Graph Embedding Aspect of IEH Graphs
HUNG-YI CHANG AND RONG-JAYE CHEN Department of Computer Science and Information Engineering
National Chiao Tung University Hsinchu, Taiwan 300, R.O.C. E-mail: [email protected]
In order to overcome the drawback of the hypercube that the number of nodes is limited to a power of two, the incrementally extensible hypercube (IEH) graph is derived for an arbitrary number of nodes [12]. In this paper, we first prove that the incomplete hypercube (IH) is a spanning subgraph of IEH. Next, we present a new method to con-struct an IEH from an IH. From the aspect of graph embedding, we determine the minimum size of the IEH that contains a complete binary tree. We then embed a torus (with a side length as power of two) into an IEH with dilation 1 and expansion 1.
Keywords: hypercubes, embedding, binary trees, meshes, incrementally extensible hy-percubes, interconnection networks
1. INTRODUCTION
Hypercube graphs are one class among the most popular topologies for
implement-ing massively parallel machines. It has many advantages: regularity, symmetry, low diameter, optimally fault tolerance, and so on [10]. However, the hypercube has one major drawback that it is not incrementally extensible. The number of nodes for hyper-cubes must be a power of two, which considerably limits the choice of the number of nodes in the graphs. To overcome this drawback, a few studies have so far tried to im-prove this situation but have caused new problems described briefly in the following. Bhuyan and Agrawal [2] proposed generalized hypercubes, which have two drawbacks: (1) the networks reduce to complete graphs when their numbers of nodes are prime, and (2) they change significantly when a new node is added. Katseff [5] proposed
incom-plete hypercubes (IHs), which suffer from the problem of fault tolerance: failure of a
sin-gle node will cause the entire network to become disconnected. Sen [11] proposed
Su-percubes, which become more irregular as the size of the networks grows; for a
super-cube with N nodes, 2n < N < 2n+1, the difference between the maximum and the minimum degrees of nodes can be n − 2. Recently, Sur and Srimani [12] have proposed a new
generalization class of hypercube graphs: incrementally extensible hypercubes (IEHs). This topology can be defined for an arbitrary number of nodes and still reserves several advantages, such as optimal fault tolerance, low diameter, a simple routing algorithm, and near regularity.
Received March 25, 1998; revised May 11 & July 6, 1998; accepted July 27, 1998. Communicated by Wen-Lian Hsu.
Graph embedding has been used to model the problem of simulating a parallel
algo-rithm in a parallel machine. It is a mapping M of a guest graph G onto a host graph H. The cost of an embedding is measured in terms of dilation, congestion, and expansion [1, 3, 4, 6-10, 13-15]. The dilation of an embedding is the maximum distance of all edges of G in H. The congestion of an embedding is the maximum number of edges of G that share an edge of H. The expansion of an embedding is the ratio of the size of H to the size of G. Intuitively, dilation measures communication performance, congestion measures queuing delay, and expansion measures processor utilization. If G can be embedded into H with dilation 1 and expansion 1, then we say the embedding is optimal [15].
However, embedding of trees and tori into IEH graphs has never been studied. In this paper, we focus on IEH graphs and obtain the following results. First, we prove that IH(N) is a spanning subgraph of IEH(N), where N is the number of nodes. Next, we present a new method to construct an IEH from an IH. From the view point of graph embedding, we determine that the minimum size of IEH is 2h+1 + 1, which contains a complete binary tree of height h as a subgraph. We then embed a torus (with a side length of 2n) into an IEH graph with dilation 1 and expansion 1.
The rest of this paper is organized as follows. In Section 2, we introduce basic terminology for hypercubes, IHs, and IEHs. In Section 3, we show the relation between IHs and IEHs. In Sections 4 and 5, we embed binary trees and tori into IEH graphs. Finally, in Section 6, we present some conclusions.
