Some Sequence of Wrapped Δ-Labellings for the Complete Bipartite Graph

The design of large disk array architectures leads to interesting combinatorial problems. Minimizing the number of disk operations when writing to consecutive disks leads to the concept of “cluttered orderings” which were introduced for the complete graph by Cohen et al. (2001). Mueller et al. (2005) adapted the concept of wrapped Δ-labellings to the complete bipartite case. In this paper, we give some sequence in order to generate wrapped Δ-labellings as cluttered orderings for the complete bipartite graph. New sequence we give is different from the sequences Mueller et al. gave, though the same graphs in which these sequences are labeled.


Introduction
The desire to speed up secondary storage systems has lead to the development of disk arrays which achieve performance through disk parallelism.While performance improves with increasing numbers of disks, the chance of data loss coming from catastrophic failures, such as head crashes and failures of the disk controller electronics, also increases.To avoid high rates of data loss in large disk arrays, one includes redundant information stored on additional disks-also called check disks-which allows the reconstruction of the original datastored on the so-called information disks-even in the presence of disk failures.These disk array architectures are known as redundant arrays of independent disks (RAID) (see [1] [2]).
Optimal erasure-correcting codes using combinatorial framework in disk arrays are discussed in [1] [3].For an optimal ordering, there are [4] [5].Cohen et al. [6] gave a cyclic construction for a cluttered ordering of the complete graph.In the case of a complete graph, there are [7] [8].Furthermore, in the case of a complete bipartite graph, Mueller et al. [9] gave a cyclic construction for a cluttered ordering of the complete bipartite graph by utilizing the notion of a wrapped Δ-labelling.In the case of a complete tripartite graph, we refer to [10].
As Figure 1, we present the case 2 =  . For example, information disk 1 is associated to the check disks a and c.A 2-dimensional parity code can be modeled by the complete bipartite graph ( ) , , , in the following way.The point set of ,

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  is partitioned into the two sets-U and V both having cardinality  .Assign the points of U to the  check bits corresponding to the rows and the points of V to the  check bits corresponding to the columns.By definition, in ,

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  any point of U is connected with any point of V exactly on edge constituting the edge set E, i.e., 2 E =  (see Figure 2).In this paper, we make label to the vertex of a bipartite graph.For example, we make label 1, 3, 0 and −1, respectively, to four vertices a, b, c and d of a bipartite graph in Figure 2. By such labelling, we get that the label of the edge { } , c d are sequences.The goal of this paper is to find new sequence in order to generate wrapped Δ-labellings as cluttered orderings for the complete bipartite graph.In Section 5, we give new sequence which we want.The new sequence we give is different from the sequences Mueller et al. [9] gave, though the same graphs in which these sequences are labeled.

A Cluttered Ordering
In a RAID system disk writes are expensive operations and should therefore be minimized.In many applications there are writes on a small fraction of consecutive disks-say d disks-where d is small in comparison to k, the number of information disks.Therefore, to minimize the number of operations when writing to d consecutive information disks one has to minimize the number of check disks-say f-associated to the d information disks.Let ( ) positive integer, called a window of G, and π a permutation on { } , called an edge ordering of G.Then, given a graph G with edge ordering π and window d, we define ,d i V π to be the set of vertices which are connected by an edge of

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  into isomorphic copies of this subgraph.Secondly, we define the concept of a (d, f)-movement which will lead to "locally" defined edge orderings of ,

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  .This principle was implicitely used in [6] in case of the complete graph.In case of the complete bipartite graph, we refer to [9].
In the following, ( ) = always denotes a bipartite graph with vertex set U which is partitioned into  two subsets denoted by V and W. Any edge of the edge set E contains exactly one point of V and W respectively.Let E =  , then a Δ-labelling of H with respect to V and W is defined to be a map , where each element of Z  occurs exactly once in the difference list Here, 1 denotes the projection on the first component.In general, Δ-labellings are a wellknown tool for the decomposition of graphs into subgraphs (see [11]).In this context a decomposition is understood to be a partition of the edge set of the graph.In case of the complete bipartite graph, one has the following proposition.
, and Δ be a Δ-labelling as defined above.Then there is a decomposition of the complete bipartite graph , K   into isomorphic copies of H.For example, Figure 3 shows Δ-labellings of a graph ( ) : : for some given a positive integer f, and a map σ is called a ( )  ϕ Σ → Σ be any bijection, then a (d, f)-movement σ from 0 Now, for each j Z ∈  one gets an automorphism j τ of the bipartite graph , K   defined by cyclic translation of the vertex set: : , , : , , Obviously, j τ induces in a natural way an automorphism of the edge set of , K   which we also denote j τ .Then,  , where we fix some arbitrary edge ordering.We denote the restriction of the cyclic translation κ τ to ( ) will be denoted as ( ) Σ consistent with the translation parameter κ .
According to Definition 1, such a (d, f)-movement is given by some permutation σ of the index set . By applying the cyclic translation i τ one gets a graph ( ) ( ) ( ) We denote the restriction of κ τ to ( ) Then σ also defines a ( ) .
Having such a consistent σ , it is easy to construct a (d, f)-cluttered ordering of , K   .In short, one orders the edges of ,

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  by first arranging the subgraphs of the decomposition along ( ) ( ) ( ) and then ordering the edges within each subgraph according to σ .Proposition 2.
, be a bipartite graph allowing some ρ -labelling, and let κ be a translation parameter coprime to  .Furthermore, let 0 E Σ ⊂ , 0 : d = Σ .If there is a (d, f)-movement from 0 Σ consistent with κ , then there also is a (d, f)-cluttered ordering for the complete bipartite graph  , K .

