On Optimal Embeddings into Incomplete Hypercubes

On Optimal Embeddings into Incomplete Hypercubes (Extended Abstract)

A. Gupta , A. Boals, and N. Sherwani Department of Computer Science Western Michigan University Kalamazoo, MI 49008

1 Introduction The hypercube has emerged as one of the most eective and popular architecture for parallel machines and several hypercube based machines (e.g; Intel IPSc and NCUBE) are commercially available. Hypercube popularity may be attributed to its regular structure and its rich interconnection topology [17]. Despite its versatility, hypercube topology requires that the number of nodes must be a power of 2. In order to alleviate this shortcoming, several `incomplete' hypercube-like architectures have been proposed. Katse proposed in [13] an n-node Incomplete Hypercube by taking nodes 0 through n ? 1 of a complete hypercube. He showed that broadcasting and nodeto-node communication algorithms for incomplete hypercubes are similar to ones of a complete hypercube. Tzeng et. al. [18] investigated a restricted version of the Katse's de nition by considering only those nnode incomplete hypercubes, where n = 2l + 2m; l > m. They investigated the capability of this architecture to simulate binary trees and two dimensional meshes. In [4, 7], we de ned a generalization of incomplete hypercubes, called Composite Hypercubes. Composite hypercubes are not restricted to the rst n nodes of a com Research

supported in part by a Fellowship from the Faculty Research and Creative Activities Support Funds WMU-FRCASF 90-15, WMU-89-225274, and by the National Science Foundation under grant USE-90-52346.

plete hypercube. The basic idea is that a composite hypercube is expressible as a collection of successively smaller complete hypercubes. These complete hypercubes are also required to satisfy an adjacency requirement which maximizes the number of edges in the architecture. Composite hypercubes are interesting because they allow upgradability to any number of nodes (in contrast, complete hypercubes need to double in size). It must be noted that all three de nitions essentially de ne subgraphs of complete hypercubes. In [4, 7] we showed composite hypercubes also retain many desirable properties of complete hypercubes. One of the main reasons for the popularity of hypercube architecture is its ability to simulate other architectures very eciently. If composite hypercubes are to be competitive as an architecture, we must demonstrate similar simulation capabilities. Graph embeddings have been used very successfully to show simulation capabilities of a guest architecture by another host architecture [1, 12]. In graph embedding techniques, host and guest architectures are viewed as graphs H and G, respectively, and then the graph G is embedded into the graph H . In the embedding of a graph G into H , we injectively map the set of nodes of G into the set of nodes of H and the edges of G to paths in H which connect the image of the nodes of G. In order to obtain ecient simulations of G by H , various cost measures of an embedding must be optimized. One such mea-

sure is the dilation of an embedding. The dilation of an edge of G is the length of the path onto which an edge of G is mapped. The dilation of the embedding is the maximum edge dilation of G. The expansion of the embedding is the ratio of the number of nodes in G to the number of nodes in H . Ideally, we would like to nd embeddings with minimal dilation and expansion. Embeddings of binary trees, two dimensional meshes, butter ies and cube-connected cycles into complete hypercubes have been studied extensively [2, 3, 5, 9, 11]. Very few results are known about the embeddings of these architectures into incomplete hypercubes. To the best of our knowledge, only Tzeng et. al. [18] have investigated the embeddings of complete binary trees, incomplete binary trees and meshes into their de nition of incomplete hypercubes. In order to simplify our discussion further, let us give the precise de nition of a composite hypercube. Let n = 2d + m, where 2d+1 > n  2d  1. An n-node composite hypercube CH (n) is de ned to be a subgraph of a complete hypercube Qd+1 consisting of two disjoint composite hypercubes CH (2d ) and CH (m), where CH (2d ) is a complete hypercube Qd. In addition to the edges of CH (2d ) and CH (m), the composite hypercube CH (n) contains the edges of Qd+1 which connect nodes of CH (2d) with the nodes of CH (m). Figure 1 shows an example of an 13 node composite hypercube. Observe that if n = 2d1 +2d2 +. . .+2d , then CH (n) consists of k di -dimensional complete hypercubes where every node in Qd is adjacent to exactly one node in each Qd ; j > i. In [4, 7] it was shown that any two n-node composite hypercubes are isomorphic and that the composite hypercubes contain maximum number of edges for n node subgraphs of complete hypercubes. Due to the fact that composite hypercubes are subgraphs of complete hypercubes, it may initially appear that the embedding strategies for complete hypercubes should carry over directly to composite hypercubes. However, since the number of nodes in a composite hypercube is arbitrary (not necessarily a power of two) and the degrees of the nodes need not be equal, some of the k

