Embedding hypercubes into torus and Cartesian product of paths and cycles for minimizing wirelength

TL;DR

This paper proves the minimum wirelength for embedding hypercubes into Cartesian products of paths and cycles, confirming Gray code embedding as an optimal strategy for such problems.

Contribution

It mathematically proves the conjecture on minimum wirelength and establishes Gray code embedding as optimal for hypercube embeddings into these graph products.

Findings

Proved the minimum wirelength for hypercube embedding into Cartesian products of paths and cycles.

Confirmed Gray code embedding as the optimal strategy for these embeddings.

Solved an open problem in the embedding of hypercubes into torus-like graphs.

Abstract

Though embedding problems have been considered for several regular graphs, it is still an open problem for hypercube into torus. In the paper, we prove the conjecture mathematically and obtain the minimum wirelength of embedding for hypercube into Cartesian product of paths and/or cycles. In addition, we explain that Gray code embedding is an optimal strategy in such embedding problems.

Equations

$$WL(G,H)=\underset{f}{\min }WL(G,H;f).$$

$$WL(Q_n,C_{2^{n_1}}\times C_{2^{n_2}})=2^{n_2}(3\cdot 2^{2n_1-3}-2^{n_1-1})+2^{n_1}(3\cdot 2^{2n_2-3}-2^{n_2-1}).$$

$$WL(Q_n,\mathcal{G})=\sum _{i=1}^k{\mathcal{L}}_i,$$

$$\mathscr{L}_{i}=\left\{\begin{array}[]{rcl}&2^{n-n_{i}}(3\cdot 2^{2n_{i}-3}-2^{n_{i}-1}),&\mbox{if}\ \ \mathscr{G}_{i}=C_{2^{n_{i}}},\\ &2^{n-n_{i}}(2^{2n_{i}-1}-2^{n_{i}-1}),&\mbox{if}\ \ \mathscr{G}_{i}=P_{2^{n_{i}}}.\end{array}\right.$$

$$\Theta (S)=\{\{u,v\}\in E(G):u\in S,v\notin S\}.$$

$$\theta (n,S_1\times S_2)=\theta (n,S_2\times S_1).$$

$$WL(Q_n,H;f)=\sum _{i=1}^m\theta (n,f^{-1}(l_i)).$$

$${\xi }_{n_1\dots n_k}(v)={\xi }_{n_1\dots n_k}(v_1\dots v_k)=({\xi }_{n_1}(v_1),\dots ,{\xi }_{n_k}(v_k)),$$

$${\xi }_{n_1{n}_2}^{-1}(F_i^{n_1}\times N_{2^{n_2}})={\xi }_n^{-1}(A_i),{\xi }_{n_1{n}_2}^{-1}(N_{2^{n_1}}\times N_j)={\xi }_n^{-1}(B_j).$$

$$\sum _{i=1}^{2^{n_1-1}}\theta (n,{\xi }_{n_1{n}_2}^{-1}(F_i^{n_1}\times N_{2^{n_2}}))=2^{n-n_1}(3\cdot 2^{2n_1-3}-2^{n_1-1}).$$

$$\sum _{j=1}^{2^{n_2}-1}\theta (n,{\xi }_{n_1{n}_2}^{-1}(N_{2^{n_1}}\times N_j))=2^{n-n_2}(2^{2n_2-1}-2^{n_2-1}).$$

$$\sum _{i=1}^{2^{n_1-1}}\theta (n,f^{-1}(F_i^{n_1}\times N_{2^{n_2}}))\ge \sum _{i=1}^{2^{n_1-1}}\theta (n,{\xi }_{n_1{n}_2}^{-1}(F_i^{n_1}\times N_{2^{n_2}})).$$

$$\sum _{j=1}^{2^{n_2}-1}\theta (n,f^{-1}(N_{2^{n_1}}\times N_j))\ge \sum _{j=1}^{2^{n_2}-1}\theta (n,{\xi }_{n_1{n}_2}^{-1}(N_{2^{n_1}}\times N_j)).$$

$$\sum _{i=1}^{2^{n_1-1}}\theta (n,f^{-1}(N_{2^{n_2}}\times F_i^{n_1}))\ge \sum _{i=1}^{2^{n_1-1}}\theta (n,{\xi }_{n_2{n}_1}^{-1}(N_{2^{n_2}}\times F_i^{n_1})).$$

