. Barth, J. Bond and A. Raspaud [6] we know that contains arc-disjoint Hamilton circuits for a prime, and have at least

(409)

(410) vertices. Hence theorem (4.3) implies that, as ,

(411) contains arc-disjoint Hamilton circuits. Similary, the second result follows from a result of R. Rowley and B. Bose [14] stating that contains Hamilton circuits. . %. ( . .

(412). 4.3 Exhaustive Search for Hamilton decomposition of. . ). As seen above the problem has been reduced to find a Hamilton decomposition of for a prime . In order to provides ideas and to strengthen our conjecture we have performed some exhaustive searches. The complexity of an exhautive search being exponential, we have restricted the set of solutions to the one such that the -th circuit is obtained from by applying the automorphism of which sends vertex on vertex . Furthermore we is Hamiltonian and the Hamilton circuits , want solutions such that are arc-disjoint. However the search space is still exponentional in and a computer search (with normal computation ressources) cannot be successful for greater than 7. So we restricted again the search space to “nearly linear” solutions. This restriction gave us solutions for small primes . For example for we found the cycle given by the following set of arcs:. . . * . . . . * + + * + * + * ,. . , + & & ( . .

(413) , $ *+ , + , . for -( .. * * + + It induce the next cyclic permutation on level . +.+ .- % .+ % + .+ + % % , + - , - - . + % %.% +. . % % % . +. Finally we looked for very special Hamilton circuits . That enables us to find a solution for . . + . + + every prime between and . More precisely, we searched for parameters and in such . . . . . . . . . that . . is given by the following set of arcs:. + + * + ,. . . . . ! . % , + ,. , * + . for . . . + ,. . . +. . One can easily check that if the ’s are arc-disjoint. So we have only to find and such that is a Hamilton circuit. In particular we should have (condition to obtain a one difactor) and (otherwise we obtain a circuit of length starting at vertex ). We conjecture that:. +. . + + . INRIA.

(414) Hamilton circuits in the directed Butterfly network. Conjecture 7 For any prime defined by:. . . there exist. 21. $ . + . , % + . . . . . . and . +. if . # + ,. such that the permutation of. is cyclic.. +-+ . . + &. . The number of possible solutions is then only . So we have been able to verify the conjecture, by a computer search for large values of ( ). Below we give some solutions for less than . .

(415)

(416)

(417)

(418) 100. . . . . . . . . . . . . . . . . . . . . .

(419) . . # , we obtain the following cyclic permutation on level 0: +.+ % % +. - % - . %,% - . + % % + , .. ,. . %

(420) + + , % . % ,. % % % , +- + ,+ ,+ % + + ,+ . For example for. . . ,. . So using corollary (4.3) we have:. $ %& $ & *,+-+.+ then and consequently % . Indeed we know that % and can be strengthened in the case of This result. % have Hamilton decomposition and we have been able to generalize lemma (4.2) for % with a odd

(421) . prime and *,+-+.+ then % and consequently Theorem 4.6 If is divisible by any number $ , & $ &

(422) for

(423) % has a Hamilton decomposition.

(424) have an Hamilton decomposition for

(425) % . As a consequence the butterflies

(426) . . Theorem 4.5 If is divisible by any number , has a Hamilton decomposition.. 5 Conclusion. . In this paper, we have shown that in a lot of cases Butterfly digraphs have a Hamilton decomposition and give strong evidence that the only exceptions should be , and . We have furthermore reduced the problem to check if has a Hamilton decomposition for prime (or equivalently that has an eulerian compatible decomposition). We have also shown that such a decomposition will follow from the solution of a problem (conjecture (7)) in number theory. Our interest came from a conjecture of D. Barth and A. Raspaud [7] concerning the decomposition of Butterfly networks into undirected Hamilton cycles. This conjecture is solved in [8] by generalizing the technics of section (3.2).. . RR n˚2925. .

(427) 22. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. Finally we have seen in proposition (4.4) that the technics can be applied to obtain results on the Hamilton decomposition of de Bruijn digraphs. In the spirit it will be interesting to solve the following problem: Problem: Determine the smallest integer composition.. .

(428) &

(429). .

(430) such that

(431) . A proof similar to that of lemma (4.2) should lead to given , equivalent to . . . . . . has a Hamilton de-.