i

j

structural symmetry is lost. As many known embedding strategies depend on the regularity or symmetry of complete hypercubes, in general these strategies can not be used for composite hypercubes. Therefore, new embedding strategies are needed. The purpose of this paper is to investigate new strategies for embeddings of binary trees, two-dimensional meshes, butter ies, and cube-connected cycles into composite hypercubes. It has been shown that an n-node complete binary tree can be embedded into an n-node composite hypercube with a dilation of two [2], since n = 2h ? 1. Several divide-and-conquer type algorithms exhibit incomplete binary tree structure, thus it is interesting and natural to investigate the classes of incomplete binary trees which can be eciently embedded into composite hypercubes. Section 2 shows the embeddings of various types of n-node incomplete binary trees into n-node or (n + 1)-node composite hypercubes with dilation of at most 2. We also present lower bound proofs showing optimality of the dilation. We characterize the class of incomplete binary trees which are subgraphs of composite hypercubes. In Section 3, we present dilation 1 embedding of a two-dimensional n-node mesh, where one dimension is a power of two, into its optimal n-node composite hypercube. When neither dimension is a power of two, it is shown that a dilation 1 embedding is not possible; thereby characterizing the class of two-dimensional meshes that can be embedded into composite hypercubes with dilation 1. All two-dimensional meshes are shown to be embeddable with dilation 1 if expansion greater than 1 but less than 2 is allowed. We also consider two types of incomplete meshes and their embeddings into their optimal composite hypercubes.

2 Embedding Trees

In this section we consider embeddings of complete binary trees and incomplete binary trees into composite hypercubes. Researchers have previously considered embeddings of the these networks into complete hypercubes [3].

An n-node complete binary tree Th of height h can be embedded into a complete hypercube Qh+1 of n + 1PEs with dilation 2. There are many embeddings which achieve these bounds on embeddings. One simple such embeddings is inorder embedding [6] which labels the nodes of Th using an inorder traversal (the leftmost leaf of Th is labeled as 0) and then assigns node with label i to PE i of Qh+1 , 0  i  n ? 1. Observe that (n ? 1)=2 edges of Th are dilated by 2 in the inorder embedding. Bhatt and Ipsen [2] have shown that a n + 1-node tworooted complete binary tree TRh are subgraphs of Qh+1 and thus can be embedded with dilation 1 (TRh is obtained from a complete binary tree Th by replacing one of the edges incident on the root r with a path of length 2, and the additional node on the path is called a root, say r0 ). Throughout this section we refer to this embedding as BI-embedding. This result gives an embedding of Th into Qh with only one edge having dilation 2 and rest of the n ? 2 edges have dilation 1. A natural question which arises is that how well composite hypercubes embed binary trees. Since composite hypercube CH (n) is a subgraph of complete hypercube Qh+1 , it is easy to see that Th can not be embedded into an n-PE composite hypercube CH (n) with a dilation of 1. (Now onwards, for clarity we will refer to the nodes of a composite hypercube as PEs, Processing Elements.) Furthermore, by using Bhatt and Ipsen embedding and by deleting the PE which does not get a node of Th assigned, complete binary tree Th can be embedded into CH (n) with dilation 2. Composite hypercubes are de ned for any arbitrary number of PEs, hence one needs to explore incomplete binary tree structures which arise in many applications along with their embeddings into composite hypercubes. We de ne three types of incomplete binary trees which are denoted as Type 1, 2 and 3. Type 1 incomplete binary trees consists of a sequence of classes of binary trees, IT 1h for every h  0 where h denotes the height of the trees in a class. The class IT 10 consists of a single node and IT 11 consists of the two binary trees of height 1. The class IT 1h, for h  2, is de ned re-