$$\sum _{j=1}^{2^{n_2}-1}\theta (n,f^{-1}(N_j\times N_{2^{n_1}}))\ge \sum _{j=1}^{2^{n_2}-1}\theta (n,{\xi }_{n_2{n}_1}^{-1}(N_j\times N_{2^{n_1}})).$$

$$V(C_{2^{n_1}}\times C_{2^{n_2}})=\{x=(x_1,x_2):1\le x_i\le 2^{n_i},i=1,2\}=N_{2^{n_1}}\times N_{2^{n_2}}.$$

$$\begin{array}[]{rcl}\mathscr{E}_{1}&=&\{\{(x_{1},x_{2}),(x_{1}^{\prime},x_{2})\}:\{x_{1},x_{1}^{\prime}\}\in E(C_{2^{n_{1}}}),x_{2}\in N_{2^{n_{2}}}\},\\ \mathscr{E}_{2}&=&\{\{(x_{1},x_{2}),(x_{1},x_{2}^{\prime})\}:x_{1}\in N_{2^{n_{1}}},\{x_{2},x_{2}^{\prime}\}\in E(C_{2^{n_{2}}})\}.\end{array}$$

$${\mathcal{P}}_{ij}=\bigcup _{\{x_i,x_i^{\prime }\}\in {\mathcal{X}}_{ij}}\{\{x,x^{\prime }\}\in {\mathcal{E}}_i\},$$

$$WL(Q_n,C_{2^{n_1}}\times C_{2^{n_2}};f)=\sum _{j=1}^{2^{n_1-1}}\theta (n,f^{-1}(F_j^{n_1}\times N_{2^{n_2}}))+\sum _{j=1}^{2^{n_2-1}}\theta (n,f^{-1}(N_{2^{n_1}}\times F_j^{n_2})).$$

$$\sum _{j=1}^{2^{n_2-1}}\theta (n,{\xi }_{n_1{n}_2}^{-1}(N_{2^{n_1}}\times F_j^{n_2}))=2^{n-n_2}(3\cdot 2^{2n_2-3}-2^{n_2-1}).$$

$$\sum _{j=1}^{2^{n_2-1}}\theta (n,f^{-1}(N_{2^{n_1}}\times F_j^{n_2}))\ge \sum _{j=1}^{2^{n_2-1}}\theta (n,{\xi }_{n_1{n}_2}^{-1}(N_{2^{n_1}}\times F_j^{n_2})).$$

$$V(\mathcal{G})=\{x=(x_1,\dots ,x_k):x_i\in N_{2^{n_i}},1\le i\le k\}=N_{2^{n_1}}\times \cdots \times N_{2^{n_k}}$$

$$\mathscr{F}_{ij}=\left\{\begin{array}[]{cl}\mathscr{X}_{ij},&\mbox{if}\quad\mathscr{G}_{i}=C_{2^{n_{i}}},\\ \mathscr{Y}_{ij},&\mbox{if}\quad\mathscr{G}_{i}=P_{2^{n_{i}}}.\end{array}\right.$$

$${\mathcal{P}}_{ij}=\bigcup _{\{x_i,x_i^{\prime }\}\in {\mathcal{F}}_{ij}}\{\{x,x^{\prime }\}\in {\mathcal{E}}_i\},$$

$$\begin{array}[]{cl}&\mathscr{B}_{1j}=\mathscr{A}_{1j}\times N_{2^{n_{2}}}\times\cdots\times N_{2^{n_{k}}},\ \ \mathscr{B}_{kj}=N_{2^{n_{1}}}\times\cdots\times N_{2^{n_{k-1}}}\times\mathscr{A}_{kj},\\ &\mathscr{B}_{ij}=N_{2^{n_{1}}}\times\cdots\times\mathscr{A}_{ij}\times\cdots\times N_{2^{n_{k}}},\ \ 1<i<k.\end{array}$$

$$WL(Q_n,\mathcal{G};f)=\sum _{i=1}^k\sum _{j=1}^{q_i}\theta (n,f^{-1}({\mathcal{B}}_{ij})).$$

$$\sum _{j=1}^{q_i}\theta (n,{\xi }_{n_1\dots n_k}^{-1}({\mathcal{B}}_{ij}))=2^{n-n_i}(3\cdot 2^{2n_i-3}-2^{n_i-1}).$$