2. PRELIMINARIES
In the research on interconnection networks, systems are modeled as graphs. In these graphs, nodes represent processors, and edges represent communication channels. A hypercube Hn is a graph G(V, E), where V is the set of 2n nodes, which are labeled as binary numbers of length n; E is the set of edges that connect two nodes if and only if they differ in exact one bit of their labels. An IH is a graph with N nodes that are la-beled as binary numbers of length log2N
.. Each edge joins two nodes that differ in exact one bit of their labels. An IEH graph, a generalized hypercube graph, is composed of several hypercubes of different sizes. These hypercubes are connected with
In-ter-Cube (IC) edges. Let IEH(N) be an IEH graph of N nodes. This graph is constructed
by the following algorithm [12].
Algorithm 1.
Input : a positive integer N Output : IEH(N)
1. Express N as a binary number (cn, …, c1, c2)2, where cn = 1. For each ci, with ci
≠ 0, construct a hypercube Hi. The edges constructed in this step are called
regular edges.
2. For all His’, label each node with a dedicated binary number 11…10bi-1…b0, where the length of leading 1s is n − i, and bi-1…b0 is the label of this node in the regular hypercube of dimension i.
i = i + 1.
While i ≤ n if ci≠ 0 then
Connect the node 11…1bjbj-1…b0 in Gj to the following i − j
nodes in Hi : - -1 1 ... 11 0 1 ... 11 − − −i i j n bjbj-1…b0, - 1 1 ... 01 0 1 ... 11 − − −i i j n bjbj-1…b0, …,
-
1
0
...
11
0
1
...
11
− − −i i j n bjbj-1…b0.Set j = i and Gi be the composed graph obtained in this step. /* Gi is
the graph which is composed of Hks’ for k ≤ i.*/ endif
i = i + 1.
endwhile
Thus, obtain Gn as the output.#
In Algorithm 1, we observe two useful properties. First, Gi is the IEH( ∑
= i k 0
ck2k)
graph. Second, any two nodes that are joined by IC edges differ in one or two bits of their labels. To illustrate, Fig. 1 shows the IEH(11) graph. Note that solid lines represent regular edges, and that dotted lines represent IC edges.
0100 0000 0101 0001 0111 0110 0011 0010 1100 1101 1110 H3 H1 H0 G1
Fig. 1. IEH(11) graph.
For convenience of discussion, we divide IC edges into two classes: 1-IC edges and 2-IC edges. A 1-IC edge connects nodes that differ in exactly one bit of their labels; and a 2-IC edge connects nodes that differ in exact two bits. Let (u, v) be an IC edge, u be in
H
i, and v be in Hj for i ≠ j. We call (u, v) a forward IC edge of u if i < j; otherwise, it is
called a backward one. Fig. 1 shows that (1100, 1110) is a forward 1-IC edge of node 1110 and (0000,1100) is a backward 2-IC edge of node 0000. Note that node u, which has forward 2-IC edges joining some nodes in Hk for k > i, has exactly one forward 1-IC
3. RELATION BETWEEN IH AND IEH
In [7], an IH was decomposed into several hypercubes of different sizes. Any pair of distinct subcubes Hk and Hj, where k > j, are only connected through links along
dimen-sion k. Applying this idea, we have the following algorithm, similar to Algorithm 1, to construct an IH.
Algorithm 2.
Input : a positive integer N Output : IH(N)
1. Express N as a binary number (cn, …, c1, c2)2, where cn = 1. This vector is called
cube vector. For each ci≠ 0, construct a hypercube Hi.
2. For all His’, label each node with a dedicated binary number
cn…ci+10bi-1…b0, where bi-1…b0 is the label of this node in the regular hyper-cube of dimension i.
3. Find minimum i where ci = 1, set Gj = Hi, and set j = i.
i = i + 1. While i ≤ n if ci≠ 0 then
Connect the node cn…cj+1bjbj-1…b0 in Gj to the node in Hi : 1 1 1 10 ... ... − − + − − + j i j i i n i n c c c c bjbj-1…b0.
Set j = i and Gi be the composed graph obtained in this step. /* Gi is
the graph which is composed of Hks’ for k ≤ i.*/ endif
i = i + 1.
endwhile
Thus, obtain Gn as the output.#
Observe Algorithm 1 and 2. We find that they both use hypercubes of the same size as subcubes. Further, let lab(x) denote node x’s label, and let (u, v) be an arbitrary edge connecting subcubes in IH(N). By relabeling IEH(N) with Step 1 and 2 of Algo-rithm 2, we can find a 1-IC edge (u’, v’) in IEH(N) such that lab(u) = lab(u’) and lab(v) = lab(v’). Thus, we have the following corollary.
