Embeddings into Crossed Cubes

Transcription

1 Embeddings into Cossed Cubes Emad Abuelub *, Membe, IAENG Abstact- The hypecube paallel achitectue is one of the most popula inteconnection netwoks due to many of its attactive popeties and its suitability fo geneal pupose paallel pocessing. An attactive vesion of the hypecube is the cossed cube. It peseves the impotant popeties of the hypecube and most impotantly educes the diamete by a facto of two. In this pape, we show the ability of the cossed cube as a vesatile achitectue to simulate othe inteconnection netwoks efficiently. We pesent new schemes to embed complete binay tees, complete quad tees, and cycles into cossed cubes. Index Tems- binay tees, cycles, dilation, embedding, expansion, hypecubes, cossed cubes. I. NTRODUCTION Hypecube achitectues ae loosely coupled paallel pocessos based on the binay cube netwok. Paallel computes based on the hypecube topology have gained widespead acceptance in paallel computing. Recently, many machines based on the hypecube have been designed and made commecially available. The hypecube offes a ich inteconnection topology with lage bandwidth, logaithmic diamete, simple outing and boadcasting of data, ecusive stuctue that is natually suited to divide and conque applications, and the ability to simulate othe inteconnection netwoks with minimum ovehead [2, 11, 14, 15, 20, 21]. Due to the populaity of the hypecube, many vaiations of the hypecube topology have been poposed to impove on its popeties and computational powe. El-Amaway and Latifi [7] poposed the folded hypecube to educe the diamete and the taffic congestion with little hadwae ovehead. Pepaata and Vuillemin [18] intoduced the cube-connected cycles in which the degee of the diamete was educed to 3. Efe [6] poposed an attactive vesion of the hypecube called the cossed cube, whee peliminay studies poved that the cossed cube peseves many of the attactive popeties of the hypecube and moe impotantly educes the diamete by a facto of two [1, 4, 6, 8, 9, 22]. This implies that the cossed cube has an advantage ove the hypecube when data communication is of majo concen. It is well known that fo paallel achitectues, data communication cost dominates computation cost. Theefoe, it is wothwhile to make compaative studies on cossed cubes and othe inteconnection netwoks, and exploe the advantages povided by them. The poblem of embedding one inteconnection netwok into anothe is vey impotant in the aea of paallel computing fo potability of algoithms acoss vaious achitectues, layout of cicuits in VLSI, and mapping logical data stuctues into compute memoies. * Depatment of Compute Science, Zaqa Pivate Univesity, Jodan, abuelub@zpu.edu.jo The impotance of binay tees comes fom the fact that this class of stuctues is useful in the solution of banded and spase systems, by diect elimination, and captues the essence of divide and conques algoithms. Embedding binay tees into othe inteconnection netwoks attacted the attention of many eseaches. Baasch et al. have consideed embedding complete binay tees into genealized hypecubes [2], while Dingle and Sudboough consideed simulation of binay tees and X-tees on pyamid netwoks [5]. Quad tees ae becoming an impotant technique in the domain of image pocessing, compute gaphics, obotics, computational geomety, and geogaphic infomation systems [3, 17, 19]. This hieachical stuctue is based on the pinciple of ecusive decomposition, which is simila to divide and conque methods. Mapping quad tees into othe inteconnection netwoks attacted the attention of many eseaches [14]. The poblem of embedding ings o cycles into othe inteconnection netwoks has been studied by many eseaches. It is well known that ings can be embedded into hypecubes using cyclic Gay codes [20]. Latifi and Zheng [12] genealized the cyclic