One-to-Many Disjoint Paths in Hypercubes

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(1)One-to-Many Disjoint Paths in Hypercubes Cheng-Nan Lai,. Gen-Huey Chen. Department of Computer Science and Information Engineering, National Taiwan University, Taipei, TAIWAN. Abstract This paper introduces a new concept called routing functions, which have a close relation to one-to-many disjoint paths in networks. By using a minimal routing function, m disjoint paths whose maximal length is minimized in the worst case can be obtained in a k-dimensional hypercube, where m≤k. Besides, there exist routing functions that can be used to construct m disjoint paths whose total length is minimized. The end nodes of these paths are not necessarily distinct. A minimal routing function can also be used to construct a maximal number of disjoint paths in the folded hypercube whose maximal length is minimized in the worst case.. 1. Introduction In the past decade, routing with internally node-disjoint paths (disjoint paths for short) has received much attention because disjoint paths have the advantages of efficiency and fault tolerance. There are three categories of disjoint paths, i.e., one-to-one, one-to-many, and many-to-many. Suppose that W is an interconnection network (network for short) with node connectivity k [2]. According to Menger's theorem [2], there exist k disjoint paths from one source node to another destination node in W. These disjoint paths belong to the one-to-one category. Many one-to-one disjoint paths constructed for a variety of networks can be found in the literature [3, 4, 6-10, 12, 22, 24]. There is an excellent survey of one-to-one disjoint paths in [19] where several related problems were also addressed. According to Theorem 2.6 in [1], there exist k disjoint paths from one source node to another k distinct destination nodes in W. These disjoint paths belong to the one-to-many category. A k-dimensional hypercube (abbreviated to a kcube) consists of 2k nodes that are labeled with 2k binary numbers from 0 to 2 k−1. Two nodes of a k-cube are adjacent if and only if their labels differ by exactly one bit. The node connectivity of a k-cube is k. In [23], k disjoint paths were constructed from one source node to another k destination nodes in a k-cube, where the k destination nodes were distinct. The maximal length is minimized in the worst case. In [15], m disjoint paths were constructed from one source node to another m destination nodes in a k-cube, where m≤k. Dyi-Rong Duh Department of Computer Science and Information Engineering, National Chi Nan University, Nantou, TAIWAN. and the m destination nodes were not necessarily distinct. The total length is minimized. One-to-many disjoint paths constructed for other networks appeared in [5, 9, 11, 20]. There were many-to-many disjoint paths constructed for the hypercube [17] and the star graph [9, 18]. In this paper, a new concept called routing functions is proposed, which is useful to derive one-to-many disjoint paths. In the next section, routing functions and their fundamental properties are introduced. In Section 3, by the aid of a minimal routing function, m disjoint paths from one source node to another m distinct destination nodes are constructed in a k-cube, where m≤k. It is shown in Section 4 that the maximal length of the m disjoint paths is minimized in the worst case. For any given m destination nodes, the maximal length of the resulting m disjoint paths is equal to dis max or min{dis max+2, k+1}, where dis max is the maximal distance from the source node to the destination nodes. In Section 5, the situation that the m destination nodes are not necessarily distinct is discussed. In Section 6, this paper concludes with some remarks on routing functions. It is indicated that a minimal routing function can be also used to derive a maximal number of disjoint paths in the folded hypercube whose maximal length is minimized in the worst case, and there are routing functions that can be used to construct m disjoint paths in a k-cube whose total length is minimized.. 2. Routing functions Suppose that s, d1, d2, … , dm are arbitrary m+1 distinct nodes of a k-cube, where m≤k. Since the hypercube is node k8 67 symmetric, we assume s= 00... 0 =0k without loss of generality. A routing function for a k-cube is a one-to-one correspondence Φ from D={d1, d2, … , dm} to I={n1, n2, … , nm}, where 1≤nj≤k for all 1≤j≤m and n1, n2, … , nm denote m distinct dimensions of a k-cube. Suppose di=d i,1di,2… di,k, and let |di| denote the number of bits 1 contained in di (i.e., the distance from s to di), where 1 ≤i≤m. We define VΦ=(v1, v2, … , vk), where vj=|{di | |di|=j, d i ,Φ ( d i ) = 0, and 1≤i≤m}| for all 1≤j≤k. For example, if m=k=5, (d1, d2, d3, d4, d5)= (00011, 00101, 00111, 00110, 11100), and (Φ(d1), Φ(d2), Φ(d3), Φ(d4), Φ(d5))=(5, 2, 3, 4, 1), then VΦ=(0, 1, 0, 0, 0). We say.