cursively as follows: the rst tree of this class is a two rooted complete binary tree TRh?1 and all the other trees are obtained by appending a tree from class IT 1m with m  h ? 2 to the root r0 of TRh?1. Figure 2 shows rst four classes of Type 1 incomplete binary trees. Observe that the number of trees jIT 1hj in the hth class is jIT 1h?1j + jIT 1h?2j, for h > 2. For example the number of trees in the rst few classes are 1; 2; 2; 4; 6; 10; 16; :::. We next show that Type 1 incomplete binary trees are subgraphs of their optimal compact hypercubes. Type 1 incomplete binary trees are interesting from two aspects: First they characterize a whole class of binary trees which are subgraphs of their optimal size composite hypercubes, and second they contain special cases of Type 2 incomplete binary trees discussed later.

Theorem 1 Every Type 1 incomplete binary tree can be embedded into a composite hypercube with dilation of 1 and expansion of 1.

Proof: We extend the ideas of Bhatt and Ipsen [2]. It is

clear that all the trees in IT 10 and IT 11 can be embedded into the optimal composite hypercubes with dilation 1. For every class IT 1h, for h > 1, the rst tree is a two rooted tree TRh?1. By using BI-embedding this tree can be embedded into its optimal composite hypercube CH (2h), which is a complete hypercube Qh , with dilation 1. In order to embed all the other trees in IT 1h, we use induction. An n-node tree T in IT 1h is composed of a two rooted complete binary tree TRh?1 and a tree, say T 0, from class IT 1m, m  h ? 2, such that the root of T 0 is connected to one of the roots r of TRh?1. Assume by induction that T 0 is embedded into its optimal composite hypercube CH (n ? 2h) with dilation and expansion 1. Let b1 b2 . . . bh+1 be the label of the PE in CH (n ? 2h) which gets root of T 0 assigned. By using BI-embedding, TRh?1 can be embedded into its optimal composite hypercube CH (2h ) with dilation and expansion 1. Since CH (2h) is also a complete hypercube, the embedding of TRh?1 into CH (2h ) can be easily translated so that the root r of TRh?1 is assigned to PE b1 b2 . . . bh+1 in CH (2h). Now by combining the embeddings of T 0 and

TRh?1 in CH (n ? 2h) and CH (2h ), respectively, the root of T 0 and r become adjacent and hence we have the composite hypercube CH (n) with an embedding of T. An n-node Type 2 incomplete binary tree IT 2(n) is a complete binary tree of height h with right most 2h+1 ? n ? 1 leaves missing, for the smallest h such that 2h+1  n. A common way to label a complete binary tree is to label the root 0 and then for any node v which is labeled i, the left child of v is labeled 2i + 1 and the right child is labeled 2i +2. It is thus easy to see that the nodes of IT 2(n) can be labeled from 0 to n ? 1 in a similar manner. Note that the IT 2(n)'s are also known as

full trees. The trees of this type were considered in [18] and the authors have shown the existence of dilation 1 embeddings for several special cases. Note that the expansion of their embeddings is greater than 1 whereas our focus is to nd embeddings into composite hypercubes with expansion very close to 1. Theorem 2 shows that IT 2(n) can be embedded into CH (n +1) with a dilation of 2. This embedding uses only one extra PE from its optimal size composite hypercube. As will be shown by Lemma 3, for n 6= 32 (2h+1 ? 1) +  when h is odd, or n 6= 34 (2h ? 1)+  when h is even, with  = ?1; 0; 1, this embedding is optimal with respect to dilation. For the cases when this embedding is not optimal, it is easy to see that Type 2 incomplete binary trees are also Type 1 incomplete binary trees and hence embeddable into CH (n) with dilation and expansion 1.