Gay code method to embed ings into twisted cubes. On the othe hand, othe eseaches addessed the poblem of embedding ings into fault-fee and faulty topologies [9, 12, 13, 14, 16, 20] o the Hamiltonicity of such stuctues in fault-fee and faulty envionments [1, 8, 10, 11]. The emainde of this pape is oganized as follows. In section 2, we establish a few peliminay definitions and notations. Section 3 explains a scheme to embed complete binay tees into cossed cubes. Section 4 pesents a ecusive technique to embed complete quad tees. In section 5, we extend the Gay code scheme to embed cycles into cossed cubes. Finally, section 6 concludes the pape and discusses some futue possible wok. II. DIFINITIONS AND NOTATIONS In this pape, we use undiected gaphs to model inteconnection netwoks. Each vetex epesents a pocesso and each edge a communication link between pocessos. The embedding of a guest gaph G = (V G, E G ) into a host gaph H = (V H, E H ) is an injective mapping f fom V G to V H, whee V G, E G and V H, E H ae the vetex and edge sets of G and H, espectively, and whee V H V G. We conside a complete binay tee of height n-1, a complete quad tee of height n-1, and a cycle of size n, denoted CB n, CQ n, and C n, espectively, as guest gaphs and a cossed cube of dimension n, denoted XQ n, as a host gaph. Two cost functions, dilation and expansion often measue the quality of an embedding. If u and v ae two nodes in G, then the distance fom u to v, d = (u, v), is the length of the shotest path fom u to v. The dilation D is the maximum distance in H between the images of

2 adjacent vetices of G, D = max {d(f(u), f(v)), whee u v E G }. The expansion E is the atio of the cadinality of the host vetex set to the cadinality of the guest vetex set, E = V H / V G. Minimizing each of these measuements has a diect implication on the quality of the simulation of the guest netwok by the coesponding host netwok. The dilation of an embedding measues how fa apat neighboing guest pocessos ae placed in the host netwok. Clealy, if adjacent guest pocessos ae placed fa apat in the host netwok, then thee will be a significant degadation in simulation due to the long length of the communication path between them. The expansion of an embedding measues how much lage is the host netwok than the guest netwok duing the simulation. We want to minimize expansion, as we want to use the smallest possible host netwok that has at least as many pocessos as in the guest netwok. In eality, we usually have a fixed size host netwok and we may have to conside many-to-one embedding fo lage guest netwoks. When the size of the guest netwok is not equal to the size of the host netwok in tems of the numbe of pocessos, then we ty to find the smallest host netwok that has at least as many pocessos as the guest netwok. Such a host netwok is efeed to as the optimal host netwok. A hypecube of dimension n, denoted by Q n, is an undiected gaph consisting of 2 n vetices labeled fom 0 to 2 n -1 and such that thee is an edge between any two vetices if and only if the binay epesentation of thei labels diffes in exactly one bit position. A complete binay tee of height n-1, denoted by CB n, is an undiected gaph consisting of 2 n -1 vetices and such that evey vetex of depth less than n-1 has exactly two sons and evey vetex of depth n-1 is a leaf. A complete quad tee of height n-1, denoted by CQ n, is an undiected gaph consisting of (4 n -1)/3 vetices and such that evey vetex of depth less than n-1 has exactly fou sons and evey vetex of depth n-1 is a leaf. A cycle of size n, denoted C n, is an undiected gaph consisting of n vetices labeled fom v 1 to v n, such that node v i is a neighbo with node v (i+1)mod n, 1 i n. A path (v o, v 1, v 2,, v n-1 ) is a sequence of nodes such that each two consecutive nodes ae adjacent. A