The proof of Lemma 17

Θ

2 2

/ −

v k =v_^t_k/2−2 +ϕ, and v_^Θ_k/2−1= v_^t_k/2−1−δ, where ϕ=

|

^{di | _{, ( )} =0

di

di_Θ and 1≤i≤t}

|

^and δ=

|

^{qi |

) 0

( , _t q_i =

qi_Ω and 1≤i≤t}

|

. If v₁^t=v₁, v₂^t=v₂, … , v_h^t₋₁=v_h−1, v <v_h^t _h for some 1≤h≤k/2−3, then VΘ<

Ω0

V , which is a contradiction because Ω0 is minimal. On the other hand, if v₁^t=v₁, v₂^t=v₂, … , v_l^t₋₁=v_l−1, v >v_l^t _l for some 1≤l≤k/2−3, then we have v₁^t=v ,^Ψ₁ v₂^t=v , … ,^Ψ₂ v_l^t₋₁=v ,^Ψ_l−1 v >_l^t v , which implies^Ψ_l V_Ψ <

V_Ωt . This is also a contradiction because Ωt is minimal. Hence we have v =v_c^t _c for all 1≤c≤k/2−3.

Similarly, if v_^t_k/2_−2 <v_k/2_−2−t, then v_^Θk_/2−2 =v_^tk_/2−2+ϕ<(v_{k/2−2}−t)+ϕ≤v_{k/2−2}, which implies

Θ <

V VΩ0, a contradiction, because v =^Θ_c v =v_c^t _c for all 1≤c≤k/2−3. If v_^t_k/2−2>v_{k /2−2}−t, then v^Ψ_{k/2−2}= v_{k /2−2}−t<v_^t_k/2−2, which implies V_Ψ <

V_Ωt , a contradiction, because v =v^Ψ_{c c}=v_c^t for all 1≤c≤k/2−3. Hence we have v_^t_k/2−2=v_{k /2−2}−t.

Finally, if v_^t_k/2−1<v_{k/2−1}, then v_^Θ_k/2−1= v_^tk_/2−1−δ<v_{k/2−1}. Since v_^Θ_k/2−2 = v_^tk_/2−2+ϕ=v_{k /2−2}−t+ϕ, we have ϕ=t because Ω0 is minimal. Then, v_^Θ_k/2−2 = v_^tk_/2−2+ϕ=^vk/2−2−t+ϕ=^vk /2−2. Since v^Θ_c = v_c^t =v_c for all 1≤c≤k/2−3, we haveV_Θ<

Ω0

V , a contradiction. If v_^t_k/2−1>v_{k/2−1}, then v^Ψ_{k/2−1} =v_{k /2−1}<v_^t_k/2−1, which implies V_Ψ <

V_Ωt , a contradiction, because v^Ψ_{k/2−2} =v_{k /2−2}−t= v_^tk_/2−2 and v^Ψ_{c =}v_c= v_c^t for all

1≤c≤k/2−3. Hence we have v_^t_k/2−1 =v_{k /2−1}. ¨

The following lemma will be proved in Section 3.2.2.

Lemma 17. Suppose that s, d₁, d₂, … , d_{k +1} are k+2 distinct nodes of a k-fcube. Procedure Paths2(s, d₁, d₂, … , d_{k +1}) can produce k+1 disjoint paths from s to d₁, d₂, … , d_{k +1} in the k-fcube whose lengths do not exceed k/2+1.

3.2.2 The proof of Lemma 17

There are two steps, i.e., Step 1 and Step 2, in Paths2(s, d₁, d₂, … , d_{k +1}). Step 1 is easy and no further explanation is needed. Step 2 constructs k+1 disjoint paths, denoted by Q₁, Q₂, … , Q_{k +1}, according to five cases. In the rest of this section, we show that Q₁, Q₂, … , Q_{k +1} are disjoint and have maximal length not greater than k/2+1.

