 # Embedding of Hypercube into Cylinder

Task mapping in modern high performance parallel computers can be modeled as a graph embedding problem, which simulates the mapping as embedding one graph into another and try to find the minimum wirelength for the mapping. Though embedding problems have been considered for several regular graphs, such as hypercubes into grids, binary trees into grids, et al, it is still an open problem for hypercubes into cylinders. In this paper, we consider the problem of embedding hypercubes into cylinders to minimize the wirelength. We obtain the exact wirelength formula of embedding hypercube Q^r into cylinder C_2^3× P_2^r-3 with r>3.

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## 1. Introduction

On one hand, a parallel program can be modeled as a task graph, in which the vertices of the graph represent a computing task, and the edges represent the communications among different tasks. On the other hand, a massive parallel computer has a large number of processing nodes that are connected together with an interconnection network. One of the key problems of efficient execution of parallel programs on these computers is how to find an optimal mapping from computing tasks to processing nodes, so that the communication overhead could be reduced when the tasks are run in parallel. Without loss of generality, this problem can also be modeled as an graph embedding problem, since task mapping is actually try to find an optimal embedding of computing task graph into the interconnection graph (network) with minimum link congestion. Unfortunately, task mapping is proved as a NP hard problem and heuristics algorithms are usually used to find an approximate solution for a given application and interconnection.

Researchers have been working on graph embedding for years and proposed a number of solution for regular graphs, such as hypercubes into grids , binary trees into grids , honeycomb into hypercubes , grids into grids 

. This paper introduce a new technique to estimate the wirelength of embedding hypercube

into cylinder with .

The rest of the paper is organized as follows. We show the existing work on graph embedding in Section 2. Section 3 introduces the gray embedding and some useful property, which is used afterwards in the wirelength calculation. Section  4 discusses some composite sets with Cartesian production structure. The wirelength calculation of hypercube into cylinder is given in section  5. Conclusion and future work appear in Section  6.

## 2. Problem definition

Let and be finite graphs with vertices. and denote the vertex sets of and respectively. and denote the edge sets of G and H respectively. An embedding  of into is defined as follows:

(i) is a bijective map from to ;

(ii) is a one-to-one map from to is a path in between and .

The edge congestion of an embedding of into is the maximum number of edges of the graph that are embedded on any single edge of . Let denote the number of edges of such that is in the path between and in , in other words,

 ECf(e)=|{(u,v)∈E(G) : e∈Pf(u,v)}|.

For any , define

 ECf(S)=∑e∈SECf(e).

The edge congestion of an embedding of into is given by

 ECf(G,H)=maxe∈E(H)ECf(e).

The minimum edge congestion of G into H

 EC(G,H)=minf:G→HECf(G,H),

where the minimum is taken over all embeddings of into .

The edge congestion problem of into is to find an embedding of into that induces minimum edge congestion .

The wirelength of an embedding of into is given by

 WLf(G,H)=∑(u,v)∈E(G)dH(f(u),f(v))=∑e∈E(H)ECf(e).

where denote the length of the path in . The wirelength of into is defined as

 WL(G,H)=minf:G→HWLf(G,H),

where the minimum is taken over all embeddings of into .

The wirelength problem of into is to find an embedding of into that induces minimum wirelength .

Manuel et al() find that the maximal subgraph problem play an important role in solving wirelength problem. For a graph and an integer ,

 IG(m)=maxA⊂V(G), |A|=m|IG(A)|,

where

 IG(A)={(u,v)∈E(G) : u,v∈V(G)}.

A subset is called optimal if .

The following lemmas are proved in . Note that a set of edges of is said to be an edge cut of , if the removal of these edges results in a disconnection of .

###### Lemma 2.1 (Congestion Lemma).

Let be an -regular graph and be an embedding of into . Let be an edge cut of such that the removal of edges of leaves into components , and let , . Also satisfy the following condition,

(i) For every edge , , has no edges in .

(ii) For every edge in with and , has exactly one edge in .

