Journal of Graph Algorithms and Applications Embeddings between Hypercubes and Hypertrees

Graph embedding problems have gained importance in the field of in-terconnection networks for parallel computer architectures. Interconnec-tion networks provide an effective mechanism for exchanging data between processors in a parallel computing system. In this paper, we embed the rooted hypertree RHT (r) into r-dimensional hypercube Q r with dilation 2, r ≥ 2. Also, we compute the exact wirelength of the embedding of the r-dimensional hypercube Q r into the rooted hypertree RHT (r), r ≥ 2.


Introduction
A suitable interconnection network is an important part for the design of a multicomputer or multiprocessor system.This network is usually modeled by a symmetric graph, where the nodes represent the processing elements and the edges represent the communication channels.Desirable properties of an interconnection network include symmetry, embedding capabilities, relatively small degree, small diameter, scalability, robustness, and efficient routing [21].One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance [22].The hypercube has many excellent features and thus becomes the first choice of topological structure of parallel processing and computing systems.The machines based on hypercubes such as the Cosmic Cube from Caltech, the iPSC/2 from Intel and Connection Machines have been implemented commercially [8].Hypercubes are very popular models for paralled computation because of their symmetry and relatively small number of interprocessor connections.The hypercube embedding problem is the problem of mapping a communication graph into a hypercube multiprocessor.Hypercubes are known to simulate other structures such as grids and binary trees [7,16].
Graph embedding is an important technique that maps a logical graph into a host graph, usually an interconnection network.Many applications can be modeled as graph embedding.In architecture simulation, graph embedding has been known as a powerful tool for implementation of parallel algorithms or simulation of different interconnection networks.A parallel algorithm can be modeled by a task interaction graph, where nodes and edges represent tasks and direct communications between tasks, respectively.Thus, the problem of efficiently executing a parallel algorithm A on a parallel computer M can be often reduced to the problem of mapping the logical graph G, representing A, on the host graph H, representing M , so that the communication overhead is minimized [15].In parallel computing, a large process is often decomposed into a set of small sub-processes that can execute in parallel with communications among these sub-processes.The problem of allocating these sub-processes into a parallel computing system can be again modeled by graph embedding [6].
The quality of an embedding can be measured by certain cost criteria.One of these criteria which is considered very often is the dilation.The dilation of an embedding is defined as the maximum distance between a pair of vertices of host graph that are images of adjacent vertices of logical graph.It is a measure for the communication time needed when simulation one network on another [15].Another important cost criteria is the wirelength.The wirelength of an embedding is the sum of the dilations in host graph of edges in guest graph.The wirelength of a graph embedding arises from VLSI designs, data structures and data representations, networks for parallel computer systems, biological models that deal with cloning and visual stimuli, parallel architecture, structural engineering and so on [14,24].Graph embeddings have been well studied for a number of networks [2,3,7,16,17,18,19,20].Even though there are numerous results and discussions on the wirelength problem, most of them deal with only approximate results and the estimation of lower bounds [2].The embedding discussed in this paper produce exact wirelength.

Preliminaries
In this section we give the basic definitions and preliminaries related to embedding problems.Definition 2.1 [2] Let G and H be finite graphs.An embedding of G into H is a pair (f, P f ) defined as follows: 1. f is a one-to-one map from V (G) → V (H) For brevity, we denote the pair (f, P f ) as f .Definition 2.2 [2] If e = (u, v) ∈ E(G), then the length of P f (u, v) in H is called the dilation of the edge e.The maximum dilation over all edges of G is called the dilation of the embedding f .The dilation of embedding G into H is the minimum dilation taken over all embeddings f of G into H and denote it by dil(G, H).
The expansion [2] of an embedding f is the ratio of the number of vertices of H to the number of vertices of G.In this paper, we consider embeddings with expansion one.Definition 2.3 [2] Let f : G → H be an embedding.For e ∈ E(H), let EC f (e) denote the number of edges (u, v) of G such that e is in the path P f (u, v) between f (u) and f (v) in H.In other words,

