The Hypercube 3.1. Section 3.1 The Hypercube

Transcription

1 392 Sectin 3.1 The Hypercube 3.1 The Hypercube In this sectin, we define the hypercube and explain why it is such a pwerful netwrk fr parallel cmputatin. Amng ther things, we will shw hw the hypercube can be used t simulate all f the netwrks discussed in Chapters 1 and 2. In fact, we will find that the hypercube cntains (r nearly cntains) all f these netwrks as subgraphs. This material is bth surprising and imprtant because it demnstrates hw all f the parallel algrithms discussed thus far can be directly implemented n the hypercube withut significantly affecting the number f prcessrs r the running time. Hence, we will quickly understand ne f the main reasns why the hypercube is s pwerful. We begin the sectin with sme definitins and a brief discussin f the hypercube's simplest prperties in Subsectin In Subsectin 3.1.2, we shw that the hypercube is Hamiltnian, and we explain the crrespndence between Hamiltnian cycles in the hypercube and Gray cdes. We als prve that any A^-nde array (f any dimensinality) is a subgraph f the A^-nde hypercube (assuming that A" is a pwer f 2). In Subsectin 3.1.3, we describe several embeddings f an (A" l)-nde cmplete binary tree in an A^-nde hypercube. Althugh the (N l)-nde cmplete binary tree is nt a subgraph f the Af-nde hypercube, we will find that a cmplete binary tree can be simulated very efficiently n the hypercube. Mre generally, we will shw that the Af-nde hypercube can efficiently simulate any binary tree in Subsectin In particular, we will shw hw t grw an arbitrary M-nde binary tree in an n-line fashin in an A'-nde hypercube s that neighbring ndes in the tree are nearby in the hypercube and s that at mst 0(M/N + 1) tree ndes are mapped t each hypercube nde with high prbability. The analysis f the treegrwing algrithm invlves an interesting relatinship between ne-errrcrrecting cdes and hypercubes that has numerus applicatins. We als define the hypercube f cliques in Subsectin and prve that it is cmputatinally equivalent t the hypercube. In Subsectin 3.1.5, we shw hw t efficiently simulate a mesh f trees n the hypercube. As a cnsequence, we will find that all f the algrithms described in Chapter 2 can be implemented withut significant slwdwn n a hypercube f apprximately the same size. We cnclude in Subsectin with a brief survey f sme related

2 3.1.1 Definitins and Prperties \ 0 00 Q N = 2 N = 4 N = 8 Figure 3-1 The N-nde hypercube fr N = 2, 4, and 8. Tw ndes are linked with an edge if and nly if their strings differ in precisely ne bit psitin. Dimensin 1 edges are shwn in bldface. netwrk cntainment and simulatin results fr the hypercube Definitins and Prperties The r-dimensinal hypercube has N = 2 ndes and r2 ~ edges. Each r r 1 nde crrespnds t an r-bit binary string, and tw ndes are linked with an edge if and nly if their binary strings differ in precisely ne bit. As a cnsequence, each nde is incident t r = lg N ther ndes, ne fr each bit psitin. Fr example, we have drawn the hypercubes with 2, 4, and 8 ndes in Figure 3-1. The edges f the hypercube can be naturally partitined accrding t the dimensins that they traverse. In particular, an edge is called a dimensin k edge if it links tw ndes that differ in the kth bit psitin. We will use the ntatin u t dente the neighbr f nde u acrss dimensin k k in the hypercube. In particular, given any binary string u = iti u i s N, the string u is the same as u except that the kth bit is cmplemented. k Mre generally, we will use the ntatin ul»>*»>-.*.} t fc Q dente the string frmed by cmplementing the kxth, k2th,..., and A;sth bits f u. Fr example, = and ' > = in a 128-nde 2 {3 4 hypercube. The dimensin k edges in a hypercube frm a perfect matching fr each k, 1 < k < lg N. (Recall that a perfect matching fr an Af-nde graph is a set f A^/2 edges that d nt share any ndes.) Mrever, remval f the dimensin k edges fr any k < lg A^ leaves tw disjint cpies f an