(2) (v1, v2, … , vk)<(v'1, v'2, … , v'k) if either v1<v'1 or v1=v'1, v2=v'2, … , vl−1=v'l−1, and vl<v'l for some 2≤l≤k. Φ is said to be minimal if VΦ≤VΦ' for every routing function Φ': D → I. A minimal Φ can be determined in O(k3) time (see Section 6). A minimal Φ is intended to derive m disjoint paths from s to d1, d2, … , dm, and there is a favorable property about a minimal Φ: if d i1 ,Φ (d i ) = d i2 ,Φ ( d i ) = ⋅ ⋅ ⋅ = 1. d ic ,Φ (d i. ) c. 2. = 1 , then there exist c disjoint shortest paths from s. to d i1 , d i2 , … , d ic , respectively, where {d i1 , d i2 , … , d ic } ⊆ D (see Section 4). Let eβ= 0β−110 k−β, β=1, 2, … , k,. denote the k adjacent nodes of s. An intuitive meaning of Φ(di)=nj is to assign the immediate successor of s in the path to di to be the node en j . In the rest of this section, some fundamental properties of Φ are introduced. Suppose that Φ': D' → I' and Φ'': D'' → I'' are two routing functions, where {D', D''} is a partition of D and {I', I''} is a partition of I. If Φ'(di)=Φ(di) for all di ∈D' and Φ''(dj)=Φ(dj) for all dj ∈D'', then Φ is said to be the union of Φ' and Φ'', denoted by Φ=Φ'∪Φ''. If Φ=Φ'∪Φ'', then VΦ=VΦ'+VΦ'', i.e., VΦ=(v'1+v''1, v'2+v''2, … , v'k+v''k) where VΦ'=(v'1, v'2, … , v'k) and VΦ''=(v''1, v''2, … , v''k). The following two lemmas are immediate. Lemma 1. Suppose Φ=Φ'∪Φ''. If d i ,Φ '( di ) = 1 for every di ∈D', then VΦ=VΦ''. Lemma 2. Suppose that Φ=Φ'∪Φ'' is minimal. Then Φ' and Φ'' are minimal. Lemma 3. Suppose. d i ,Φ ( di ) = 0 , d j ,Φ ( d j ) = 0 , and. d i ,Φ ( d j ) = 1 , where 1≤i≤m, 1≤j≤m, and i≠j. Then Φ is not minimal. Proof. We define Ψ: D → I as follows: Ψ(di)=Φ(dj), Ψ(dj)=Φ(di), and Ψ(dr)=Φ(dr) for all r ∈{1, 2, … , m}−{i, j}. Suppose VΦ=(v1, v2, … , vk) and VΨ=(v'1, v'2, … , v'k). If |di|≠|dj|, then v'|d i | = v|d i | −1, v'|d j | ≤ v|d j | , and v'l=vl for all l ∈{1, 2, … , k}−{|di|, |dj|}. If |di|=|dj|, then v'|d i | ≤ v|d i | −1 and v'l=vl for all ¨. 1≤l≤k and l≠|di|. Hence VΨ<VΦ. Lemma 4.. Suppose. d j ,Φ ( d j ) = 0 ,. d j ,Φ ( d i ) = 1 ,. Proof. We define Ψ, VΨ, and VΦ all the same as the above. Then v' |d i | = v |di | − 1 and v'l=vl for all 1≤l<|di|. Hence we ¨. have VΨ<VΦ. Lemma 6. If Φ is minimal and d i ,1 d i ,2 ... d i,Φ ( di ) −11d i,Φ ( di ) +1 ...d i , k ∉{d 1,. d i ,Φ ( di ) = 0 , then. d2,. … ,. ( d i ,1 d i ,2 ... d i ,Φ (d i )−1 1d i ,Φ (d i )+1 ...d i ,k is the node obtained by changing the bit d i ,Φ (d i ) of di to 1). Proof. Suppose conversely =d for some 1≤r≤m and d i ,1 d i ,2 ... d i,Φ ( di ) −11d i,Φ ( di ) +1 ...d i , k r r≠i. That is, d r ,Φ ( d i ) = 1 and d r, j = d i , j for all 1≤j≤k and j≠Φ(di). Then Φ(dr) ∈{n1, n2, … , nm}. If d r ,Φ ( d r ) = 0 , then Φ is not minimal by Lemma 3, which is a contradiction. If d r ,Φ ( d r ) = d i ,Φ ( d r ) = 1 , then Φ is not minimal by Lemma 4, which is again a contradiction.. In this section, a recursive procedure, named Paths, is proposed. With inputs minimal Φ, m, k, D={d1, d2, … , dm}, and I={n1, n2, … , nm}, the