(iii) is optimal.

Then is minimum and

###### Lemma 2.2 (Partition Lemma).

Let be an embedding. Let be a partition of such that each is an edge cut of . Then

 WLf(G,H)=p∑i=1WLf(Si).

We will discuss the embedding of following graphs.

, the graph of the -dimensional hypercube, has vertex-set , the -fold Cartesian product of . Thus . has an edge between two vertices (-tuple of s and s) if they differ in exactly one entry.

The -dimensional grid with vertices is denoted as . The -dimensional cycle with vertices is denoted as . The -dimensional grid is defined as , where is an integer for each . The cylinder , where and , is a grid with a wraparound edge in each column (see e.g., Figure 2). The torus , where , is a grid with a wraparound edge in each column and a wrapround edge in each row.

It is conjectured that the wirelength of embedding hypercube into cycle is . It is called CT conjecture [4, 2, 3, 7]. It is also conjectured in  such that the wirelength of embedding hypercube into cylinder with positive integers is

 2r1(22r2−1−2r2−1)+2r2(3⋅22r1−3−2r1−1),

and the wirelength of embedding hypercube into torus with positive integers is

 2r1(3×22r2−3−2r2−1)+2r2(3⋅22r1−3−2r1−1).

Manuel et al () verified the case of embedding into cylinder for . We prove in this paper that

###### Theorem 2.3.

For any , the wirelength of embedding into cylinder is

 2r1(22r2−1−2r2−1)+2r2(3⋅22r1−3−2r1−1),

where and .

Remark 1. Our argument for Theorem 2.3 also valid for embedding of hypercube into torus . So

 WL(Q6,C8×C8)=2×8×20=320.

## 3. Gray Embedding

Grid embedding plays an important role in computer architecture, and researchers believe that gray embedding minimize wirelength of emdedding hypercube into cycles, cylinders and torus [2, 6].

To construct gray embedding, we give first the bijection of to .

Given and under Gray code list of bits, every code corresponding to a number, e.g.,

 gd(0d)=0, gd(0d−11)=1, gd(0d−211)=2,⋯, gd(10r−1)=2d−1.

Define an embedding from into with and . The vertices of have coordinates of the form , for , . Every vertex of correspond to a string in . Take any , write , where , , define

 gray(w)=(gr1(u),gr2(v)),

which corresponding to a unique vertex in .

As an example, we illustrate the embedding from to . By the above construction

 g2(00)=0, g2(01)=1, g2(11)=2, g2(10)=3.

So we can directly get the map as shown in Figure 1.

###### Lemma 3.1.

Fix and with . Define a bijection from to as above. For any edge in hypercube , and are in the same row or the same column of .

###### Proof.

Without loose of generality, let . Write , , with and . Since and are different only in one bit, either or . Hence either the x-coordinate or the y-coordinate of and are equal. The result of the lemma follows.∎∎

Let and (respectively ). Now we can construct graph embedding from to . We have already define a vertices bijection from to . For any edge in , define be the path in between and with minimal number of edges.

## 4. Structure of a class of composite sets

In our paper we will discuss some composite sets with Cartesian product structure.

A -subcube of the -cube is the subgraph of induced by the set of all vertices having the same value in some coordinates.

For any , write

 k=m∑i=12ci,0≤c1

If is a subgraph of which is a disjoint union of -subcubes, , such that each -subcube lies in a neighborhood of every -subcube for any , then is called a composite set (or cubal, see [1, 5]).

Let be a subgraph of with . It is proved in  that is a composite set if and only if it is optimal, or equivalently,

 IQr(S)=IQr(k).

For any graph and , recall that the Cartesian product of is, , and if and only if

 (u1,v1)∈E(S), u2=v2∈V(T)oru1=v1∈V(S), (u2,v2)∈E(T).

Take any with . By definition of Cartesian product of graphs, it is direct to know that . If is a -subcube of , is a -subcube of , then is a -subcube of . Hence by definition of composite set, we see that if is a subcube of , is a composite set of , then is a composite set of . So we get the following lemma

###### Lemma 4.1.

Let and are composite sets of and respectively. Suppose further that at least one of and is a subcube, then is a composite set of .

