Embedding Complete Binary Trees into Locally Twisted Cubes

The locally twisted cube is a newly introduced interconnection network for parallel computing, which possesses many desirable properties. In this paper, the problem of embedding complete binary trees into locally twisted cubes is studied.Let LTQn(V;E) denote the n-dimensional locally twisted cube.We find the following result in this paper: for any integern ≥ 2,we show that a complete binary tree with 2n—1 nodes can be embedded into the LTQn with dilation 2.


Introduction
An interconnection network can be represented by a graph G = (V, E), where V represents the node set and E represents the edge set. One of the important properties of interconnection networks is graph embedding ability. Given a host graph G 2 = (V 2 , E 2 ), which represents the network into which other networks are to be embedded, and a guest graph G 1 = (V 1 , E 1 ), which represents the network to be embedded, the problem is to find a mapping from each node of G 1 to a node of G 2 , and a mapping from each edge of G 1 to a path in G 2 . Two common measures of effectiveness of an embedding are the dilation and expansion. The dilation of embedding ψ is defined as dil(G 1 , G 2 , ψ) = max{dist(G 2 , ψ(u), ψ(v))|(u, v) ∈ E 1 }, where dist(G 2 , ψ(u), ψ(v)) denotes the distance between the two nodes ψ(u) and ψ(v) in G 2 . The smaller the dilation of an embedding is, the shorter the communication delay that the graph G 2 simulates the graph G 1 [1]. The expansion of embedding is defined as exp(G 1 , G 2 , ψ) = |V (G 2 )|/|V (G 1 )|, which measures the processor utilization. The smaller the expansion of an embedding is, the more efficient the processor utilization that the graph G 2 simulates the graph G 1 . Graph embedding has good applications in transplanting parallel algorithms developed for one network to a different one, and allocating concurrent processes to processors in the network. Path, cycle and mesh are three fundamental networks for parallel computing, and much work about path, cycle and mesh embedding [3], [4], [6] appeared in the literature.
Trees are another common interconnection structures used in parallel computing. It is important to study the problem of how to embed different kinds of trees into a host graph. Recently, many tree embedding problems have been studied [9], [7], [10].
The locally twisted cube LT Q n is a variant of hypercube, proposed by Yang et al. [11]. It has many attractive features superior to those of the hypercube, such as the diameter is only about half of that of Q n . In particular, Yang et al. [12] showed that LT Q n is Hamiltonian connected and contains a cycle of every length from 4 to 2 n for n ≥ 3. Furthermore, LT Q n was proved to be (n − 2)-pancyclic [2], for any integer n ≥ 3. And some other properties of locally twisted cubes were discussed [8], [7], [5].
In this paper, the problem of embedding complete binary trees into locally twisted cubes is studied. We find for any integer n ≥ 2, a complete binary tree with 2 n − 1 nodes can be embedded into the LT Q n with dilation 2.

Preliminaries
A binary string x of length n is denoted by x 1 x 2 ...x n−1 x n , where x 1 is the most significant bit and x n is the least significant bit. Similar to Q n , LT Q n is an n-regular graph of 2 n nodes. Every node of LT Q n is identified by a unique binary string of length n. LT Q n can be recursively defined as follows. Definition 1 [11]. For n ≥ 2, an n-dimensional locally twisted cube, LT Q n , is defined recursively as follows: (1) LT Q 2 is a graph consisting of four nodes labeled with 00, 01, 10, and 11, respectively, connected by four edges (00, 01), (00, 10), (01, 11), and (10, 11).
(2) For n ≥ 3, LT Q n is built from two disjoint copies of LT Q n−1 with the following steps. Let LT Q 0 n−1 denote the graph obtained by prefixing the label of each node of one copy of LT Q n−1 with 0, and LT Q 1 n−1 denote the graph obtained by prefixing the label of each node of the other copy of n−1 with an edge, where ′ + ′ represents the modulo 2 addition. We use CBT n to denote the complete binary tree with 2 n − 1 nodes which can be embedded into LT Q n . For any integer i ∈ {0, 1}, CBT i n denotes to prefix the node labels of CBT n with i. |P | is the length of path P .

Embedding complete binary trees into locally twisted cubes
Lemma 1. A complete binary tree with 3 nodes can be embedded into LT Q 2 with dilation 1 rooted at any node of LT Q 2 .
Proof. Obviously, we can embedded a complete binary tree rooted at any node of LT Q 2 into LT Q 2 with dilation 1, the lemma holds.
2 Considering the symmetric properties of LT Q 3 , we can intuitively find the symmetric properties of LT Q 3 as shown in the following lemma.
A complete binary tree with 7 nodes can be embedded into LT Q 3 with dilation 1 rooted at any node of LT Q 3 .
Proof. By Lemma 2, we only need to consider the following two nodes: 000 and 100. Figure 1 demonstrates two complete binary trees which can be embedded into LT Q 3 rooted at 000 and 100, respectively. It is easy to verify that a complete binary tree can be embedded into LT Q 3 rooted at any node of LT Q 3 with dilation 1, see Figure 1, every node of CBT 3 is mapping to a node in LT Q 3 and every edge of CBT 3 is mapping to an edge in LT Q 3 , the lemma holds. 2