procedure can produce m disjoint shortest paths, denoted by P1, P2, … , P m, in a k-cube that connect { e n1 , e n2 , … , e nm } and {d1, d2, … , dm}, where Pi (1≤i≤m) is the path to di. By augmenting P1 , P2, … , P m with links (s, e n1 ), (s, e n2 ), … , (s, e nm ), we have m disjoint paths from s to d1, d2, … , dm, respectively. In the procedure, ∗k−11 and ∗k−10 represent two disjoint (k−1)-cubes whose nodes have rightmost bits 1 and 0, k −18 67 k −1 respectively, where ∗ ∈{0, 1} and ∗ = ∗ ∗ ⋅ ⋅ ⋅ ∗ (usually, a k-cube is represented with ∗k). For each node x=x1x2...xk of a k-cube, define x(k)=x1x2...xk−1(1−xk), i.e., x differs from x(k) only in the bit xk. The following is a formal description of the procedure. Procedure Paths(Φ, m, k, D, I). Step 1.. and. If k=2, then { Construct P1, P2, … , Pm as the m disjoint paths in a 2-cube that connect { en1 , e n2 , … , e nm } and {d 1, d 2, … , d m}.. minimal. Proof. We define Ψ, VΨ, and VΦ all the same as the above. If d i ,Φ (d i ) = 0 , then Φ is not minimal by Lemma 3. If. Return P1, P2, … , Pm. } Step 2.. d i ,Φ ( di ) = 1 , then v'|d j | = v|d j | − 1 and v'l=vl for all 1≤l≤k and. Determine dc so that |dc|≤|di| for all 1≤i≤m, where 1≤c≤m. If there are multiple candidates for dc, then select arbitrary one with d c,Φ (d c ) = 1 , or select any if they all have d c,Φ ( dc ) = 0 . Without. ¨. loss of generality, we assume c=1, nc=n1=k, and Φ(di)=ni for all 1 ≤i≤m.. Lemma 5. Suppose d i ,Φ ( di ) = 0 , d i ,Φ (d j ) = 1 , and |di|<|dj|, where 1≤i≤m, 1≤j≤m, and i≠j. Then Φ is not minimal.. ¨. 3. A procedure to produce disjoint paths. d i ,Φ (d j ) = 1 , where 1≤i≤m, 1≤j≤m, and i≠j. Then Φ is not. l≠|dj|, which implies VΨ<VΦ.. dm}. Step 3.. Partition D into D' and D'', where D'={dj |.

(3) d j ,Φ ( dc ) = d j ,Φ ( d1 ) = d j ,k = 0. D''={dj. |. and 1≤j≤m} and. d j ,Φ ( dc ) = d j ,Φ ( d 1 ) = d j ,k = 1. and. 1≤j≤m}. Step 4.. Construct P1, P2, … , Pm according to the following four cases. Case 1. d1,k=0. /* Without loss of generality, suppose D'={d1, d2, … , dr} and D''={dr+1, dr+2, … , dm}, where 1≤r≤m. Define Ψ': {d2, d3, … , dr} → {n2, n3, … , nr} as follows: Ψ'(di)=Φ(di)=ni for all 2≤i≤r, and Ψ'': { d 1(k ) , dr+1, dr+2, … , dm} → {u, nr+1, nr+2, … , nm} as follows: Ψ''( d 1(k ) )=u and Ψ''(di)=Φ(di)=ni for all r+1≤i≤m, where 1≤u<k and d1,u=1. */ Construct P2, P3, … , Pr in ∗k −1 0 by executing Paths(Ψ', r−1, k−1, {d2, d3, … , dr}, {n2, n3, … , nr}). /* P2, P3, … , Pr connect {e n2 , en 3 , … , en r } and {d 2, d 3, … , d r}. */. Construct P'1, Pr+1, Pr+2, … , Pm in ∗k − 1 1 by executing Paths(Ψ'', m−r+1, k−1, { d 1(k ) , dr+1, dr+2, … , dm}, {u, nr+1, nr+2, … , nm}), where P'1 is the path to d 1(k ) . /* P'1, Pr+1, Pr+2, … , Pm ) (k ) connect { eu( k ) , en( kr +)1 , e (nkr +)2 , … , e (k nm } and { d 1 ,. dr+1, dr+2, … , dm}. */ Construct P1 as P'1 augmented with the link ( d 1(k ) , d1). Augment P1, Pr+1, Pr+2, … , Pm with links (ek, eu( k ) ), ) ( enr +1 , e (nkr +)1 ), ( e nr +2 , e (nkr +)2 ), … , ( e nm , e (k nm ).. Case 2. d1,k=1 and |d1|=1. /* Suppose D'={d2, d3, … , dr} and D''={d1, dr+1, dr+2, … , dm}, where 1≤r≤m. Define Ψ': {d2, d3, … , dr} → {n2, n3, … , nr} as follows: Ψ'(di)=Φ(di)=ni for all 2≤i≤r, and Ψ'': {dr+1, dr+2, … , dm} → {nr+1, nr+2, … , nm} as follows: Ψ''(di)=Φ(di)=ni for all r+1≤i≤m. */ Construct P2, P3, … , Pr in ∗k −1 0 by executing Paths(Ψ', r−1, k−1, {d2, d3, … , dr}, {n2, n3, … , nr}). /* P2, P3, … , Pr connect {e n2 , en 3 , … , en r } and {d 2, d 3, … , d r}. */. Construct Pr+1, Pr+2, … , Pm in ∗k − 1 1 by executing Paths(Ψ'', m−r, k−1, {dr+1, dr+2, … , dm}, {nr+1, nr+2, … , nm}). /* Pr+1, Pr+2, … , Pm connect { en( kr +)1 , ) e (nkr +)2 , … , e (k nm } and {d r+1, d r+2, … , d m}. */. Construct P1 as (ek, d1). /* P1 has length 0. */ Augment Pr+1, Pr+2, … , Pm with links ( enr +1 ,. ) en( kr +)1 ), ( e nr + 2 , e (nkr +)2 ), … , ( e nm , e (k nm ).. Case 3. d1,k=1, |d1|>1, and d1,α=1 for some α ∈{1, 2, … , k−1}−{k, nr+1, nr+2, … , nm}. /* Suppose D'={d2, d3, … , dr} and D''={d1, dr+1, dr+2, … , dm}, where 1≤r≤m. Define Ψ': {d2, d3, … , dr} → {n2, n3, … , nr} as follows: Ψ'(di)=Φ(di)=ni for all 2≤i≤r, and Ψ'': {d1, dr+1, dr+2, … , dm} → {α, nr+1, nr+2, … , nm} as follows: Ψ''(d1)=α and Ψ''(di)=Φ(di)=ni for all r+1≤i≤m. */ Construct P2, P3, … , Pr in ∗k −1 0 by executing Paths(Ψ', r−1, k−1, {d2, d3, … , dr}, {n2, n3, … , nr}). /* P2, P3, … , Pr connect {e n2 , en 3 , … , en r } and {d 2, d 3, … , d r}. */ k −1. 1 by Construct P1, Pr+1, Pr+2, … , Pm in ∗ executing Paths(Ψ'', m−r+1, k−1, {d1, dr+1, dr+2, … , dm}, {α, nr+1, nr+2, … , nm}). /* P1, Pr+1, Pr+2, … , ) Pm connect { eα( k ) , en( kr +)1 , e (nkr +)2 , … , e (k nm } and {d 1, dr+1, dr+2, … , dm}. */ Augment P1, Pr+1, Pr+2, … , Pm with links (ek, eα( k ) ), ) ( enr +1 , en( kr +)1 ), ( e nr + 2 , e (nkr +)2 ), … , ( e nm , e (k nm ).. Case 4. d1,k=1, |d1|>1, and d1,α=0 for all α ∈{1, 2, … , k−1}−{k, nr+1, nr+2, … , nm}. /* Suppose D'={d2, d3, … , dr} and D''={d1, dr+1, dr+2, … , dm}, where 1≤r≤m. Define Ψ'': {dr+1, dr+2, … , dm} → {nr+1, nr+2, … , nm} as follows: Ψ''(di)=Φ(di)=ni for all r+1≤i≤m. */ Construct Pr+1, Pr+2, … , Pm in ∗k − 1 1 by executing Paths(Ψ'', m−r, k−1, {dr+1, dr+2, … , dm}, {nr+1, nr+2, … , nm}). /* Pr+1, Pr+2, … , Pm connect { en( kr +)1 , ) e (nkr +)2 , … , e (k nm } and {d r+1, d r+2, … , d m}. */. /* Define Ω'': {dr+1, dr+2, … , dm} → {nr+1, nr+2, … , nm} as follows: Ω''(di)=nj if Pi begins at en j for (k ). all r+1≤i≤m, where r+1≤j≤m. */ If d1 ∉Pi for all r+1≤i≤m, then { /* Define Ψ': { d1( k ) , d2, d3, … , dr} → {u, n2, n3, … , nr} as follows: Ψ'( d 1(k ) )=u and Ψ'(di)=Φ(di)=ni for all 2≤i≤r, where 1≤u<k and d1,u=1. */ Construct P'1, P2, P3, … , Pr in ∗k −1 0 by executing Paths(Ψ', r, k−1, { d 1(k ) , d2, d3, … , dr}, {u, n2, n3, … , nr}), where P'1 is the path to d 1(k ) . /* P'1, P2, P3 , … , Pr connect {eu, e n2 , en 3 , … , en r } and { d 1(k ) , d2, d3, … , dr}. */ Construct P1 as P'1 augmented with the link.