Theorem 2 An n-node Type 2 incomplete binary tree IT 2(n) is embeddable in a n + 1-node composite hypercube CH (n + 1) with a dilation of 2.

The proof of this theorem is omitted for sake of brevity.

Lemma 3 An n-node Type 2 incomplete binary tree

IT 2(n) of height h can not be embedded into an nPE composite hypercube CH (n) with a dilation of 1. The only values of n where an n-node Type 2 incomplete binary tree of height h can be embedded into an n-PE composite hypercube with a dilation of 1 are n = 4 h 2 h+1 3 (2 ? 1) +  when h is odd, and n = 3 (2 ? 1) + 

when h is even, with  = ?1; 0; 1, for the smallest h such that 2h+1 > n > 2h . Proof: The proof relies on the following fact about bipartite graphs: The partite sets of a connected bipartite graph are unique. We know that an n-PE composite hypercube CH (n) is a connected bipartite graph in which the number of PEs in two partite set diers by at most one. Any n-node Type 2 incomplete binary tree IT 2(n) is a connected bipartite graph and, in general, the two partite sets of IT 2(n) dier by at least 2. Since the sizes of partite sets of CH (n) and IT 2(n) do not match, the lemma follows. An n-node Type 3 incomplete binary tree IT 3hk(n) of height h consists a complete binary tree Tk of height k and 2k?1 complete binary trees, each of height h ? k, with their roots as the alternate leaves of Tk . An example of a 39-node Type 3 incomplete binary tree of height 5 with k = 3 is shown in Figure 3. Type 3 incomplete binary trees are of interest in the situations where divide-and-conquer type of algorithms, such that every process at a node up to level k ? 1 in a complete binary tree Th of height h spawns two processes and then only one of the two children of every node at level k ? 1 spawns two processes up to level h in Th, are abstracted as a binary tree and these algorithms are then simulated by composite hypercubes. We next show that except when n = 5 (with h = 2 and k = 1) or n = 11 (with h = 3 and k = 2), a tree IT 3hk(n) can not be embedded into an n-PE composite hypercube CH (n) with a dilation of 1. We also describe an embedding that embeds IT 3hk(n) into CH (n) with a dilation of 2 for n 6=; 5; 11. Note that when n = 5 or 11, tree IT 3hk(n) can be easily embedded into CH (n) with a dilation of 1.

Lemma 4 An n-node Type 3 incomplete binary tree IT 3hk(n) is not a subgraph of an n-PE composite hypercube CH (n) whenever n 6= 5; 11; i.e., any embedding of IT 3hk(n) into CH (n) must have a dilation of at least 2.

Proof: The basic idea of the proof is similar to the one

used in Lemma 3, namely the uniqueness of the number

of nodes in the bipartite sets of a connected bipartite graph. The details are omitted. We now describe an optimal dilation 2 embedding of IT 3hk(n) into CH (n), for n 6= 5; 11. For brevity henceforth we drop the subscript and superscript from IT 3hk(n) and simply refer to it as IT 3(n). The basic idea in the embedding is to assign the nodes at levels k through h of IT 3(n) to the PEs of CH (n) which form a complete hypercube Qh of 2h PEs and then assign the nodes at levels 0 through k ? 1 to PEs in CH (n) which form a composite hypercube CH (2k ? 1). We use a combination of ideas of the BI-embedding and the inorder embedding to embed IT 3(n) with dilation 2. Further details are omitted.

Theorem 5 An n-node Type 3 incomplete binary tree IT 3(n) can be embedded into CH (n) so that the dilation is at most 2. Proof: Omitted.