Lemma 4.
A complete binary tree with 15 nodes can be embedded into LT Q 4 with dilation 1 rooted at 1000, while another complete binary tree with 15 nodes can be embedded into LT Q 4 with dilation 2 rooted at 1010.
Proof. We can embed a complete binary tree rooted at 1000 into LT Q 4 with dilation 1, see Figure  2 (a), every node of CBT 4 is mapping to a node in LT Q 4 and every edge of CBT 4 is mapping to an edge in LT Q 4 . While another complete binary tree rooted at 1010 can be embedded into LT Q 4 with dilation 2, see Figure 2 (b). In the second embedding, edge (1010,0000) in the complete binary tree is mapping to a path P : 1010− > 0010− > 0000 of LT Q 4 . Every node of CBT 4 is mapping to a node in LT Q 4 and every edge of CBT 4 except edge (1010,0000) is mapping to an edge in LT Q 4 . By the definition of dilation, the dilation of the second embedding is 2.

Lemma 5.
A complete binary tree with 2 5 − 1 nodes can be embedded into LT Q 5 with dilation 2, whose root is 00010.
Proof. By the proof of Lemma 4, we have two complete binary trees which can be embedded into LT Q 4 . we can embedded CBT 5 into LT Q 5 with dilation 2, the proof process is as follows. We prefix the label of each node of CBT 4 in the Figure 2 (a) with 0, and prefix the label of each node of CBT 4 in the Figure 2 (b) with 1, and then, we link the roots of this two complete binary trees with two edges (00010,01000) and (00010,11010). It upper steps can construct CBT 5 . Edge (00010,01000) of CBT 5 is mapping to the path P 1 : 00010− > 01010− > 01000 of LT Q 5 and edge (00010,11010) of CBT 5 is mapping to the path P 2 : 00010− > 10010− > 11010 of LT Q 5 , see Figure 3. By the definition of dilation, the dilation of the embedding is 2.
Every node of CBT 5 is mapping to a node in LT Q 5 and every edge of CBT 5 is mapping to an edge in LT Q 5 , except edges (1010,0000),(00010,01000) and (00010,11010), while these three edges are mapping to a path of length 2 in LT Q 5 , respectively. 2

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Information Technology for Manufacturing Systems III Lemma 6. A complete binary trees with 2 6 − 1 nodes can be embedded into LT Q 6 with dilation 2, whose root is 010010.
Proof. By the proof of Lemma 5, we have two complete binary trees which can be embedded into LT Q 5 . we can embedded CBT 6 into LT Q 6 with dilation 2, the construction process is as follows. To show the embedded tree clearly, Figure 4 omits some edges in LT Q 6 . We prefix the label of each node of CBT 5 in the Figure 3 with 0, and prefix the label of each node of CBT 5 in the Figure 3 with 1, and then, we link the roots of this two complete binary trees with two edges (000010,010010) and (100010,010010). The upper steps can construct CBT 6 . Edge (000010,010010) of CBT 6 is mapping to the edge (000010,010010) of LT Q 6 and edge (010010,100010) of CBT 6 is mapping to the path P : 010010− > 110010− > 100010 of LT Q 6 , see Figure 5. By the definition of dilation, the dilation of the embedding is 2.

Lemma 7.
For any integer n ≥ 6, a complete binary tree with 2 n − 1 nodes can be embedded into LT Q n with dilation 2, whose root is 01 n−5 0010.
Proof. We prove this lemma by induction on the dimension n of LT Q n . According to Lemmas 6, this lemma holds when n=6. Supposing that the lemma holds for n = τ (τ ≥ 6), we will prove that the lemma holds for n = τ + 1.
According to the induction of hypothesis, for any integer τ ≥ 6, for LT Q τ , a complete binary tree with 2 τ − 1 nodes can be embedded into LT Q τ with dilation 2, whose root is 01 τ −5 0010.

Theorem 1.
A complete binary tree CBT n with 2 n − 1 nodes can be embedded into LT Q n with dilation 2.
Proof. By Lemma 1 and Lemmas 3 -7, the theorem holds obviously. 2 Since the node number of LT Q n is 2 n , and CBT n has 2 n −1 nodes, the expansion of this embedding is 2 n −1 2 n . When n is big enough, the expansion is almost 1.

Conclusions
In this paper, the problem of embedding complete binary trees into locally twisted cubes is studied. We find the following result in this paper: for any integer n ≥ 2, a complete binary tree with 2 n − 1 nodes can be embedded into the LT Q n with dilation 2.