(432) &

(433) . Conjecture (6) is, for a. Acknowledgements We thank very much J. Bond, R. Klasing and P. Paulraja for their remarks and comments which improve this paper.. References [1] B. Alspach. Research problem 59. Discrete Mathematics, 50:115, 1984. [2] B. Alspach, J-C. Bermond, and D. Sotteau. Decomposition into cycles I: Hamilton decompositions. In G. Hahn et al., editor, Cycles and Rays, Proceeding Colloquium Montreál, 1987, NATO ASI Ser. C, pages 9–18, Dordrecht, 1990. Kluwer Academic Publishers. [3] J. Aubert and B. Schneider. Dećomposition de la somme carte´sienne d’un cycle et de l’union de deux cycles en cycles hamiltoniens. Discrete Mathematics, 38:7–16, 1982. [4] R. Balakrishnan, J-C. Bermond, and P. Paulraja. On the decomposition of de bruijn graphs into hamiltonian cycles. Manuscript. [5] D. Barth. Re´seaux d’interconnection: structures et communications. PhD thesis, Universite´ de Bordeaux I, 1994. [6] D. Barth, J. Bond, and A. Raspaud. Compatible eulerian circuits in Mathematics, 56:127–136, 1995.. .

(434).

(435) . Discrete Applied. [7] D. Barth and A. Raspaud. Two edge-disjoint hamiltonian cycles in the Butterfly graph. Information Processing Letters, 51:175–179, 1994. [8] J-C. Bermond, E. Darrot, O. Delmas, and S. Perennes. Hamilton cycle decomposition of the Butterfly network. Technical Report RR-2920, Inria - Sophia Antipolis - Projet SLOOP (CNRS/INRIA/UNSA), June 1996. To appear in Parallel Processing Letters. [9] J-C. Bermond, O. Favaron, and M. Maheo. Hamiltonian decomposition of Cayley graphs of degree 4. Journal of Combinatorial Theory, Series B, 46(2):142–153, 1989.. INRIA.

(436) Hamilton circuits in the directed Butterfly network. 23. [10] J-C. Bermond and C. Peyrat. Broadcasting in de Bruijn networks. In Proceedings of the 19th S-E conference on Combinatorics, editor, Congressus Numerantium 66, pages 283–292. Graph theory, and Computing, Florida, 1988. [11] S.J. Curran and J.A. Gallian. Hamilton cycles and paths in Cayley graphs and digraphs - a survey. To appear in Discrete Mathematics. [12] H. Fleischner and B. Jackson. Compatible euler tours in eulerian digraphs. In G. Hahn et al., editor, Cycles and Rays, Proceeding Colloquium Montreál, 1987, pages 95–100. NATO ASI Ser. C, Kluwer Academic Publishers, Dordrecht, 1990. [13] F. Thomson Leighton. Introduction to Parallel Algorithms and Architectures: Arrays . Trees . Hypercubes. Computer Science, Mathematics Electrical Engineering. Morgan Kaufmann Publishers, 1992. [14] R. Rowley and B. Bose. On the number of arc-disjoint hamiltonian circuits in the de Bruijn graph. Parallel Processing Letters, 3(4):375–380, 1993. [15] Jean de Rumeur. Communication dans les re´seaux de processeurs. Collection Etudes et Recherches en Informatique. Masson, 1994. (English version to appear). [16] M. Syska. Communications dans les architectures a` me´moire distribueé. PhD thesis, Universite´ de Nice Sophia Antipolis, 1992. [17] T. Tillson. A Hamiltonian decomposition of Series B, 29:68–74, 1980.. .

(437) . . . . Journal of Combinatorial Theory,. Table of Contents 1 Introduction and notations 1.1 Butterfly networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Other definitions and general results . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Results for the Butterfly networks . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 3 4 5. 2 Circuits and Permutations 2.1 More definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hamilton circuits and permutations . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7 7 7.

(438) . 3 Recursive construction 8 . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1 Recursive decomposition of 3.2 Iterative Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. RR n˚2925.

(439) 24. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. . 4 Decomposition of into Hamilton circuits 4.1 Line digraphs and conjunction . . . . . . . . . . 4.2 Reduction to the case where is prime . . . . . . 4.2.1 Initial constructions . . . . . . . . . . . . 4.2.2 Induction step . . . . . . . . . . . . . . . 4.3 Exhaustive Search for Hamilton decomposition of. . 5 Conclusion. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 12 12 16 17 18 20 21. INRIA.

(440) Unite´ de recherche INRIA Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE`S NANCY Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex 1 Unite´ de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unite´ de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. E´diteur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) ISSN 0249-6399.

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