$$\begin{array}[]{rcl}\xi_{n_{1}\ldots n_{k}}^{-1}(\mathscr{B}_{ij})&=&\xi_{n_{1}\ldots n_{k}}^{-1}(N_{2^{n_{1}}}\times\cdots\times F_{j}^{n_{i}}\times\ldots\times N_{2^{n_{k}}})\\ &=&V(Q_{n_{1}})\times\ldots\times\xi_{n_{i}}^{-1}(F_{j}^{n_{i}})\times\ldots\times V(Q_{n_{k}})\\ &=&V(Q_{n_{1}+\ldots+n_{i-1}})\times\xi_{n_{i}}^{-1}(F_{j}^{n_{i}})\times V(Q_{n_{i+1}+\ldots+n_{k}}).\end{array}$$

$$\begin{array}[]{rcl}&&\theta(n,V(Q_{n_{1}+\ldots+n_{i-1}})\times\xi_{n_{i}}^{-1}(F_{j}^{n_{i}})\times V(Q_{n_{i+1}+\ldots+n_{k}}))\\ &=&\theta(n,\xi_{n_{i}}^{-1}(F_{j}^{n_{i}})\times V(Q_{n_{i+1}+\ldots+n_{k}})\times V(Q_{n_{1}+\ldots+n_{i-1}}))\\ &=&\theta(n,\xi_{n_{i}}^{-1}(F_{j}^{n_{i}})\times V(Q_{n-n_{i}})).\end{array}$$

$$\sum _{j=1}^{q_i}\theta (n,{\xi }_{n_1\dots n_k}^{-1}({\mathcal{B}}_{ij}))=2^{n-n_i}(2^{2n_i-1}-2^{n_i-1}).$$

$$WL(Q_n,\mathcal{G};{\xi }_{n_1\dots n_k})=\sum _{i=1}^k{\mathcal{L}}_i,$$

$$\sum _{i=1}^k\sum _{j=1}^{q_i}\theta (n,f^{-1}({\mathcal{B}}_{ij}))\ge \sum _{i=1}^k\sum _{j=1}^{q_i}\theta (n,{\xi }_{n_1\cdots n_k}^{-1}({\mathcal{B}}_{ij})).$$

Full text

Key words:  embedding wirelength; hypercube; torus; Cartesian product; Gray code embedding.

1. Introduction

Task mapping in modern high performance parallel computers can be modeled as a graph embedding problem. Let $G(V,E)$ be a simple and connected graph with vertex set $V(G)$ and edge set $E(G)$. Graph embedding[1, 2, 3] is an ordered pair $<f,P_{f}>$ of injective mapping between the guest graph $G$ and the host graph $H$ such that

• (i)

$f:V(G)\rightarrow V(H)$, and

• (ii)

$P_{f}:E(G)\rightarrow$ $\{P_{f}(u,v):$ $P_{f}(u,v)$ is a path in $H$ between $f(u)$ and $f(v)$ for $\{u,v\}\in E(G)\}$.

It is known that the topology mapping problem is NP-complete[5]. Since Harper[6] in 1964 and Bernstein[7] in 1967, a series of embedding problems have been studied[8, 9, 10, 11, 12]. The quality of an embedding can be measured by certain cost criteria. One of these criteria is the wirelength. Let $WL(G,H;f)$ denote the wirelength of $G$ into $H$ under the embedding $f$. Taking over all embeddings $f$, the minimum wirelength of $G$ into $H$ is defined as

[TABLE]

Hypercube is one of the most popular, versatile and efficient topological structures of interconnection networks[13]. More and more studies related to hypbercubes have been performed[14, 15, 4, 16]. Manuel[4] et al. computated the minimum wirelength of embedding hypercube into a simple cylinder. In that paper, the wirelenth for hypercube into general cylinder and torus were given as conjectures. Though Rajan et al.[17] and Arockiaraj et al.[2] studied those embedding problems, the two conjectures are still open. We recently gave rigorous proofs of hypercubes into cycles[18] and cylinders (the first conjecture)[19] successively. Using those techniques and process, we try to settle the last conjecture for torus. In the paper, we also generaliz the results to other Cartesian product of paths and/or cycles.