Then the edge congestion of
, where the maximum is taken over all edge e of H.
The edge congestion of G into H is defined as EC(G, H) = min EC f (G, H), where the minimum is taken over all embeddings f : G → H. On the other hand, if S is any subset of If we think of G as representing the wiring diagram of an electronic circuit, with the vertices representing components and the edges representing wires connecting them, then the edge congestion EC(G, H) is the minimum, over all embeddings f : V (G) → V (H), of the maximum number of wires that cross any edge of H [3]. See Figure 1.
Definition 2.4 [16] The wirelength of an embedding f of G into H is given by The wirelength of G into H is defined as where the minimum is taken over all embeddings f of G into H.
The wirelength problem [2,3,16,18] of a graph G into H is to find an embedding of G into H that induces the minimum wirelength W L(G, H).The following two versions of the edge isoperimetric problem of a graph G(V, E) have been considered in the literature [4], and are N P -complete [10].
Problem 1 : Find a subset of vertices of a given graph, such that the edge cut separating this subset from its complement has minimal size among all subsets of the same cardinality.Mathematically, for a given m, if θ G (m) = min Problem 2 : Find a subset of vertices of a given graph, such that the number of edges in the subgraph induced by this subset is maximal among all induced subgraphs with the same number of vertices.Mathematically, for a given m, if For a given m, where m = 1, 2, . . ., n, we consider the problem of finding a subset A of vertices of G such that |A| = m and |θ G (A)| = θ G (m).Such subsets are called optimal.We say that optimal subsets are nested if there exists a total order O on the set V such that for any m = 1, 2, . . ., n, the first m vertices in this order is an optimal subset.In this case we call the order O an optimal order [4,12].This implies that W L(G, P n ) = n m=0 θ G (m) [10].
Further, if a subset of vertices is optimal with respect to Problem 1, then its complement is also an optimal set.But, it is not true for Problem 2 in general.However for regular graphs a subset of vertices S is optimal with respect to Problem 1 if and only if S is optimal for Problem 2 [4].In the literature, Problem 2 is defined as the maximum subgraph problem [10].
Lemma 2.5 (Congestion Lemma) [16] Let G be an r-regular graph and f be an embedding of G into H.Let S be an edge cut of H such that the removal of edges of S leaves H into 2 components H 1 and H 2 and let . Also S satisfies the following conditions: (iii) G 1 is an optimal set.
Then EC f (S) is minimum and Lemma 2.6 (Partition Lemma) [16] Let f : G → H be an embedding.Let {S 1 , S 2 , . . ., S p } be a partition of E(H) such that each S i is an edge cut of H satisfying the conditions of Congestion Lemma.Then Definition 2.8 [24] For r ≥ 1, let Q r denote the r-dimensional hypercube.The vertex set of Q r is formed by the collection of all r-dimensional binary strings.Two vertices x, y ∈ V (Q r ) are adjacent if and only if the corresponding binary strings differ exactly in one bit.
Equivalently if n = 2 r then the vertices of Q r can also be identified with integers 0, 1, . . ., n − 1 so that if a pair of vertices i and j are adjacent then i − j = ±2 p for some p ≥ 0. Definition 2.9 [13] An incomplete hypercube on i vertices of Q r is the subcube induced by {0, 1, . . ., i − 1} and is denoted by

Definition 2.10 [11]
The basic skeleton of a hypertree is a complete binary tree T r .Here the nodes of the tree are numbered as follows: The root node has label 1.The root is said to be at level 1. Labels of left and right children are formed by appending a 0 and 1, respectively, to the label of the parent node.See Figure 2(a).The decimal labels of the hypertree in Figure 2(a) is depicted in Figure 2(b).Here the children of the node x are labeled as 2x and 2x + 1.Additional links in a hypertree are horizontal and two nodes are joined in the same level i of the tree if their label difference is 2 i−2 .We denote an r-level hypertree as HT (r).It has 2 r − 1 vertices and 3 (2 r−1 − 1) edges.The rooted hypertree RHT (r) is obtained from the hypertree HT (r) by attaching to its root a pendant edge.The new vertex is called the root of RHT (r), r ≥ 2.
Theorem 2.11 [12] Let Q r be an r-dimensional hypercube.For 1 ≤ i ≤ 2 r , L i is an optimal set on i vertices.