Construction of Cluttered Orderings of ( )
;

H h t
In this section, we define an infinite family of bipartite graphs which allow (d, f)-movements with small f.In order to ensure that these (d, f)-movements are consistent with some translation parameter κ , we impose an additional condition on the Δ-labellings also referred to as wrapped-condition.
Let h and t be two positive integers.For each parameter f and t, we define a bipartite graph denoted by ( ) ( ) and consists of the following ( ) : .
shows the edge partition of ( ) The parameter κ is also referred to as translation parameter of the wrapped

Δ-labelling.
For the graphs ( ) Furthermore, in the following we only consider wrapped Δ-labellings relative to X and Y for which the stronger condition , 0 and , 0 , hold for 0 i h ≤ < .Suppose we have such labelling Δ satisfying condition (7).Now, 4. Sequences of Wrapped Δ -Labellings for H(1; t), H(2; t) and H(h; 1) In this section, we construct some infinite families of such wrapped Δ-labellings.By applying Proposition 2 we get explicite (d, f)-cluttered orderings of the corresponding bipartite graphs.For these results in this section, we refer to [9].

A Sequence for H(1; t)
We define a wrapped Δ-labelling of ( ) , 0 , for 0 , 1, 0 , for , 1 ,1 , for 0 , ( ) 1,1 , for , where the integers in the first components are considered modulo 3t.We now compute the difference list ( ) E ∆ of δ defined as in (1).Hence each element of 3t Z appears in ( ) and the difference condition holds.Figure 3 illustrates the definition for the case t = 1.
Obviously, the wrapped-condition (7) relative to holds as well and the translation parameter + is coprime to 3t for any t.Therefore, Δ defines the desired wrapped Δ-labelling of ( ) ( )

A Sequence for H(2; t)
We define a wrapped Δ-labelling of ( ) 2; H t for any positive integer t.
( ) ( ) vertices and 10t edges.For a fixed t, a labelling Δ is a map on the vertex set U V W = ∪ .We specify the second component of Δ on the vertices ( ) sequentially by the following list of 2t + 2 numbers: , , , t W w w w + =  by, similarly, where we set + are coprime for all t and that the wrapped-condition (7)  ( ) ( )

A Sequence for H(h; 1)
We define in this section a wrapped Δ-labelling for ( ) : by specifying the first component of Δ on the vertices ( ) sequentially by the following list of 2h numbers:  ) All integers are considered modulo ( ) and κ are coprime for any posi- tive integer h and the wrapped-condition ( 7) is fulfilled.Figure 7 illustrates the definition for the case 3 h = .All numbers ( ) Z + appear exactly once as difference of Δ which hence defines a wrapped Δ-labelling.Theorem 9. ([9]) Let h be a positive integer.For all h there is a (d, f)-cluttered ordering of the complete bipartite graph

Our Result:
A Sequence of a Wrapped ∆ -Labelling for ( )

3; H t
In this section, we define a wrapped Δ-labelling of ( ) : We specify the second component of Δ on the vertices ( ) sequentially by the following list of 3 3 t + numbers: and, on the vertices ( ) + are coprime for all positive in- teger t and that the wrapped-condition ( 7) is obviously fulfilled.Figure 8 illustrates the definition for the case t = 1.

Conclusion
In conclusion, we give a new sequence for construction of wrapped Δ-labellings.Figure 7 and Figure 8 are the same as a graph, but they are different as a sequence.Cluttered orderings given by two sequences construct the different orderings for the complete bipartite graph 21,21 K .

, a c is 1 0 1 − 1 −
=; the label of the edge { } , of the lower vertices [ ] , where indices are considered modulo m.The cost of accessing a subgraph of d consecutive edges is measured by the number of its vertices.An upper bound of this cost is given by the d-maximum access cost of G defined as , maxd i i V π .An ordering π is a (d, f)-cluttered ordering, if it has d-maximum access cost equal to f.We are interested in minimizing the parameter f.Let  be a positive integer and let , K   denote the complete bipartite graph with 2 vertices and 2  edges.In the following, we identify the vertex set of , vertices are connected by an edge iff they have different second components in 2 Z Z ×  .The construction of (d, f)-cluttered orderings for , K   with small positive integer f is based on two fundamental concepts.Firstly, we introduce the well-known concept of a Δ-labelling of a suitable bipartite subgraph from which one gets a decomposition of ,

Figure 1 .
Figure 1.2-dim.parity code and its parity check matrix.
order to assemble such (d, f)-movements of certain subgraphs to a (d, f)-cluttered ordering, we need some notion of consistency.Let

Figure 3 .
Figure 3.A Δ-labelling of a graph

Figure 8 .
Figure 8.Some wrapped Δ-labelling of [6]here the last h vertices of V and W respectively of one copy are identified with the first h vertices of V and W respectively of the next copy.Traversing these copies with increasing s will define a (d, f)-movement of -movement consistent with some translation parameter κ .To this means, we impose an additional con- dition on the Δ-labelling.The following definition generalizes and adapts the notion of a wrapped Δ-labelling to the bipartite case, which was introduced in[6]for certain subgraphs of the complete graph. is called a wrapped Δ-labelling of H relative to X and Y if there exists a From this one easily checks that the twenty-two lists cover all numbers in 21t Z exactly once.Thus, Δ defines a wrapped Δ-labelling and by applying Proposition 4 we get the following result.Theorem 10.Let t be a positive integer.For all t there is a (d, f)-cluttered ordering of the complete bipartite graph 21 ,21 one gets the following theorem by enlarging the window d.Theorem 11.Let t be a positive integer.For all t there is a (d, f)-cluttered ordering of the complete bipartite graph 21 ,21