Corollary 1. IEH(N) contains IH(N) as a subgraph. Proof: This corollary is proved by the above argument.#
Since IHs are subgraphs of IEHs, many good results for IHs are immediately avail-able in IEHs. For example, there is a deadlock-free routing algorithm for IHs [5]; thus, this result can be used to implement a wormhole routing algorithm for IEHs. Moreover, many parallel algorithms for IHs [3, 6, 9, 13, 15] will adapt to IEHs with slight modifica-tion.
In another topological view, we can construct IEH(N) from IH(2n-1), where 2n-1≤ N
node v in Gj connects nodes in Hi that are different in two bits from v; one is the ith bit,
and the other is the kth bit, where j < k < i. Thus, IEH(N) graphs can be obtained as fol-lows. First, construct IH(2n-1). Second, let N = (cn, …, c1, c2)2, where cn = 1. Consider
each node u in Hl, where cl = 0, and its backward IC edge from Hk’ for k’ < l and ck’ = 1.
Connect u’s backward IC-edge to its forward IC-edge with respect to Hk, where k is the
minimum integer for ck = 1 and n ≥ k > l. Third, delete u but keep the edges constructed
in the second step left. For example, Fig. 2 shows how to construct IEH(9) from IH(15). In this figure, gray cycles represent exist nodes, and dashed lines represent IC edges in IEH(9). Note that two forward 2-IC edges, (1110, 0100) and (1110, 0010), are composed of paths as 1110-1100-0100 and 1110-1010-0010 in IH(15), respectively.
0100 0000 0101 0001 0111 0110 0011 0010 1100 1101 1110 1000 1010 1001 1011
Fig. 2. Construct IEH(9) from IH(15).
4. EMBEDDING COMPLETE BINARY TREES INTO IEHS
In this section, we will show how to optimally embed complete binary trees in IEHs. We will now give some necessary definitions and explain our work.
Definition 1. [8] A double-rooted binary tree DRBTd, where d is the height of the tree, is
a complete binary tree with the root replaced by a path of length two.#
Definition 2. A twin binary tree TBTd, where d is the height of the tree, is a complete
binary tree with the root removed and the two level-one nodes are joined.#
To illustrate, Fig. 3 (a) shows DRBT3, and Fig. 2 (b) shows TBT2. We still need the following two lemmas for ease of reference.
Lemma 1. [8] A double-rooted tree of height h can be embedded into a (h+1) -dimensional hypercube with edge adjacency reserved.#
(a) (b)
Fig. 3. DRBT3 and TBT2.
It seems that we can easily embed a TBT2 from a DRBT3 into H4 by removing the edge of roots and joining the two nodes in the second level. However, by this method, it is impossible to embed TBT1 from DRBT2 in H3 with edge adjacency since every node's degree is three. Thus, the following lemma is necessary.
Lemma 2. A twin binary tree of height h can be embedded into a (h+2)-dimensional hy-percube with edge adjacency reserved.
Proof. It is trivial that TBT1 can be embedded in H3 as Fig. 4 (a) shows. Consider the embedding of TBT2 in H4. H4 is divided into two H3: one contains TBT1 and the other contains DRBT2 as Fig. 4 (b) shows. Obviously, TBT2 can be embedded in H4. By way of induction, we assume TBTk can be embedded into Hk+2 for k > 2. Consider the case of
k+1. By Lemma 1 and the above hypothesis, we partition Hk+3 into two Hk+2: one contains
a DRBTk+1 as a subgraph and the other contains a TBTk as a subgraph as Fig. 5 shows.
By adding necessary edges (i.e., the dot lines) and deleting the redundant one (i.e., the dash line), this lemma is proved.#
Observe that a complete binary tree CBTd has 2d+1− 1 nodes. Under the condition of
expansion 1, we have the following theorem for embedding a complete binary tree into an IEH with the same size.