cycle o a cicuit is called a Hamiltonian cicuit if it taveses evey node of G exactly once. The cossed cube is defined ecusively as follows. Let G be any undiected labeled gaph, then G b is obtained fom G by pefixing evey vetex label with b. Two binay stings x = x 1 x 0 and y = y 1 y 0, each of length two, ae paielated if and only if (x, y) {(00, 00), (10, 10), (01, 11), (11, 01)}. Now, we define a cossed cube of dimension n, denoted XQ n, as an undiected gaph consisting of 2 n vetices labeled fom 0 to 2 n -1 and defined ecusively as following: 1. XQ 1 is the complete gaph on two vetices with labels 0 and Fo n > 1, XQ n consists of two copies of XQ n-1 one pefixed by 0, XQ 0 n-1, and the othe by 1, XQ1 n-1. Two vetices u = 0u n-2...u 0 XQ 0 n-1 and v = 1v n-2...v 0 XQ 1 n-1 ae adjacent, if and only if: a. u n-2 = v n-2, if n is even, and b. Fo 0 i (n-1)/2, u 2i+1 u 2i and v 2i+1 v 2i ae paielated. Figue 1 shows cossed cubes of dimension 3. XQ n is constucted ecusively based on the constuction of XQ n-1 by pasting togethe a copy of XQ 0 n-1 and the mio image of XQ 1 n-1, then adding the appopiate links between the two copies accoding to the pai-elated elationship. Fo claity, we view the cossed cube XQ n as a [2 x 2 n-1 ] gid. If the gid is patitioned hoizontally into two equal pats, then all nodes above the hoizontal line have a 0 as a pefix, while all nodes below the hoizontal line have a 1 as a pefix Figue 1: The cossed cube XQ 3. III. EMBEDDING COMPLETE BINARY TREES This section descibes ou scheme to embed a complete binay tee CB n into a cossed cube XQ n with dilation two and unit expansion. Ou scheme is based on the inode labeling to embed CB n into XQ n in a staight fowad way. The inode embedding is constucted by Algoithm Embedding Complete Binay Tee (ECBT). Algoithm ECBT Label the nodes of the complete binay tee based on the inode tavesal using binay epesentation. Map each node of the complete binay tee to the node in the cossed cube with the coesponding binay epesentation. Theoem 1: Fo all n, the inode labeling of the complete binay tee embeds CB n within the cossed cube XQ n with dilation two. Poof: Let β k be the binay sting of length k with 1 in all positions. Fo n 3, the inode embedding is shown in Figue 2. Fo n > 3, we pove the theoem by induction on the height of the binay tee. Ou induction basis is CB 3, a dilation two embedding of CB 3 into XQ 3 is shown in Figue 2. Assume the theoem is tue fo an embedding of CB n-1 into XQ n-1. We now pove that the theoem is tue fo the embedding of CB n into XQ n. In XQ n, conside the two subcubes XQ 0 n-1 and XQ 1 n-1. By induction hypothesis, we can embed CB n-1 into XQ 0 n-1 and XQ 1 n-1, with dilation two. Since the numbe of nodes in CB n-1 is less than the numbe of nodes in XQ n-1 by one, then XQ 0 n-1 and XQ 1 n-1 contain two exta-unused nodes located at addesses 01β n-2 and 11β n-2, espectively. We can use the exta-unused node in XQ 0 n-1, the CB n-1 of XQ 0 n-1, and the CB n-1 of XQ 1 n-1 to constuct the complete binay tee CB n with 2 n -1 nodes. Next, we pove that the dilation of this embedding is two. We again use the outing algoithm of [12] to show that the length of the shotest path fom the oot of CB n to any of its childen is of length two. Let l be the shotest path fom the oot of CB n to the oot l of the left complete