Recall that the (binary-reflected) Gray code list for bits can be generated recursively from the list for bits by reflecting the list (i.e. listing the entries in reverse order), concatenating the original list with the reversed list, prefixing the entries in the original list with a binary , and then prefixing the entries in the reflected list with a binary .

By this kind of recursive structure, we see that if , then is a composite set, where we use the matlab notation for any to indicate the set of consecutive integers.

In fact, write

 k=m∑i=12ci,0≤c1

Let , and for ,

 ai=m∑k=i2ck.

Then for any , is a -subcube in . These subcubes are disjoint, and each -subcube lies in a neighborhood of every -subcube for any . So we get the following lemma

###### Lemma 4.2.

For any and , is a composite set of hypercube .

## 5. Hypercube into Cylinder

Now we can prove Theorem 2.3 by computing for , . Denote and .

To apply Congestion Lemma, we need to construct suitable edge cuts to form a partition. For , define edge cut

 Bj={((i,j−1),(i,j)) : i=0,1,⋯,7}.

Define edge cuts

 A1={((0,j),(1,j)),((3,j),(4,j)) : j=0,1,⋯,2r2−1}A2={((1,j),(2,j)),((6,j),(7,j)) : j=0,1,⋯,2r2−1}A3={((2,j),(3,j)),((5,j),(6,j)) : j=0,1,⋯,2r2−1}A4={((4,j),(5,j)),((7,j),(0,j)) : j=0,1,⋯,2r2−1}

For any , disconnects into two components and . For any , disconnects into two components and . We illustrate the case , in Figure 2.

We discuss first .

Let and be the inverse images of and under respectively. By Lemma 3.1, the edge cut satisfies condition (ii) of the Congestion Lemma.

Now we show that satisfies condition (i). Since for any edge , is the shortest path connecting and , we only need to check whether there is a path from to for any

. Notice that, if the Hamming distance of two codes is odd, then the gray coding distance is odd, and vice versa. This implies that the Hamming distance between

and is even. So there are no edge path connecting and .

Notice that for some . It is direct to see that is a composite set, and is a -subcube. This implies that the subgraph is Cartesian product of a composite set and a subcube. By Lemma 4.1, is a composite set, and hence optimal. Thus by the Congestion Lemma, is minimum.

The argument for , and are analogous. For , we see that is a composite set, and there are no edge path connecting and . For , we see that is a composite set, and there are no edge path connecting and . For , we see that is a composite set, and there are no edge path connecting and . Thus by the Congestion Lemma, , and are also minimum.

Fix any . Let and be the inverse images of and under respectively. The edge cut satisfies conditions (i) and (ii) of the Congestion Lemma. Note that . It is direct to see that is a -subcube. And by Lemma 4.2, is a composite set. This implies that is Cartesian product of a sub-hypercube of order with a composite set. By Lemma 4.1, is also a composite set, and hence is optimal. Thus by the Congestion Lemma, is minimum.

The Partition lemma implies that is minimum.

It is direct to compute that for any

 WLgray(Qr,C23×P2r−3)=2r1(22r2−1−2r2−1)+2r2(3⋅22r1−3−2r1−1),

where and .

This proves Theorem 2.3.

## 6. Conclusion

Manuel et al get the exact wirelength of embedding of hypercube into cylinder with . We prove in this paper that gray embedding minimizes the wirelength of embedding hypercube into cylinder with , and hence get the exact wirelength of this case. We also get the exact wirelength of embedding hypercube into torus .

We ever apply this method to study the case of embedding hypercube into cylinder for any . We tried many embeddings(including gray embedding), but we can’t get a partition to apply Congestion Lemma.

Acknowledgements Liu and Wang are supported by the National Natural Science Foundation of China, No. 11371055. Ji is supported by the National Natural Science Foundation of China, No. 61300010.

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