It is seen that the grid, cylinder and torus are Cartesian product of graphs. In the past, the vertices of those graphs are labeled by a series of nature numbers[15, 4, 2, 19]. But it is not convenient for some higher dimensional graphs. To describe a certain embedding efficiently, we apply tuples to lable the vertices in the paper. By the tool of Edge Isoperimetric Problem(EIP)[20], we estimate and explain the minimal wirelength for hypercube into torus and other Cartesian product of graphs.

Notation. For $n\geq 1$, we define $Q_{n}$ to be the hypercube with vertex-set $\{0,1\}^{n}$, where two $0-1$ vectors are adjacent if they differ in exactly one coordinate [21].

Notation. An $r_{1}\times r_{2}$ grid with $r_{1}$ rows and $r_{2}$ colums is represented by $P_{r_{1}}\times P_{r_{2}}$ where the rows are labeled $1,2,\ldots,r_{1}$ and the columns are labeled $1,2,\ldots,r_{2}$ [15]. The torus $C_{r_{1}}\times C_{r_{2}}$ is a $P_{r_{1}}\times P_{r_{2}}$ with a wraparound edge in each column and a wrapround edge in each row.

Main Results

Theorem 1.1**.**

For any $n_{1},\ n_{2}\geq 2,\ n_{1}+n_{2}=n$. The minimum wirelength of hypercubes into torus is

[TABLE]

Moreover, Gray code embedding is an optimal embedding.

Notation. Cartesian product of paths and/or cycles is denoted by $\mathscr{G}=\mathscr{G}_{1}\times\mathscr{G}_{2}\times\cdots\times\mathscr{G}_{k},$ where $\mathscr{G}_{i}\in\{P_{2^{n_{i}}},C_{2^{n_{i}}}\},1\leq i\leq k$.

Theorem 1.2**.**

For any $k>0$, $n_{i}\geq 2$, and $\sum_{i=1}^{k}n_{i}=n$. The minimum wirelength of hypercubes into Cartesian product $\mathscr{G}$ is

[TABLE]

where

[TABLE]

Moreover, Gray code embedding is an optimal embedding.

The paper is organized as follows. In Section 2, some definitions and elementary properties are introduced. In Section 3, we explain the Gray code embedding is an optimal strategy for hypercube into torus. Section 4 is devoted to Cartesian products of paths and/or cycles.

2. Preliminaries

EIP has been used as a powful tool in the computation of minimum wirelength of graph embedding[20]. EIP is to determine a subset $S$ of cardinality $k$ of a graph $G$ such that the edge cut separating this subset from its complement has minimal size. Mathematically, Harper denotes

[TABLE]

For any $S\subset V(Q_{n})$, use $\Theta(n,S)$ in place of $\Theta(S)$ and let $|\Theta(n,S)|$ be $\theta(n,S)$.

Lemma 2.1**.**

Take a subcube $Q_{n_{1}}$ of $Q_{n}$, and $S_{1}\subset V(Q_{n_{1}}),S_{2}\subset V(Q_{n-n_{1}})$, then

[TABLE]

Proof.

By the definition of hypercube $Q_{n}$, there is an edge connected in $S_{1}\times S_{2}$ if and only if there is an edge connected in $S_{2}\times S_{1}$. ∎

The following lemma is efficient technique to find the exact wirelength.

Lemma 2.2**.**

[19]** Let $f$ be an embedding of $Q_{n}$ into $H$. Let $(L_{i})_{i=1}^{m}$ be a partition of $E(H)$. For each $1\leq i\leq m$, $(L_{i})_{i=1}^{m}$ satisfies:

• (A1)

$L_{i}$* is an edge cut of $H$ such that $L_{i}$ disconnects $H$ into two components and one of induduced vertex sets is denoted by $l_{i}$;*

• (A2)

$|P_{f}(u,v)\cap L_{i}|$* is one if $\{u,v\}\in\Theta(n,f^{-1}(l_{i}))$ and zero otherwise for any $\{u,v\}\in E(Q_{n})$.*

Then

[TABLE]

Notation. $N_{n}=\{1,2,\cdots,n\}$, and $F_{i}^{n}=\{i,i+1,\cdots,i+2^{n-1}-1\},\quad 1\leq i\leq 2^{n-1}$.

