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The main contribution of this paper is to show the effectiveness of routing functions in constructing one-to-many disjoint paths in networks. The hypercube and the folded hypercube were taken as two illustrative examples. For both, any minimal routing function can be used to produce a maximal number of disjoint paths whose maximal length is minimized. As a byproduct, the strong Rabin number of the hypercube and the Rabin number of the folded hypercube were computed. The latter has been left as an open problem in [18]. Besides minimal routing functions, there are some other routing functions with different characteristics which can be used to produce disjoint paths with different properties. For example, the disjoint paths produced for the hypercube will have minimum total length if an implicit routing function in [14] is used. Some existing methods (e.g., [14, 20]) for constructing disjoint paths involved the use of an implicit routing function.

In [14], n non-empty subsets X1, X2, … , Xn of {1, 2, … , n} were used to represent n nodes d1, d2, … , dn so that for all 1≤u≤n and 1≤w≤n, w∈Xu if and only if du,w=1. A set of m distinct integers t1

k1

X , t2

k2

X , … , tmXkm is called a partial System of Distinct Representatives (SDR for short) for {X1, X2, … , Xn} if k1, k2, … , km are all distinct, where m≤n. Further, {t1, t2, … , tm} is maximum if m is maximized, and is called an SDR if m=n. A maximum partial SDR {t1, t2, … , tm} can be used to construct a maximal number of disjoint paths whose total length is minimized, if there is no j∈{1, 2, … , n}−{k1, k2, … , km} satisfying the following two: (C1) XjXki for some 1≤i≤m and (C2) there exists an SDR for {X ,k1

k2

X , … ,

1

ki

X , X ,j

+1

ki

X , … ,

km

X }. Such a maximum partial SDR can be determined in O(n2.5) time (see [14]).

Actually, a maximum partial SDR {t1, t2, … , tm} can be regarded as a routing function from {X1, X2, … , Xn} to {1, 2, … , n} which maps

ki

X to ti for all 1≤i≤m. Suppose LΦ={u| du,Φ(du) =1 and 1≤u≤n}.

It follows that Paths1(Φ, n, n, {d1, d2, … , dn}, {1, 2, … , n}) will produce a maximal number of disjoint paths whose total length is minimized, if |LΦ| is maximized and there is no j∈{1, 2, … , n}−LΦ satisfying the following two: (C1') dj≠dl and dj,w≤dl,w for some l∈LΦ and all 1≤w≤n and (C2') there exists a routing function Ψ with dj,Ψ(dj) =dv,Ψ(dv) =1 for all v∈LΦ−{l}. Here, Φ:{d1, d2, … , dn} →{1, 2, … , n} with maximum |LΦ| corresponds to a maximum partial SDR (LΦ corresponds to {k1, k2, … , km} and {Φ(du)|u

∈LΦ} corresponds to {t1, t2, … , tm}). Besides, (C1') and (C2') correspond to (C1) and (C2), respectively. A

routing function Φ with maximum |LΦ| so that there is no j∈{1, 2, … , n}−LΦ satisfying (C1') and (C2') can be determined with the same time complexity as a maximum partial SDR so that there is no j∈{1, 2, … , n}{k1, k2, … , km} satisfying (C1) and (C2).

There exist other routing functions that can result in m disjoint paths from s to d1, d2, … , dm whose total length is minimized. For example, if (C1') is changed to "|dj|<|dl| for some l∈LΦ", then the resulting Φ also serves the purpose. The method of [20] for constructing disjoint paths also involved the use of an implicit routing function. It is worth while exploring more relations between the characteristics of routing functions and the properties of disjoint paths.

Although both [20] and [14] also constructed disjoint paths in the hypercube, their construction methods cannot be used to obtain Theorem 1. The reason is that the correctness of Lemma 16 cannot be assured by their construction methods.

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