(4) ( d 1(k ) , d1). /* Suppose u=nh for some r+1≤h≤m. */ Augment Pr+1, Pr+2, … , Pm with links ( enr +1 , en( kr +)1 ), ( e nr + 2 , e (nkr +)2 ), … , ( e nh −1 , e (nkh −)1 ), (ek, ) eu( k ) ), ( enh +1 , en( kh +)1 ), … , ( e nm , e (k nm ). }. If d1 ∈Pl for some r+1≤l≤m and d l(k ) ∉{d2, d3, … , dr}, then { /* Define Θ': { d l(k ) , d2, d3, … , dr} → {Ω''(dl), n2, n3, … , nr} as follows: Θ'( d l(k ) )=Ω''(dl) and Θ'(di)=Φ(di)=ni for all 2≤i≤r. */ Construct P'l, P2, P3, … , Pr in ∗k −1 0 by executing Paths(Θ', r, k−1, { d l(k ) , d2, d3, … , dr}, {Ω''(dl), n2, n3, … , nr}), where P'l is the path to d l(k ) . /* P'l, P2, P3, … , Pr connect {e Ω ''( dl ) , e n2 , en 3 , … , en r } d l(k ). and { , d2, d3, … , dr}. */ Construct P1 as the subpath of Pl from e (Ωk'')( dl ) to d 1. Reconstruct Pl as P'l augmented with the link ( d l(k ) , dl). Augment P1, Pr+1, Pr+2, … , Pl−1, Pl+1, … , Pm with links ( enr +1 , en( kr +)1 ), ( e nr + 2 , e (nkr +)2 ), … , ( e Ω' ' ( dl )−1 ,. (k ) eΩ ' ' ( dl )−1 ),. (ek,. e (Ωk'')( dl ). ),. (k ) (k ) ( e Ω' ' ( dl )+1 , e Ω ' ' ( dl )+1 ), … , ( e nm , e nm ). }. If d1 ∈Pl for some r+1≤l≤m and d l(k ) ∈{d2, d3, … , dr}, then /* Suppose d l(k ) =dh for some 2≤h≤r. Define Ψ: {d1, d2, … , dm} → {n1, n2, … , nm} as follows: Ψ(dl)=Φ(dh), Ψ(dh)=Ω''(dl), Ψ(d1)=Φ(d1)=k, Ψ(di)=Φ(di) for all 2≤i≤r and i≠h, and Ψ(dj)=Ω''(dj) for all r+1≤j≤m and j≠l. */ Construct P1, P2, … , Pm all the same as Case 3. /* Substitute Ψ for the Φ in Case 3. */ Step 5.. Return P1, P2, … , Pm.. Intuitively, the m disjoint paths from s to d1, d2, … , dm can be obtained by first including m links (s, eΦ ( d1 ) ), (s, eΦ ( d2 ) ), … , (s, eΦ ( d m ) ) and then constructing m disjoint. paths, i.e., P1, P2, … , P m, that connect {eΦ ( d1 ) , eΦ ( d2 ) , … , eΦ ( d m ) } and {d 1, d 2, … , d m}. The latter can be done by. executing Paths(Φ, m, k, D, I). In Step 2, the k-cube was. divided into two (k−1)-cubes ∗k −1 0 and ∗ k −1 1 , according to the dimension Φ(dc)=k. In Step 3, {d1, d2, … , dm} was partitioned into D' and D'' which contain nodes belonging to ∗k −1 0 and ∗ k −1 1 , respectively. In Step 4, P1, P2, … , P m were constructed according to four cases in which paths in ∗k −1 0 and ∗ k −1 1 need to be constructed in a recursive manner. In Case 1, |D'|−1 disjoint paths in ∗k −1 0 that connect { eΦ ( d2 ) , eΦ ( d3 ) , … , eΦ ( dr ) } and D'−{d1} and |D''|+1 disjoint paths in ∗ k −1 1 that connect {eu( k ) , eΦ( k()dr +1 ) , eΦ( k()dr +2 ) , … , eΦ(k()d m ) } and D''∪{ d 1(k ) } were constructed. It. should be noted that ek in ∗ k −1 1 corresponds to s=0k in ∗k −1 0 (refer to Figure 1). In Case 2, |D'| disjoint paths in ∗k −1 0 that connect { eΦ ( d2 ) , eΦ ( d3 ) , … , eΦ ( dr ) } and D' and |D''|−1 disjoint paths in ∗ k −1 1 that connect { eΦ( k()dr +1 ) , eΦ( k()dr +2 ) , … , eΦ(k()d m ) } and D''−{d 1} were constructed. Since. d1=ek, P1 has length 0. In Case 3, |D'| disjoint paths in ∗k −1 0 that connect {eΦ ( d2 ) , eΦ ( d3 ) , … , eΦ ( dr ) } and D' and |D''| disjoint paths in ∗ k −1 1 that connect { eα( k ) , eΦ( k()dr +1 ) , eΦ( k()dr +2 ) , … , eΦ(k()d m ) } and D'' were constructed.. In Case 4, |D''|−1 disjoint paths in ∗ k −1 1 that connect { eΦ( k()dr +1 ) , eΦ( k()dr +2 ) , … , eΦ(k()d m ) } and D''−{d1} were first constructed. The subsequent construction depends on whether d1 is contained in these |D''|−1 paths or not. If d1 is not contained in these |D''|−1 paths, then |D'|+1 disjoint paths in ∗k −1 0 that connect {eu, eΦ ( d2 ) , eΦ ( d3 ) , … , eΦ ( dr ) } and D'∪{ d 1(k ) } were constructed. If d1 is contained in one of these |D''|−1 paths which connects e (Ωk'')( dl ) and dl and d l(k ) ∉D', then |D '|+1 disjoint paths in ∗k −1 0 that connect. { e Ω ''( dl ) , eΦ ( d2 ) , eΦ ( d3 ) , … , eΦ ( dr ) } and D'∪{ d l(k ) } were constructed. If d1 is contained in one of these |D''|−1 paths that connects e (Ωk'')( dl ) and dl and d l(k ) ∈D', then after substituting Ψ for Φ, the situation is the same as Case 3 and P1, P2, … , Pm can be obtained likewise. The maximal length of the m disjoint paths from s to d1, d2, … , dm is computed as follows. Theorem 1. Suppose that s, d1, d2, … , dm are arbitrary m+1 distinct nodes of a k-dimensional hypercube, where m≤k and k≥2. There are m disjoint paths from s to d1, d2, … , dm, respectively, whose maximal length is not greater than k+1 if m=k, and not greater than k if m<k. The maximal length is minimized in the worst case. The proof of Theorem 1 is presented in the next section..

(5) ∗k −1 1. ∗k −1 0 eΦ( k()d r +1 ). eΦ ( d r +1 ). eΦ( k()d r +2 ). eΦ ( d r+2 ) eΦ ( d m ). eΦ( k()d m ). eu. eu(k ). ek. s. Figure 1. ek in ∗k −1 1 corresponds to s = 0k in ∗k −1 0.. 4. Proof of Theorem 1 We need to show two properties of P1, P2, … , Pm: (P1) P1, P2, … , Pm are disjoint, and (P2) for all 1≤i≤m, if d i ,Φ ( di ) = 1 , then Pi has length |di|−1, and if d i ,Φ (d i ) = 0 , then Pi has length |di|+1 and |ti|=|di|+1, where ti is the immediate predecessor of di in Pi. According to (P2), Pi has length k−1 if |di|=k, and at most k if |di|<k. Hence, Pi has length at most k. However, when m<k, Pi has length at most k−1, as explained below. If 1k ∈ {d1, d2, … , dm}, then Pi has length at most k−1, for otherwise (Pi has length k) we have |ti|=|di|+1=(k−1)+1= k (i.e., ti=1k) according to (P2). This contradicts to (P1). On the other hand, if 1k ∉{d1, d2, … , dm}, then after adding a new node dm+1=1k to D and a new dimension u ∈{1, 2, … , k}−I to I, Paths(Φ, m+1, k, D, I) can produce P1, P2, … , Pm+1. Since 1k ∈{d1, d2, … , dm+1}, P1, P2, … , Pm+1 have lengths at most k−1. Consequently, the m disjoint paths from s to d1, d2, … , dm have maximal length k+1 if m=k, and k if m<k. Since the diameter of a k-cube is k, the maximal length of the m disjoint paths is at least k in the worst case. It was shown in [11] that the maximal length is at least k+1 in the worst case when m=k. Hence the maximal length in Theorem 1 is minimized in the worst case. (P1) and (P2) can be verified by induction on k. The detailed proof my be found in [21].. 5. When {d1, d2, … , dm} is a multiset A multiset is a collection of elements in which multiple occurrences of the same element are allowed [16]. Paths(Φ,. m, k, D, I) can deal with a multiset D, if Step 4 is modified as follows. Let T={dj | dj=ek and r+1≤j≤m}. In Case 2, Ψ'' is changed to Ψ'': {dr+1, dr+2, … , dm}−T → {nr+1, nr+2, … , nm}−{Φ(dj) | dj ∈T and r+1≤j≤m} so that Ψ''(di)=Φ(di)=ni for all r+1≤i≤m and di≠ek. Instead of executing Paths(Ψ'', m−r, k−1, {dr+1, dr+2, … , dm}, {nr+1, nr+2, … , nm}), we execute Paths(Ψ'', m−r−|T|, k−1, {dr+1, dr+2, … , dm}−T, {nr+1, nr+2, … , nm}−{Φ(dj) | dj ∈T and r+1≤j≤m}), in order to construct m−r−|T| disjoint paths in ∗k−11 that connect { en( kr +)1 , e (nkr +)2 , … , ) (k ) e (k nm }−{ eΦ ( d ) | d j ∈T and r+1≤j≤m} and {d r+1, d r+2, … , j. dm}−T. Additionally, we construct another |T| disjoint paths of length one that connect eΦ( k()d ) and dj for all dj ∈T. j. In Case 4, the conditions for the second and third situations are changed to "d1 ∈Pl for some r+1≤l≤m, d1 ∉{dr+1, dr+2, … , dm}, and d l(k ) ∉{d2, d3, … , dr}" and "d1 ∈Pl for some r+1≤l≤m, d1 ∉{dr+1, dr+2, … , dm}, and d l(k ) ∈{d2, d3, … , dr}", respectively. One more situation whose condition is "d1 ∈Pl for some r+1≤l≤m and d1 ∈{dr+1, dr+2, … , dm}" needs to be added. The new situation constructs P1, P2, … , Pr in ∗k −1 0 all the same as the first situation (i.e., d1 ∉Pi for all r+1≤i≤m). Both (P1) and (P2) remains correct after the modifications above, as shown in [21]. When {d1, d2, … , dm} is a multiset, Theorem 1 can be rewritten as follows. Theorem 2. Suppose that s, d1, d2, … , dm are arbitrary m+1 nodes of a k-dimensional hypercube so that s ∉{d1, d2, … , dm} and {d1, d2, … , dm} is a multiset, where m≤k and k≥2. There are m disjoint paths from s to d1, d2, … , dm, respectively,.

(6) whose maximal length is not greater than k+1 if m=k, and not greater than k if m<k. The maximal length is minimized in the worst case.. 6. Discussion and conclusion The main contribution of this paper is to show the effectiveness of routing functions in deriving one-to-many disjoint paths in networks. By using a minimal routing function, m disjoint paths whose maximal length is minimized in the worst case can be obtained in a k-cube, where m≤k. The problem of finding a minimal routing function can be reduced to the problem of finding a maximum matching in a weighted bipartite graph. Suppose G=(U, V, E) is a weighted bipartite graph, where U and V are two partite sets of nodes and E is the set of weighted links (each link of G is assigned a weight). A subset of E forms a matching in G if no two of them share a common node. A matching in G is maximum if its total weight is maximum. A maximum matching in G can be found in O(|U∪V|(|E|+|U∪V|log|U∪V|)) time (see [14]). Suppose Φ: {d1, d2, … , dm} → {n1, n2, … , nm} is a routing function, where {d1, d2, … , dm} is a multiset. Let U={d1, d2, … , dm}, V={n1, n2, … , nm}, and E={(di, nj) | 1≤i≤m and 1≤j≤m}. We assign each link (di, nj) a weight ( k + 1) k −|di |+1 if d i ,n j = 1 , and 1 if d i ,n j = 0 . A maximum matching M in G contains m links and its total weight can be expressed as a1∗(k+1)k+a2∗(k+1)k−1+ … +ak∗(k+1)+ak+1, where 0≤ar≤m for all 1≤r≤k+1. That is, there are ar links in M whose weights are ( k + 1) k − r +1 (or equivalently, M contains av links (di, nj) of weight ( k + 1) k − v +1 with d| i|=v and d i ,n j = 1 for all 1≤v≤k, and ak+1 links (di, nj) of weight 1 with d i ,n j = 0 ). For all 1≤v≤k, let rv denote the number of di's with |di|=v and di ∈U. A minimal Φ with VΦ= (r1−a1, r2−a2, … , rk−ak) can be obtained from M as follows: Φ(di)=nj if (di, nj) ∈M. If Φ is not minimal, then there exists VΦ'<VΦ for some routing function Φ': {d1, d2, … , dm} → {n1, n2, … , nm}, which implies another matching in G whose total weight is greater than the total weight of M. This is a contradiction. Besides minimal routing functions, there are some other routing functions that can be used to produce disjoint paths with different properties. For