Notation. Let $(i,j)$ denote a vertice in row $i$ and column $j$ of cylinder $C_{2^{n_{1}}}\times P_{2^{n_{2}}}$, where $1\leq i\leq 2^{n_{1}}$ and $1\leq j\leq 2^{n_{2}}$. Then $V(C_{2^{n_{1}}}\times P_{2^{n_{2}}})=N_{2^{n_{1}}}\times N_{2^{n_{2}}}.$ It is seen that the vertex sets $F_{i}^{n_{1}}\times N_{2^{n_{2}}}=\{(x_{1},x_{2}):x_{1}\in F_{i}^{n_{1}},\ x_{2}\in N_{2^{n_{2}}}\}$ and $N_{2^{n_{1}}}\times N_{j}=\{(x_{1},x_{2}):x_{1}\in N_{2^{n_{1}}},\ x_{2}\in N_{j}\}$ are equivalent to $A_{i}$ and $B_{j}$ defined in [19] , respectively. See Fig.1 and Fig.2 for examples.

Now we generalize Gray code map $\xi_{n}:\{0,1\}^{n}\rightarrow\{1,2,\cdots,2^{n}\}$ defined in [18, 19]. Define $k$-order Gray code map $\xi_{n_{1}\ldots n_{k}}$ corresponding to $k$ components.

Definition 2.3**.**

$k$-order Gray code map $\xi_{n_{1}\ldots n_{k}}$ is given by $\xi_{n_{1}\ldots n_{k}}:\{0,1\}^{n}\rightarrow N_{2^{n_{1}}}\times\cdots\times N_{2^{n_{k}}}$, i.e.,

[TABLE]

where $n_{1}+\ldots+n_{k}=n$, and $v=v_{1}\ldots v_{k}\in\{0,1\}^{n},v_{i}\in\{0,1\}^{n_{i}},1\leq i\leq k$.

For example, $\xi_{32}(11011)=(\xi_{3}(110),\xi_{2}(11))=(5,3)$.

According to the rule of Gray code map, we have that

[TABLE]

Together with (12) and (13) in [19], we have that

[TABLE]

Let $f:\{0,1\}^{n}\rightarrow N_{2^{n_{1}}}\times N_{2^{n_{2}}}$ be an embedding of $Q_{n}$ into $C_{2^{n_{1}}}\times P_{2^{n_{2}}}$. Theorems 5.2 and 5.1 in [19] is rewritten as

[TABLE]

Cylinder $C_{2^{n_{1}}}\times P_{2^{n_{2}}}$ can also be observed as $P_{2^{n_{2}}}\times C_{2^{n_{1}}}$. Let $f:\{0,1\}^{n}\rightarrow N_{2^{n_{2}}}\times N_{2^{n_{1}}}$ be an embedding of $Q_{n}$ into $P_{2^{n_{2}}}\times C_{2^{n_{1}}}$, then (2) is rewritten as

[TABLE]

Remark 2.4**.**

It is seen that $\xi_{n_{1}n_{2}}^{-1}(F_{i}^{n_{1}}\times N_{2^{n_{2}}})=\xi_{n_{1}}^{-1}(F_{i}^{n_{1}})\times V(Q_{{n_{2}}})$. Then, by Lemma 2.1, we get that $\theta(n,\xi_{n_{1}n_{2}}^{-1}(F_{i}^{n_{1}}\times N_{2^{n_{2}}}))=\theta(n,\xi_{n_{2}n_{1}}^{-1}(N_{2^{n_{2}}}\times F_{i}^{n_{1}}))$.

3. hypbercubes into torus

In this section, we prove Theorem 1.1 in the following procedures.

$\bullet$ Labeling.  Let a binary tuple set denote the vertex set of torus $C_{2^{n_{1}}}\times C_{2^{n_{2}}}$ , that is

[TABLE]

The edge set $E(C_{2^{n_{1}}}\times C_{2^{n_{2}}})$ is composed of $\mathscr{E}_{1}$ and $\cup\mathscr{E}_{2}$, where

[TABLE]

$\bullet$ Partition.  Construct a partition of the edge set of torus.

Step 1. For each $i=1,2$, $j=1,\ldots,2^{n_{i}-1}$, let $\mathscr{X}_{ij}$ be an edge cut of the cycle $C_{2^{n_{i}}}$ such that $\mathscr{X}_{ij}$ disconnects $C_{2^{n_{i}}}$ into two components where the induced vertex set is $F_{j}^{n_{i}}$.