3 394 Sectin 3.1 The Hypercube y-nde hypercube. Cnversely, an TV-nde hypercube can be cnstructed frm tw y-nde hypercubes by simply cnnecting the ith nde f ne y- nde hypercube t the ith nde f the ther fr 0 < i < y. Fr example, see Figure 3-2. In additin t a simple recursive structure, the hypercube als has many f the ther nice prperties that we wuld like a netwrk t have. In particular, it has lw diameter (lgiv) and high bisectin width (N/2). The bund n the diameter is easily prved by bserving that any tw ndes u = uxu2 u\gn and v = VyD? V\gN are cnnected by the path «i«a '' "lg N * Viu2 n l g N > Vi^ 2U3 "lg N»» VlV2 Ui 0g N-lU\gN > VXV2 V\gN. The bund n bisectin width is established by shwing that the smallest bisectin cnsists f the edges in a single dimensin. The prf fllws as a special case f Therem 1.21 frm Sectin 1.9. As an interesting aside, it is wrth nting that a hypercube can be bisected by remving far fewer than y ndes, even thugh y edges are required t bisect the A-n d e hypercube. Fr example, cnsider the p a r t i tin frmed by remving all ndes with size and p2t ". ( T h e size, r weight, f a nde in the hypercube is the number f Is cntained in its binary string.) A simple calculatin reveals that remval f these ndes frms a bisectin with G (N/ \/lg N) ndes, which is the best pssible. The details f these and sme related results are left t the exercises (see Prblems ). It is als wrth nting that the hypercube pssesses many symmetries. Fr example, it is nde and edge symmetric. In ther wrds, by just relabelling ndes, we can map any nde nt any ther nde, and any edge nt any ther edge. Mre precisely, fr any pair f edges (u, v) and (u 1, v') in an A-nde hypercube H, there is an autmrphism a f H such that ~(u) = u' and (v) = v'. (An autmrphism f a graph is a ne-t-ne mapping f the ndes t the ndes such that edges are mapped t edges.) In fact, there are many such autmrphisms. Fr example, let u = uxu2 U\gN, u' = u[u'2 u[ g N, k be the dimensin f (u, v), and k' be the dimensin f (u 1, v'). Then fr any permutatin n n {1,2,..., lg A"} such that n(k') = k, we can define an autmrphism a with the desired prperty by setting

4 3.1.1 Definitins and Prperties 395 Figure 3-2 Cnstructin f a fur-dimensinal hypercube (b) frm tw threedimensinal hypercubes (a). Dashed edges frm a matching between the tw threedimensinal cubes.

5 396 Sectin 3.1 The Hypercube Figure 3-3 Tw labellings f the 8-nde hypercube. By relabelling apprpriately, we culd have mapped edge e = (000,001) t any psitin in the netwrk. a(xxx2 - X ] g N ) = 0tl*(i) u[) (xn(2) «*(2) «2) I " * I (av(lg/v) ^(lcgat) U\gN)- ( 3 -l) (Here and thrughut Chapter 3, we use the ntatin a \ 8 t dente the cncatenatin f a and B.) It is a simple exercise t check that a is an autmrphism f the hypercube with the desired prperties. (See Prblem 3.10.) As an example, we have illustrated tw labellings f the 8-nde hypercube in Figure 3-3. In the example, we have mapped the edge (000,001) t edge (110,100) using the autmrphism a(x1x2x3) = (xi 1) I (x3 1) I x2. In general, we can rearrange the dimensins f the edges in any rder that we want (by varying n) withut altering the netwrk. (See Prblems ) We will use such symmetries rutinely in the chapter t simplify explanatins Cntainment f Arrays One f the mst interesting prperties f the A-nde hypercube is that it cntains every A^-nde array as a subgraph. This result hlds true even fr high-dimensinal arrays and even if wraparund edges are allwed. Fr example, the embedding f a 4 x 4 array in a 16-nde hypercube is shwn