3 binay subtee LCB n-1 and be the shotest path fom the oot of CB n to the oot of the ight complete binay subtee RCB n-1. will appea at addess 01β n-2, l at addess 00β n-2, and at addess 10β n-2. Notice that, l, and ae identical except fo the left most two bits. By using the outing algoithm, the shotest paths fom 01β n-2 to 00β n-2 and fom 01β n-2 to 10β n-2 ae of length two. So, the dilation of this embedding is two. As an illustation to the esulted embedding, in the lowest level, each edge fom a left child to its paent is mapped to the coesponding cossed cube edge between the images of the two nodes, while the edge between a ight child to its paent is mapped to a path of length two, fom the ight child to the left child and fom the left child to the paent. In the highe level, each edge fom a left child, o a ight child, to its paent is mapped to the coesponding cossed cube edge between the images of the two nodes. In all highe levels, each edge fom a left child, o a ight child, to its paent is mapped to a path of length two. Notice that the inode embedding is vey simple and staight fowad. 001 Figue 2: The inode embedding of CB 3 into XQ 3. IV. EMBEDDING QUAD TREES This section descibes ou ecusive scheme to embed a complete quad tee CQ n into its optimal cossed cube XQ 2nwith dilation two and unit expansion. We poceed in fou 1 steps. In the fist step, CQ n is decomposed into a fou complete quad sub tees; a left complete quad sub tee ACQ n-1 with oot a, a left middle complete quad sub tee BCQ n-1 with oot b, a ight middle complete quad sub tee CCQ n-1 with oot c, a ight complete quad sub tee DCQ n-1 with oot d, and a oot. In the second step, XQ 2n-1 is decomposed into fou sub cubes XQ 00, XQ01, XQ11 2n-, and 3 XQ10. In the thid step, ACQ is embedded into XQ 00, BCQ is embedded into XQ01, CCQ is embedded into XQ 11, DCQ n-1 is embedded into XQ10, and the oot is embedded into one of the unused nodes in XO 00. In the last step, we constuct CQ by finding the n paths a, b, c, and d, each of at most length two. The embedding pocess is continued ecusively by decomposing the complete quad sub tees and the cossed sub cubes, epeating the above steps, until we each the leaves of the complete quad tee. At the bottom level of the complete quad tee, each complete quad sub tee with 5 nodes is mapped into a cossed sub cube of dimension 3. Next, we pesent Algoithm Embed Complete Quad Tee (ECQT) that uses a ecusive divide and conques technique to embed a complete quad tee CQ n into its optimal cossed cube XQ 2n-1. Algoithm ECQT Let δ i be the binay sting of length n with a 1 in position i and 0 in all othe positions, θ k be the binay sting of length k with 0 in all positions, and be the xo opeato. Decompose CQ n to ACQ n-1, BCQ n-1, CCQ n-1, DCQ n-1, and. Decompose XQ 2n-1 to XQ 00, XQ01, XQ 11, and XQ10. Map the quad sub tees into the cossed sub cubes as follows: a. Embed ACQ n-1 into XQ 00, BCQ n-1 into XQ 01, CCQ n-1 into XQ11, and DCQ n-1 into XQ 10. a, b, c, and d will appea at addesses 000θ 2n-4, 010θ 2n-4, 110θ 2n-4, and 100θ 2n-4,, espectively. b. Tanslate the embeddings in XQ 00 and XQ 10 by complementing the ()th bit of each node. Fomally, if a tee node was mapped to addess x then afte the tanslation it will appea at addess x δ. Afte the tanslation, the left oot a and the ight oot d will appea at addesses 001θ and 101θ 2n-4, espectively. Theefoe, the final position of a, b, c, and d ae 001θ 2n-4, 010θ 2n-4, 110θ 2n-4, and 101θ 2n-4,, espectively. c. Map the oot into the node with label 0 in XQ 00. Constuct CQ n fom ACQ n-1, BCQ n-1, CQ n-1, DCQ n-1, and by finding the fou paths ~a, ~b, ~c, and ~d. The edges -a and -b of CQ n ae mapped to paths of length one in XO 2n-1, while the edges -c and -d ae mapped to paths of length two. The shotest paths fom to c and fom to d ae 000θ 2n-4-010θ 2n-4-110θ 2n-4 and 000θ 2n-4-100θ 2n-4-101θ 2n-4,espectively. Theoem 2: Fo all n, Algoithm ECQT maps the complete quad tee CQ n within the cossed cube XQ 2n-1 with dilation two and unit expansion. Poof: Fo n = 1, CQ 1 can be easily embedded into XQ 1. Fo n = 2, the existence of a mapping with dilation two is shown in Figue 4. Since the cossed cube is vetex symmetic, then we can elabel the cossed cube and any of the cone nodes of the sub cube XQ 3 can be the oot.