3.2.2.1 Case 1: |d_k+1|≤k/2−2

Q₁, Q₂, … , Q_k were first obtained by executing Paths1(Φ, k, k, {d1, d₂, … , d_k}, {1, 2, … , k}). Lemma 8 assures their disjoint property. Besides, since |d_i|≤|dk +1|≤k/2−2 for all 1≤i≤k as a consequence of Step 1, each Q_i has length at most |d_i|+2≤k/2. Then Q_{k +1} was constructed and Q_l was changed, depending on whether d_{k +1}∈Ql for some 1≤l≤k or not.

When d_{k +1} ∈Ql, Q_{k +1} was constructed as the subpath of Q_l from s to d_{k +1}, and then Q_l was reconstructed as the combination of a link (s, s), a shortest path from s to d_l in a k-cube, and a link (d_l , d_l). Q_{k +1} has length at most k/2−1, and (new) Ql has length (k−|dl |)+2≤k/2. Qk +1 is disjoint with Q₁, … , Q_l−1, Q_l+1, … , Q_k. Suppose that α is a node in Q₁, … , Q_l−1, Q_l+1, … , Q_{k +1} and β ∉{s, dl} is a node in (new) Q_l. (New) Q_l is disjoint with Q₁, … , Q_l−1, Q_l+1, … , Q_{k +1}, provided α≠β. We have α≠β because

|α|≤|dk +1|+1≤k/2−1<|dl |≤|β|.

When d_{k +1}∉Qi for all 1≤i≤k, Qk +1 was constructed as the combination of a link (s, s), a shortest path from s to d_k+1 in a k-cube, and a link (d_k+1, d_{k +1}). Q_{k +1} has length (k−|d_k+1|)+2≤k/2. Suppose that γ is a node in Q₁, Q₂, … , Q_k and η ∉{s, dk +1} is a node in Q_{k +1}. Q_{k +1} is disjoint with Q₁, Q₂, … , Q_k because |γ|≤

|d_{k +1}|+1≤k/2−1<|dk+1|≤|η|.

3.2.2.2 Case 2: |d_k+1|=k/2−1

Q₁, Q₂, … , Q_{k +1} were obtained, depending on whether k is even or odd. If k is even, then Q₁, Q₂, … , Q_{k +1} were constructed all the same as in Case 1. Similarly, they are disjoint to each other, and their maximal length is at most|d_{k +1}|+2=k/2+1. If k is odd, then Q1, Q₂, … , Q_k were obtained by executing Paths1(Φ, k, k, {d₁, d₂, … , d_k}, {1, 2, … , k}). By Lemma 8, they are disjoint to each other, and their maximal length is at most |d_{k +1}|+2=k/2+1. Qk +1 was constructed, depending on whether d_{k +1}∈Ql for some 1≤l≤k or not.

When d_{k +1}∈Ql, Q_{k +1} was constructed as a subpath of Q_l from s to d_{k +1}. Q_{k +1} has length at most k/2

and is disjoint with Q₁, … , Q_l−1, Q_l+1, … , Q_k. Then Q_l needs to be changed, depending on whether d_{k +1} is the immediate predecessor of d_l in Q_l or not.If d_{k +1} is the immediate predecessor of d_l in Q_l, then Q_l was reconstructed as the combination of a link (s, s), a shortest path from s to d_l in a k-cube, and a link (d_l , d_l). Since d_l and d_{k +1} are adjacent and |d_l|≤|dk +1|, we have |d_l|=|d_{k +1}|−1=k/2−2. Hence, (new) Ql has length (k−|dl |)+2=k/2. Define α and β all the same as in Section 3.2.2.1. (New) Q_l is disjoint with Q₁, … , Q_l−1, Q_l+1, … , Q_{k +1} because |α|≤|dk +1|+1=k/2<k/2+2=|d_l |≤|β|.