(a) (b) Fig. 4. Embed TBT1 and TBT2 into H3 and H4.
0000 1000 0100 0010 1100 1001 0101 1101 1010 1111 1110 1011 0001 000 100 001 010 101 110 0110
Theorem 1. A complete binary tree CBTd can be embedded into IEH(2d+1-1) with
dila-tion two, congesdila-tion one, and expansion one.
Proof. By Corollary 1, IEH(2d+1 − 1) is an IH(2d+1− 1) as well as Hd+1\(11…1).
Con-sider the base case for d is one or two. As Fig. 6 shows, CBT1 and CBT2 can be embed-ded into IEH(3) and IEH(7) with dilation one and two, respectively. By way of induction, we assume CBTk, where k > 2, can be embedded into IEH(2k+1-1) with dilation two.
Con-sider IEH(2k+2 − 1) is composed of Hk+1 and IEH(2k+1-1) by Algorithm 1. Further,
IEH(2k+1-1) is isomorphic to Hk+1\(011…10). Thus, by the hypothesis we can embedded
CBTk+1 into IEH(2k+2− 1) by Fglocating the root at (011…10). And the root has (11…10)
and (11…10) as its sons. (For illustration, Fig. 7 shows how to embed CBT3 into IEH(15).)# 00 10 01 110 010 100 001 011 000 101 (a) (b)
Fig. 6. Embed CBT1 and CBT2 into IEH(3) and IEH(7).
Under the condition of congestion 1, Tzeng et al. [13] presented an embedding of CBTn into IH(2n + 2n-1) with dilation 1 and expansion about 3/2. They also showed that
no embedding of CBTn into IH(2n+2i) with dilation 1 where i < n − 1. However, for IEHs, we show an optimal embedding of CBTn into the IEH(2n+1 + 1) with dilation 1 and
ex-pansion 1+2/(2n+1− 1). This result is superior to that of IH since we have better processor utilization in IEH.
0000…
0100…
1000… 1100…
0111 0011 0101 0000 0010 0001 0100 1110 1010 1100 1001 1011 1000 1101 0110
Fig. 7. Embed CBT3 into IEH(15).
Theorem 2. The minimal size of IEHs that contains a CBTd as a subgraph is 2d+1 + 1 for
d > 0.
Proof. Since IEH(2d+1 − 1) and IEH(2d+1) are IH(2d+1 − 1) and Hd+1, respectively, it is
impossible to embed a CBTd into them with edge adjacency reserved [13]. Observe that
IEH(2d+1+1) is a composition graph of Hd+1 and H0. By Lemma 2, Hd+1 contains a TBTd-1
as a subgraph. Since Hd+1 is symmetric, let two roots of this tree be
-0 1 ... 11 0 d and 0 1 ... 01 0 d
. Adding H0 and IC edges, a CBTd is obtained for H0 (i.e.,
-0 1 ... 11 1 + d ) as the root, and 011-...10 d and 001...10 d
are its sons. Hence, the proof.#
In [1], Supercubes contained complete binary trees as spanning subgraphs. How-ever, there is a drawback for supercubes that not all supercubes of size N, where N > 2d+1-1 contains a CBTd as a subgraph [1]. Without this drawback, IEH(N) contains a
CBTd as a subgraph when N ≥ 2d+1 + 1.
Theorem 3. IEH(N) contains CBTd as a subgraph when N ≥ 2d+1 + 1. Proof. Consider two cases.
Case 1. 2d+1 < N < 2d+1 + 2d
Because IEH(N) has Hd+1 as a subcube, we have a TBTd-1 in this subcube by Lemma 2.
Observe that a node v not in Hd+1 will have 2-IC edges connecting to nodes in Hd+1. By
adding v and its forward IC edges, our claim is found to be true in this case.