3 Embedding Meshes

Many useful algorithms, in particular, linear algebra algorithms, can be eciently performed on two or higher dimensional meshes. This led to an interest in simulating meshes on hypercubes. Meshes can be embedded with dilation one into complete hypercubes if all dimensions of the mesh are powers of two. If some dimensions are not powers of two then dilation of two is necessary and sucient [5]. The main technique for embedding meshes in complete hypercubes is based on assigning the nodes of each dimension a Binary Re ected Gray Code(BRGC). For example, if a 2p  2q mesh is to be embedded into a Qp+q , then a p-bit BRCG is assigned to the dimension with length 2p and q-bit BRCG is assigned to the other dimension. The address of a node in a hypercube is obtained by concatenation of the p bits from one dimension with q bits from the other dimension. The minimum size hypercube needed to embed a mesh has been characterized by Havel and Moravek [10]. It follows from [10] that expansion is in the range of 1 to 2k . When dilation one embedding is not possible then

dilation two and dilation three has been achieved using variety of techniques including step embedding [1], folding [15], line compression [1], modi ed line compression [5] and graph decomposition [11]. In this section, we investigate embeddings of two dimensional meshes into composite hypercubes. For these embeddings the main diculty lies in the fact that the number of nodes in a composite hypercube is not a power of two. Moreover, due to the very nature of composite hypercubes, it is interesting to consider embeddings of n node mesh onto n node composite hypercubes, that is, we use exactly the same number of processors. We characterize the class of two dimensional meshes that can be embedded into a composite hypercube with dilation one. We also investigate embedding of incomplete meshes and composite meshes and present dilation one embeddings for these architectures.

3.1 Embedding Meshes with Dilation 1.

If both dimensions of a mesh are powers of two then clearly gray code embedding is optimal i.e., dilation 1. In this section we show that a n node two dimensional meshes can be embedded into composite hypercubes if one of the dimension is a power of two. In the case when both dimensions are not powers of two, we show that no dilation 1 embedding exists.

Lemma 6 A l1  l2 mesh M can be embedded into a CH (l1  l2 ) with dilation 1 if either l1 or l2 is a power of two.

Proof: Without loss of generality assume that l2 =

2d and l1 = 2d1 + . . . + 2d , d1 > . . . > dk . We can view any l1  l2 mesh as a disjoint union of k dierent meshes Mi = 2d  l2 with additional edges. We call this decomposition of M a k-component decomposition. Clearly each Mi can be embedded with dilation 1 as both dimensions of Mi are powers of two. Our general strategy for embedding M is by assigning d-bit binary re ected gray code to l2 dimension and d1 +1-bit binary re ected gray code to the l1 dimension. An example embedding of a 3  4 mesh into a CH (12) is shown in k

i

Figure 5. We omit further details of the embedding from this extended abstract. It is clear from the gray code assignment that after renumbering, each 1  l2 mesh forms a d-dimensional hypercube. Moreover if we consider rst 2d1 rows of the l1  l2 mesh under the embedding, it is clear that it forms a (d + d1 )-dimensional hypercube.

Lemma 7 If an l1  l2 mesh can be embedded into a CH (l1  l2 ) with dilation 1, then either l1 or l2 is a

power of two. Proof: This proof is based on the fact that the restricted number of edges between the d1 -dimension subcube and the rest of the composite hypercube limits the ways in which a mesh may be partitioned for embedding. As per de nition of a composite hypercube, one may consider a CH (n) as consisting of two components; a d1 -dimensional complete hypercube and CH (n ? 2d1 ). These components will be called the red and black components respectively. Any node of M embedded in the red(black) component will be called red(black) node. It is clear that 2d1 nodes of M must be colored red and l1  l2 ? 2d1 nodes must be colored black. Assume that a node mi;j is colored black, then at most one of its neighbors can be colored red. This is due to the fact that each node in the black component is adjacent to exactly one node in the red component. This implies that if mi;j is colored black and mi+1;j is colored red then either the entire ith row must be colored black. Similarly if mi;j+1 is colored red then the entire j th column must be colored black. Without loss of generality, assume that mi+1;j exists and is colored red. Then mi+1;r ; r = 1; 2; . . . ; l2 are colored red, since mi;j is colored black. Moreover, if nodes mi?1;r ; r = 1; l2 exist they all must be colored black, due to the fact that already one neighbor, namely mi+1;j , of mi;j is colored red. This implies that the smallest mesh that can be colored red or black is either a r  l1 or a r  l2 mesh, where r  2. Since 2d1 nodes of M are colored red, it follows from the argument above that either 2d1 = r l1 or 2d1 = r l2 ,

for some r  2. Without loss of generality assume that 2d1 = r  l1 . As l1 6= 2x, there exist no r which satis es 2d1 = r  l1 . This implies that there exists no partition of M which can be colored red without violating the adjacency requirement of the two components in CH (m). Lemmas 6 and 7 give us the complete characterization of 2-dimensional meshes that are embeddable into composite hypercubes of equal size with dilation 1.