Main Results
In this section, we embed the rooted hypertree RHT (r) into r-dimensional hypercube Q r with dilation 2. Further we compute the minimum wirelength of embedding Q r into RHT (r).The concept of embedding is widely studied in the area of fixed interconnection parallel architectures.A parallel architecture is embedded into another architecture to simulate one on another.An important feature of an interconnection network is its ability to efficiently simulate programs written for other architectures [15].
A tree is a connected graph that contains no cycles.The most common type of tree is the binary tree.It is so named because each node can have at most two JGAA, 19(1) 361-373 (2015) 367 descendents.A binary tree is said to be a complete binary tree if each internal node has exactly two descendents.These descendents are described as left and right children of the parent node.Binary trees are widely used in data structures because they are easily stored, easily manipulated, and easily retrieved.Also, many operations such as searching and storing can be easily performed on tree data structures.Furthermore, binary trees appear in communication pattern of divide-and-conquer type algorithms, functional and logic programming, and graph algorithms [24].
There are several useful ways in which we can systematically order all nodes of a tree.The three most important ordering are called preorder, inorder and postorder.To achieve these orderings the tree is traversed in a particular fashion.Starting from the root, the tree is traversed counter clockwise staying as close to the tree as possible.For preorder, we list a node the first time we pass it.For inorder, we list a node the second time we pass it.For postorder, we list a node the last time we pass it [9].
For any non-negative integer r, the complete binary tree of height r − 1, denoted by T r , is the binary tree where each vertex has exactly two children and all the leaves are at the same level.Clearly, a complete binary tree T r has r levels and level i, 1 ≤ i ≤ r, contains 2 i−1 vertices.Thus, T r has exactly 2 r − 1 vertices.The rooted complete binary tree RT r is obtained from a complete binary tree T r by attaching to its root a pendant edge.The new vertex is called the root of RT r and is considered to be at level 0 [24].
A hypertree is a hypergraph H if there is a tree T such that the hyperedges of H induce subtrees in T [5].In the literature, hypertree is also called a subtree hypergraph or arboreal hypergraph [5,23].
A hypertree is an interconnection topology for incrementally expansible multicomputer systems, which combines the easy expansibility of tree structures with the compactness of the hypercube; that is, it combines the best features of the binary tree and the hypercube.These two properties make this topology particulary attractive for implementation of multiprocessor networks of the future, where a complete computer with a substantial amount of memory can fit on a single VLSI chip [11].
Algorithm Dilation (Hypertree, Hypercube) Input : The rooted hypertree RHT (r) and the r-dimensional hypercube Q r , r ≥ 2. Algorithm : Removal of the horizontal edges in rooted hypertree RHT (r) leaves a rooted complete binary tree RT r .Label the vertices of RT r using binary codes corresponding to the inorder labeling [9].Label the vertices of Q r by using binary code corresponding to the lexicographic order [2] from 0 to 2 r − 1. See Figure 3. Output : An embedding f of RHT (r) into Q r given by f (x) = x with dilation 2. See Figure 3.  Proof: Label the vertices of RHT (r) and Q r using Dilation Algorithm.RHT (r) and Q r are not isomorphic, since RHT (r) contains a cycle of length 3 and Q r is a bipartite graph.Hence the dilation of RHT (r) into Q r is ≥ 2. Consider any edge e = (u, v) in RHT (r).We have the following two cases.
Case 1 (e ∈ RT r ): Since, the children of the node u in level i, 1 ≤ i ≤ r − 1 are labeled as u−2 r−(i+1) and u+2 r−(i+1) , the binary codes of u and u−2 r−(i+1) will differ in exactly one position and the binary codes of u and u + 2 r−(i+1) will differ in exactly two positions.Suppose u is the root of RT r , then the binary code of u and v will differ in exactly one position.
Case 2 (e / ∈ RT r ): By the labeling of RHT (r), the binary codes of u and v will differ in exactly one position.
Hence the distance between f (u) and f (v) in Q r is not larger than 2 in both the cases.
Next, we compute the exact wirelength of embedding r-dimensional hypercube Q r into rooted hypertree RHT (r).For proving the main result, we need the following Lemmas.
Thus the binary representation of two numbers x and y differ in exactly one bit ⇔ the binary representation of ϕ(x) and ϕ(y) differ in exactly one bit.Therefore (x, y) is an edge in N cutS 2 i i ⇔ (ϕ(x), ϕ(y)) is an edge in L 2 i .Hence N cutS 2 i i and L 2 i are isomorphic.By Theorem 2.11, N cutS Proof: By Theorem 2.11, the set {0, 1, 2, . . ., 2 i − 2} is optimal and by Lemma 3.2, the set {2 r−1 , 2 r−1 + 1, . . ., 2 r−1 + 2 i − 2} is optimal in Q r .Also the binary representation of k and 2 r−1 + k, 0 ≤ k ≤ 2 i − 2, differ exactly in one bit. Therefore But by Lemma 2.12, E(Q r [L 2(2 i −1) ]) = (i + 1)2 i − 2i − 1 and hence by Theorem 2.11, N cutS i 1 is an optimal set in Q r .
Removal of the horizontal edges in rooted hypertree RHT (r) leaves a rooted complete binary tree RT r .Label the vertices of RT r using inorder labeling [9].See Figure 4.
Output : An embedding f of Q r into RHT (r) given by f (x) = x with optimal wirelength.

Concluding Remarks
We provide an embedding of the rooted hypertree RHT (r) into r-dimensional hypercube Q r with dilation 2. Further, we compute the exact wirelength of embedding r-dimensional hypercube Q r into rooted hypertree RHT (r), r ≥ 2. Finding the dilation of embedding hypercube into rooted hypertree is under investigation.

Figure 1 :
Figure 1: Wiring diagram of a grid G into path H with EC f (G, H) = 4

(
ii) For every edge (a, b) in G with a ∈ G 1 and b ∈ G 2 , P f (a, b) has exactly one edge in S.

Theorem 3 . 1
The rooted hypertree RHT (r) can be embedded into the r-dimensional hypercube Q r with dilation 2, r ≥ 2.