Case 2. N ≥ 2d+1 + 2d
Recall that IH is a spanning subgraph of IEH. Hence, in this case, our claim is found to be true [15].#
5. EMBEDDING MESHES AND TORI INTO IEHS
Linear arrays and rings are 1*n meshes and tori, respectively. Our previous work [4] proved that IEHs are Hamiltonian if the size of IEH is not 2n− 1 for all n ≥ 2. Next, we showed that for an IEH of size N, an arbitrary cycle of even length Ne, where 3 < Ne <
N, is found. We also found an arbitrary cycle of odd length No, where 2 < Ne < N, if and
only if a node of this graph has at least one forward 2-IC edge. It would be interesting to know how many numbers we can choose for construction of IEHs such that they contain not only even cycles, but also odd cycles. Surprisingly, there are very few integers for constructing IEHs containing only even cycles as we will show in the following theorem.
Theorem 4. Let M = {N | IEH(N) contain only even cycles, where 2n≤ N < 2n+1}. Then, the size of set M, denoted by |M|, is n + 1.
Proof. Consider an IEH(N) which contains no odd cycles. Thus, this graph has no 2-IC
edges from the above facts. Observe the only case in which N = ∑ = n j i i 2 , where j = 0,1, ..., n. We obtain |M| = n + 1. Hence, the proof. #
In [1, 6], IHs and supercubes both contained 2k*m meshes as spanning subgraphs where k ≥ 0 and m ≥ 1. Since IHs are spanning subgraphs of IEHs, a corollary is obtained immediately.
Corollary 2. IEH(N) contains 2k*m meshes as a spanning subgraph.#
However, no embedding of tori in IHs and supercubes has been studied. In the following theorem, we will show that IEH(2k*m) contains a 2k*m tori as a subgraph if and only if m ≠ 2n− 1 for all n ≥ 2.
Theorem 5. For all integers k ≥ 0 and m ≥ 1, IEH(2k*m) contains a 2k*m torus if and only if m ≠ 2n− 1 for all n ≥ 2.
Proof. It is trivial to verify this assertion when m is one or two. For m > 2, recall that
IEH(m) is Hamiltonian if and only if m ≠ 2n-1 for all n ≥ 2 [4]. Further, observe that IEH(2k*m) is a product graph of a k-dimension hypercube and an IEH(m) graph. Because a 2k*m torus is isomorphic to a product graph of a 2k ring and an m ring and a
k-dimension hypercube contains a 2k ring, this theorem is proved.#
6. CONCLUSIONS
In this paper, we have shown that IHs are spanning subgraphs of IEHs. Next, a complete binary tree of size N can be embedded into an IEH(N+2) graph with edge adja-cency reserved and expansion near 1. We can then embed a torus of size 2k*m into an IEH with dilation 1 and expansion 1 if and only if m ≠ 2n− 1 for all n ≥ 2. Our main
re-sults are summarized in Table 1. These rere-sults support the assertion that the IEH graph is a good alternative to the hypercube for constructing an interconnection network.
Table 1. Main results. The minimum size to
contain CBTd as a subgraph. If a 2k*m mesh is a spanning subgraph. If a 2k*m tori is a spanning subgraph. IH 2d+1+2d Yes No
Supercube 2d+1-1 Yes NA( still open)
IEH 2d+1+1 Yes Yes, when m ≠ 2n-1
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Hung-Yi Chang (»Ì) was born in Taiwan in 1970. He re-ceived his B.S. and M.S. degrees in Computer Science and Information Engineering from National Chiao-Tung University, Hsinchu, Taiwan, Taiwan, in 1992 and 1994, respectively. He is currently a Ph.D. candidate in the Department of Computer Science and Information Engineering in National Chiao-Tung University, Hsinchu, Taiwan. His research interests include interconnection networks, parallel architecture, graph theory, and reliability.
Rong-Jaye Chen (81) was born in Taiwan in 1952. He re-ceived his B.S. degree in Mathematics from National Tsing-Hua University, Taiwan, in 1977, his Ph.D. degree in Computer Science from the University of Wisconsin-Madison in 1987. He is now a professor in the Department of Computer Science and Information Engineering at National Chiao-Tung University, Hsinchu, Taiwan. Professor Chen is a member of IEEE. His research interests include algorithm design, theory of computation, DNA computing,
interconnection networks, mobile computing, network optimization, and combinatorial optimization.