Theorem 8 A l1  l2 mesh can be embedded into a CH (l1  l2 ) with adjacencies preserved if and only if l1

or l2 is a power of 2. Proof: Follows directly from Lemmas 6 and 7. . We conclude this section by observing that if expansion more than one is allowed then all two dimensional meshes can be embedded with dilation one. This can be done by rst embedding a l1  l2 mesh M into a mesh M 0 of size m0 = 2dlog l1 e  l2 , and then embedding M 0 into CH (m0 ). This leads to the following corollary.

Corollary 1 A l1  l2 mesh can be embedded into a CH (m) with adjacencies preserved, where m = min(2dlog l1 e  l2 ; l1  2dlog l2 e ).

3.2 Embedding Incomplete and Composite Meshes with Dilation 1 In many application, a mesh algorithm may work on a part of the mesh. In such cases, it is interesting to consider meshes, which we call Incomplete Meshes. Also it is possible that several mesh based algorithms are working together, each on a dierent size mesh. In order to capture this idea, we introduce the concept of a composite mesh. That is, a mesh consisting of several smaller dierent sized meshes. In this section we show that incomplete and composite meshes can be embedded into composite hypercubes with dilation one. We de ne an incomplete mesh IMl1 l2 ;l3 to be a l1 l2 dimensional mesh with l3  l2 right most nodes missing in the l1th row. See Figure 6.

Theorem 9 An incomplete mesh IM (l1  l2; l3), 0  l3  l2 can be embedded into CH (l1  l2 ? l3 ), if l2 = 2r

with dilation one. Proof: We provide an outline of the proof, details are omitted for the sake of brevity. First note the fact that each row of a l1 l2 mesh is embedded in a d-dimensional hypercube, l2 = 2d, according to the strategy explained in Lemma 6. If rightmost l3 nodes are missing in the last, it follows from the de nition of the composite hypercube that the last row is embedding into a l2 ?l3 -node composite hypercube. The union of this composite hypercube with the l1 ? 1  l2 mesh gives us the desired result. Figure 6 shows an example of an embedding of a IM (7  4; 2) into a CH (26). We de ne a composite mesh CM (l1 ; l2 ; l3 ; . . . ; lk ) as a collection of k meshes M1 ; M2 ; . . . ; Mk of sizes l1  l1 ; l2 l2 ; . . . ; lk lk respectively. In addition, rst row of nodes of Mi+1 are adjacent to leftmost nodes of the last row of Mi . More precisely, node mi (di ; r) is adjacent to mi+1 (1; r) for r = 1; . . . di+1 and 1  i < k. Where mi (p; q) denotes the node in the pth row and qth column of mesh Mi .

Theorem 10 A composite mesh CM (l1; l2; . . . ; lk ) 0  l3  l2 can be embedded into and a composite hypercube with same number of nodes, CH (l12  l22  . . .  lk2 ), with dilation one, if each li is power of two and li > li+1 . Proof: We provide an outline of the proof, details are

omitted for the sake of brevity. Note that each Mi can be embedded into a optimal size hypercube Qd . That is, each Mi maps to complete sub-hypercubes in the composite hypercube. Moreover gray code embedding ensures that nodes in the the last row of Mi are adjacent to rst row Mi+1 if they exist. This follows from theorem 9 as we may consider the rst row of Mi+1 along with Mi as IM (li  li ; li+1 ). Figure 7 shows an example of an embedding of a CM (4; 2; 1) into a CH (21). . i

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