4 Figues 4-b and 4-c show two diffeent kinds of possible embeddings that might be used duing the embedding pocess. Fo n > 2, we pove this by induction on the height of the complete quad tee CQ n. Ou induction basis is CQ 3, a dilation two mapping of CQ 3 into XQ 5 is shown in Figue 5. Note that the figue shows only the positions of the oots, as the est of the embedding follows diectly fom Figue 4. Assume the theoem is tue fo a mapping of CQ n-1 in XQ. Now, we pove that the theoem is tue fo the embedding of CQ n in XQ 2n-1. In XQ 2n-1, conside a b the fou sub cubes XQ 00, XQ01, XQ11, and XQ10 2n-. By induction hypothesis, thee exist a dilation two 3 mapping fom CQ n-1 to XQ 00, XQ01, XQ11, and XQ 10. Since the numbe of nodes in CQ is less than the numbe of nodes in XQ, then XQ 00, XQ01, XQ 11, and XQ10 contain exta unused nodes. Now, we can use the unused node with label 0 in XQ 00, the CQ n-1 of XQ 00, the CQ n-1 of XQ01, the CQ n-1 of XQ 11, and the CQ of XQ 10 to constuct the complete quad tee CQ n. To pove that the dilation of this mapping is two, we need to show that the length of the shotest path fom the oot to any of its fou childen is at most two. Clealy, the length of the paths ~a and ~b is one since they ae mapped diectly to edges in the cossed cube. Let ~c be the shotest path fom the oot of CQ n to the oot c of the ight middle complete quad sub tee CCQ n-1 and ~d be the shotest path fom the oot of CQ n to the oot d of the ight complete quad sub tee DCQ n-1. will appea at addess 000θ 2n-4, c will appea at addess 110θ 2n-4, and d will appea at addess 101θ 2n-4. Notice, that if we goup the addesses of, c, and d into pais of bits, fom ight to left, then they ae pai-elated except fo the left most thee bits. By using the outing algoithm of [6], the shotest path fom 000θ 2n-4 to 110θ 2n-4 is 000θ 2n-4-010θ 2n-4-110θ 2n-4 and fom 000θ 2n-4 to 101θ 2n-4 is 000θ 2n-4-100θ 2n-4-101θ 2n-4. So, the length of the paths ~c and ~d ae two. Theefoe, the dilation of this embedding is two. a b c d d Figue 5: Embedding CQ 3 into XQ 5. V. EMBEDDING CYCLES Given a cycle C 2 n with 2 n nodes, conside the poblem of assigning the cycle nodes to the nodes of the cossed cube XQ n such that adjacency is peseved. Now, given any two adjacent nodes in the cycle, thei images by this embedding should be neighbos in the cossed cube though some dimension i, whee 1 i n. We can view such an embedding as a sequence of dimensions cossed by adjacent nodes. We call such a sequence the embedding sequence, denoted by ES = (d 1, d 2,..., d 2 n), whee d i {1,..., n} fo all 1 i 2 n. Figue 6 shows two diffeent embeddings of the cycle C 2 3 into the cossed cube XQ 3. It is moe convenient to view the embedded cycle as will as the cossed cube in the way shown in Figue 6. The embedding sequence of C 2 3 is ES = (1, 3, 1, 2, 1, 3, 1, 2). Fo example, in Figue 6-a, notice that nodes 000 and 001 ae connected by a link though dimension 1, 001 and 111 ae connected by a link though dimension 3, 111 and 110 ae connected by a link though dimension 1, 110 and 100 ae connected by a link though dimension 2, and so on. The embedding sequence ES can be geneated using Algoithm ES. Algoithm ES Let n be the dimension of the cossed cube and the vetical ba be the concatenation opeato. ES 1 Fo i 3 to n do ES ES i ES ES ES 2 ES 2 c a (a) b b c d c a d (b) (c) Figue 4: Embedding CQ 1 into the sub cube XQ 3. The embedding sequence is geneated by applying Algoithm ES on n, whee n is the dimension of the cossed cube. The numbe of nodes in the cossed cube is equal to the numbe of nodes in the embedded cycle, which is 2 n nodes. Thus, the embedding sequence of the cycle C 2 4 is ES = (1, 3, 1, 4, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 1, 2) and the embedding sequence of the cycle C 2 5 is ES = (1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 4, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 4, 1, 3, 1, 2). The embedding sequence coesponds to the extended binay-eflected Gay code embedding of a cycle into a cossed cube. The binay-eflected Gay code is the