example, m disjoint paths whose total length is minimized can be also produced in a kcube, if an implicit routing function in [15] is used. In [15], m non-empty subsets X1, X2, … , Xm of {1, 2, … , k} were used to represent m nodes d1, d2, … , dm, respectively, so that for all 1≤u≤m and 1≤w≤k, w ∈Xu if and only if du,w=1. A set of c distinct integers t1 ∈ X h1 , t2 ∈ X h2 , … , tc ∈ X hc is called a partial System of Distinct Representatives (SDR for short) for {X1, X2, … , Xm} if h1, h2, … , hc are all distinct, where c≤m and 1≤hi≤m for all 1≤i≤c. Further, the partial SDR {t1, t2, … , tc} is maximum if c is maximized. When c=m, {t1, t2, … , tc} is called an SDR. A maximum partial SDR {t1, t2, … , tc} can. be used to construct m disjoint paths in a k-cube whose total length is minimized, if there is no j ∈{1, 2, … , m}− {h1, h2, … , hc} satisfying the following two conditions: (C1) Xj ⊂ X hi for some 1≤i≤c, and (C2) there exists an SDR for { X h1 , X h2 , … , X hi −1 , X j , X hi +1 , … , X hc }. Such a maximum partial SDR can be determined in O(k2.5) time (see [15]). Actually, a maximum partial SDR {t1, t2, … , tc} can be regarded as a routing function Φ: D → I so that {t1, t2, … , tc} ⊆ I and Φ( d hi )=ti for all 1≤i≤c. Suppose LΦ={u | d u,Φ ( d u ) = 1 and 1≤u≤m}. It follows that Paths(Φ, m, k, D, I). can result in m disjoint paths from s to d1, d2, … , dm whose total length is minimized, provided |LΦ| is maximized and there is no j ∈{1, 2, … , m}−LΦ satisfying the following two conditions: (C1 ') dj≠dl and dj,w≤dl,w for some l ∈LΦ and all 1≤w≤k, and (C2') there exists a routing function Ψ with d j ,Ψ ( d j ) = d v Ψ , (d v ) = 1 for all v ∈LΦ −{l}. Here, a routing function Φ: D → I with maximum L | Φ| corresponds to a maximum partial SDR (LΦ corresponds to {h1, h2, … , hc} and {Φ(du) | u ∈LΦ} corresponds to {t1, t2, … , tc}). 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, … , m}−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, … , m}−{h1, h2, … , hc} 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. We have shown in Theorem 2 that the maximal length of the m disjoint paths from s to d1, d2, … , dm is minimum for the worst-case scenario. According to (P2), each path from s to di has length |di| (i.e. shortest) if d i ,Φ (d i ) = 1 , and |d i|+2 (i.e., second shortest) if d i ,Φ (di ) = 0 , where 1≤i≤m. It follows that for any given d1, d2, … , dm, the maximal length of the m disjoint paths from s to d1, d2, … , dm is equal to max{|di| | 1≤i≤m} or min{max{|di| | 1≤i≤m}+2, k+1}. It should be noted that P1, P2, … , Pm are all shortest. The node eΦ ( di ) (or eΨ (di ) for the situation of d1 ∈Pl for some r+1≤l≤m and d l(k ) ∈{d2, d3, … , dr} in Case 4 of Step 4) is the immediate successor of s in the path to di. When d i ,Φ (d i ) = 1 , eΦ ( di ) (or eΨ (di ) ) is contained in a shortest path from s to di. When d i ,Φ (di ) = 0 , eΦ ( di ) is not contained in any shortest path from s to di. A k-dimensional folded hypercube [13] is basically a kcube augmented with 2k−1 complement links. It was shown in [21] that using a minimal routing function, k+1 disjoint paths whose maximal length is minimized in the worst case can be constructed in a k-dimensional folded hypercube, where k+1 is the node connectivity. It is worth while exploring more.

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