Step 2. For $i=1,2$, denote

[TABLE]

then $\{\mathscr{P}_{ij}:1\leq i\leq 2,1\leq j\leq 2^{n_{i}-1}\}$ is the partition of $E(C_{2^{n_{1}}}\times C_{2^{n_{2}}})$.

$\bullet$ Computation.  Notice that for each $i,j$, $\mathscr{P}_{ij}$ is an edge cut of the torus $C_{2^{n_{1}}}\times C_{2^{n_{2}}}$. $\mathscr{P}_{1j}$ disconnects the torus into two components where the induced vertex set is $F_{j}^{n_{1}}\times N_{2^{n_{2}}}$, and $\mathscr{P}_{2j}$ induces vertex set $N_{2^{n_{1}}}\times F_{j}^{n_{2}}$. See Fig.3 for an example.

Let $f:\{0,1\}^{n}\rightarrow N_{2^{n_{1}}}\times N_{2^{n_{2}}}$ be an embedding of $Q_{n}$ into $C_{2^{n_{1}}}\times C_{2^{n_{2}}}$. Under the partition $\{\mathscr{P}_{ij}:1\leq i\leq 2,1\leq j\leq 2^{n_{i}-1}\}$ and Lemma 2.2, the wirelength is written as a summation related to function $\theta$, i.e.,

[TABLE]

According to Lemma 2.1 and (1a), we have that

[TABLE]

According to Lemma 2.1 and (3a), we have that

[TABLE]

Combining above three fomulas and (1a),(2a), Theorem 1.1 holds.

4. hypercubes into Cartesian product of paths and/or cycles

In this section, we prove Theorem 1.2 in three parts. The first part follows the analogous process as Section 3. Then we obtain the wirelength under Gray code embedding. In the end, we conclude that Gray code embedding is an optimal strategy.

4.1. Compuation of embedding wirelength

$\bullet$ Labeling.  Let

[TABLE]

be the vertex set of Cartesian product $\mathscr{G}$ of $k$ paths and/or cycles. The edge set $E(\mathscr{G})$ of Cartesian product $\mathscr{G}$ is composed of all edges $\mathscr{E}_{i}$ correspongding to $k$ paths and/or cycles, denoted by $E(\mathscr{G})=\bigcup_{i=1}^{k}\mathscr{E}_{i}$.

$\bullet$ Partition.  Construct a partition of the edge set of Cartesian product $\mathscr{G}$.

Step 1. For each $i=1,\ldots,k$, $j=1,\ldots,2^{n_{i}-1}$, $\mathscr{X}_{ij}$ is described earlier in Section 3. For each $i=1,\ldots,k$, $j=1,\ldots,2^{n_{i}}-1$, let $\mathscr{Y}_{ij}$ be an edge cut of the path $P_{2^{n_{i}}}$ such that $\mathscr{Y}_{ij}$ disconnects $P_{2^{n_{i}}}$ into two components where the induced vertex set is $N_{j}$.

Notation. For $1\leq i\leq k$, let $q_{i}$ be $2^{n_{i}-1}$ if $\mathscr{G}_{i}=C_{2^{n_{i}}}$ and $2^{n_{i}}-1$ otherwise $\mathscr{G}_{i}=P_{2^{n_{i}}}$. For $j=1,\ldots,q_{i}$, denote

[TABLE]

Step 2. For $i=1,\ldots,k$, $j=1,\ldots,q_{i}$, denote

[TABLE]

then $\{\mathscr{P}_{ij}:1\leq i\leq k,1\leq j\leq q_{i}\}$ is a partition of $E(\mathscr{G})$.

$\bullet$ Computation.  Notice that for each $i,j$, $\mathscr{P}_{ij}$ is an edge cut of Cartesian product $\mathscr{G}$. Define a vertext set $\mathscr{A}_{ij}$ to be $F_{j}^{n_{i}}$ if $\mathscr{G}_{i}=C_{2^{n_{i}}}$ and $N_{j}$ otherwise $\mathscr{G}_{i}=P_{2^{n_{i}}}$.

Notation.