5 most common technique to embed a cycle into a fault-fee hypecube. Notice that the same embedding sequence may esult in diffeent embeddings of C 2 n into XQ n depending on the cossed cube node that initiates the cycle constuction. Among all diffeent embeddings, we ae inteested in one kind. The embedding when the node that initiates the cycle constuction in the cossed cube is the uppe leftmost node, node with label 0. This will not violate the genealization of the technique since the cossed cube is node and vetex symmetic, which means that we can elabel the nodes, whee any node can be labeled as node 0, and hence initiates the constuction of the cycle. In Figue 6-a, the cycle is initiated by node 000, while in Figue 6-b the cycle is initiated by node (a) Node 000 initiating the embedding sequence. (b) Node 001 initiating the embedding sequence. Figue 6: The embedding sequence. Theoem 3: Fo evey n, Algoithm ES will geneate the embedding sequence to constuct a cycle of size 2 n in a cossed cube of dimension n. Poof: We pove this by induction on the dimension of the cossed cube. Ou induction basis is XQ 2, it is tivial that a cycle of size 4 can be easily constucted in XQ 2 using the embedding sequence ES = (1, 2, 1, 2). Assume the theoem is tue fo the constuction of a cycle of size 2 n-1 in a cossed cube of dimension n-1. We now pove that the theoem is tue fo the constuction of C 2 n in XQ n. Let G be any undiected labeled gaph, then G b is obtained fom G by pefixing evey vetex label with b. Conside the two cossed sub cubes XQ 0 n-1 and XQ 1 n-1. By induction hypothesis, we can constuct a cycle of size 2 n-1 in both XQ 0 n-1 and XQ 1 n-1. Let thei embedding sequence be ES = S n-1 2 S n-1 2, whee, S n is a sequence of dimensions ecusively defined as follows: S 2 = 1 and S n-1 = S n-2 n S n- 2. Now, we combine two cycles, each of size 2 n-1, to come up with a cycle of size 2 n. This is done by eplacing the fist link that goes though dimension 2 of the fist cycle and the second link that goes though dimension 2 of the second cycle by two links that go though dimension n. The embedding sequence of the new cycle C 2 n is ES = S n- 1 n S n-1 2 S n-1 n S n-1 2 = S n 2 S n 2, which is the same embedding sequence geneated by Algoithm ES Next, we pesent Algoithm Hamiltonian Cicuit (HC) that uses a ecusive divide and conques technique to embed a cycle C 2 n into a cossed cube XQ n. Algoithm HC Patition XQ n into 2 n-3 disjoint cossed cubes, each of dimension 3. Embed the cycle C 2 3 into each sub cube using the embedding sequence ES = (1, 3, 1, 2, 1, 3, 1, 2). Connect the 2 n-3 cycles, each of size 8, though the uppe, o lowe, links to come up with a cycle of size C 2 n. Theoem 4: Fo evey n, Algoithm HC will embed a Hamiltonian cycle of size 2 n in a cossed cube of dimension n. Poof: We pove this by induction on the dimension of the cossed cube. Ou induction basis is XQ 3, the embedding of a cycle of size 2 3 into a cossed cube of dimension 3 is shown in Figue 6. Assume the theoem is tue fo the constuction of a cycle of size 2 n-1 in a cossed cube of dimension n-1. We now pove that the theoem is tue fo the constuction of C 2 n in XQ n. Conside the two cossed sub cubes XQ 0 n-1 and XQ 1 n-1. By induction hypothesis, we can constuct a cycle of size 2 n-1 in both XQ 0 n-1 and XQ 1 n-1. Now, we combine two cycles, each of size 2 n-1, to come up with a cycle of size 2 n. This is done by eplacing the fist link that goes though dimension 2 of the fist cycle constucted in XQ 0 n-1 and the fist link that goes though dimension 2 in of the second cycle constucted in XQ 1 n-1 by links that go though dimension n. Note the use of the uppe links of dimension n when the embedding sequence is geneated by node 0, while the lowe cossed links of dimension n ae used when the embedding sequence is geneated by node 1, as shown in Figue 7. VI. CONCLUSIONS The poblem of embedding one inteconnection netwok into anothe is vey impotant in the aea of paallel computing fo potability of algoithms acoss vaious achitectues, layout of cicuits in VLSI, and mapping logical data stuctues into compute memoies. The impotance of tees comes fom the fact that this class of stuctues is useful in the solution of banded and spase systems, by diect elimination, and captues the essence of divide and conques algoithms. The impotant of cycles and ings comes fom the fact that it is the task gaph of many basic opeations in paallel computing. The hypecube is one of the most popula inteconnection netwoks due to many of its attactive popeties and its suitability fo geneal pupose paallel pocessing. An attactive vesion of the hypecube is the cossed cube. It peseves many popeties of the hypecube and most impotantly educes the diamete by a facto of two. This pape has pesented diffeent schemes to show the ability of the cossed cube as a vesatile achitectue to simulate othe inteconnection netwoks efficiently. We pesent new schemes to embed complete binay tees, complete quad tees, and cycles into cossed cubes. A good poblem will be to impove the dilation on embedding tees into cossed

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