[TABLE]

It is seen that $\mathscr{P}_{ij}$ disconnects $\mathscr{G}$ into two components where the induced vertex set $\mathscr{B}_{ij}$. Let $f:\{0,1\}^{n}\rightarrow N_{2^{n_{1}}}\times\cdots\times N_{2^{n_{k}}}$ be an embedding of $Q_{n}$ into $\mathscr{G}$. Under the partition $\{\mathscr{P}_{ij}:1\leq i\leq k,1\leq j\leq q_{i}\}$ and Lemma 2.2, the wirelength is written as a summation related to function $\theta$, i.e.,

[TABLE]

4.2. The wirelength under Gray code embedding

We deal with the wirelength under Gray code embedding in two cases: one is that $\mathscr{G}_{i}$ is cycle $C_{2^{n_{i}}}$, and the other is that $\mathscr{G}_{i}$ is path $P_{2^{n_{i}}}$. In the following, set $1\leq i\leq k,1\leq j\leq q_{i}$.

Lemma 4.1**.**

If $\mathscr{G}_{i}$ is cycle $C_{2^{n_{i}}}$, then we have that

[TABLE]

Proof.

By the Notation (7), we have that

[TABLE]

Moreover, by Lemma 2.1, we have that

[TABLE]

Therefore, Lemma 4.1 follows from (1a). ∎

Similarly, we write the following lemma.

Lemma 4.2**.**

If $\mathscr{G}_{i}$ is path $P_{2^{n_{i}}}$, then we have that

[TABLE]

Combining (8), Lemma 4.1 and Lemma 4.2, we get the wirelength under Gray code embedding of hypercube into Cartesian product of paths and/or cycles. That is

[TABLE]

where $\mathscr{L}_{i}$ is defined in Theorem 1.2.

4.3. Minimum wirelength

We show that Gray code embedding wirelength is the lower bound of wirelength for hypercube into Cartesian product of paths and/or cycles. According to (8), it is sufficient to prove that

Lemma 4.3**.**

Let $f:\{0,1\}^{n}\rightarrow N_{2^{n_{1}}}\times\cdots\times N_{2^{n_{k}}}$ be an embedding of $Q_{n}$ into $\mathscr{G}$, then

[TABLE]

Proof.

To prove this lemma, we only consider that $i=1$, since a similar argument works for the other $2\leq i\leq k$.

$\bullet$ Case 1.  $\mathscr{G}_{1}=C_{2^{n_{1}}}$.

For $1\leq j\leq q_{1}=2^{n_{1}-1}$, $f^{-1}(\mathscr{B}_{1j})=f^{-1}(F_{j}^{n_{1}}\times N_{2^{n_{2}}}\times\cdots\times N_{2^{n_{k}}})$. Define a bijective map $f_{1}$ from $N_{2^{n_{1}}}\times N_{2^{n_{2}}}\times\cdots\times N_{2^{n_{k}}}$ to $N_{2^{n_{1}}}\times N_{2^{n-n_{1}}}$, where

[TABLE]

It is clear that $f_{1}(F_{j}^{n_{1}}\times N_{2^{n_{2}}}\times\cdots\times N_{2^{n_{k}}})=F_{j}^{n_{1}}\times N_{2^{n-n_{1}}}$. Moreover, we have that

[TABLE]

Notice that $f_{1}\circ f$ is an arbitrary map from $\{0,1\}^{n}$ to $N_{2^{n_{1}}}\times N_{2^{n-n_{1}}}$, then, by (2a), we have that $\sum_{i=1}^{2^{n_{1}}-1}\theta(n,f^{-1}(\mathscr{B}_{1j}))\geq\sum_{i=1}^{2^{n_{1}}-1}\theta(n,\xi_{n_{1}}^{-1}(F_{j}^{n_{1}})\times V(Q_{n-n_{1}}))$. Therefore, we conclude that

[TABLE]

$\bullet$ Case 2.  $\mathscr{G}_{1}=P_{2^{n_{1}}}$. By a similar analysis, we also get (9).

Combining Case 1 and Case 2, the case for $i=1$ is proved. Thus the lemma holds. ∎

Proof of Theorem 1.2. Theorem 1.2 follows from Subsection 4.1 to 4.3.

Acknowledgements The author is grateful to Prof. Qinghui Liu for his thorough review and suggestions. This work is supported by the National Natural Science Foundation of China, No.11871098.