On the other hand, if d_{k +1} is not the immediate predecessor of d_l in Q_l, then Q_l was reconstructed as the combination of a link (s, s), a shortest path from s to x in a k-cube, and a link (x, d_l), where x is the immediate predecessor of d_l in (old) Q_l. (New) Q_l has length k/2+1, as explained below. Since dk +1∈(old) Q_l, (old)Q_l has length at least |d_{k +1}|+2=k/2+1. Also, (old) Ql has length at most |d_l|+2≤|dk +1|+2=k/2+1 by Lemma 8. Hence, (old) Q_l has length exactly k/2+1. Again, by Lemma 8, |dl|=k/2−1 and |x|=|dl|+1=

k/2. The length of (new) Ql is equal to (k−|x|)+2=k/2+1. Suppose that α has the same meaning as in Section 3.2.2.1 and τ∉{s, x, dl} is a node in (new) Q_l. Clearly, we have α≠x. Since |α|≤|dk +1|+1=k/2=|x|<

|τ|, we have α≠τ. Hence (new) Ql is disjoint with Q₁, … , Q_l−1, Q_l+1, … , Q_{k +1}.

When d_{k +1}∉Qi for all 1≤i≤k, Qk +1 was constructed as the combination of a link (s, s) and a shortest path from s to d_{k +1} in a k-cube, which has length (k−|dk +1|)+1=k/2+1. In case Qk +1 intersects with Q_h for some 1≤h≤k, they need to be changed as follows. Qk +1 was reconstructed as the combination of the subpath of Q_h from s to y and a shortest path from y to d_{k +1}, where y (≠s) is the intersection of (old) Q_{k +1} and Q_h. Q_h was reconstructed as the combination of a link (s, s), a shortest path from s to z in a k-cube, and a link (z, d_h), where z is an adjacent node of d_h so that |z|=|d_h|+1 and z∉(old) Q_i for all 1≤i≤k. In the following, we show the existence of z.

We first show |d_h|=k/2−1 as follows. Since y ∈(old) Q_h, we have |y|≤|dh|+1≤|dk +1|+1. Further, we have y≠dk +1 because d_{k +1}∉(old) Q_h. On the other hand, since y∈(old) Q_{k +1} and the subpath of (old) Q_{k +1} from s to d_{k +1} is shortest, we have |y|≥|dk +1|+1. Consequently, we have |y|=|d_{k +1}|+1, and hence |d_h|=|d_{k +1}|=

k/2−1. Suppose that qt's, 1≤t≤k−|dh|+1, are the k−|dh|+1 distinct adjacent nodes of d_h in a k-fcube with

|q_t|=|d_h|+1 (one of q_t's is d_h ). A node q_w∉(old) Q_i for all 1≤i≤kcan serve as z, where 1≤w≤k−|dh|+1. Since

|d_v|≤|dk +1|=|d_h|<|d_h|+1=|q_t|, each (old) Q_v with dh_,Φ(d_v) =0 contains at most one q_t, where 1≤v≤k. On the other hand, as explained below, each (old) Q_u with d_h,Φ(du) =1 does not contain any q_t, where 1≤u≤k.

Since |y|=|d_h|+1, we have d_h,Φ(dh) =0 as a consequence of Lemma 8. Lemma 11 assures |d_u|≤|dh| and

) 1

(

, d_u =

du_Φ . By Lemma 8, (old) Q_u is shortest with length |d_u|. Since |d_u|≤|dh|<|d_h|+1=|q_t|, (old) Q_u does not contain q_t. Since there are only k−|dh| (old) Q_v's with d_h,Φ(dv) =0, a q_was required can be found.

Since y is adjacent to d_{k +1} and y≠dh, (new) Q_{k +1} has length at most 1+(|d_h|+1)=k/2+1. (New) Qh has length (k−|z|)+2=k/2+1. (New) Qk +1 is disjoint with Q₁, Q₂, … , Q_h−1, Q_{h +1}, … , Q_k. Suppose that λ is a node in Q₁, … , Q_h−1, Q_h+1, … , Q_k, (new) Q_{k +1} and ρ ∉{s, z, dh} is a node in (new) Q_h. (New) Q_h is disjoint

with Q₁, Q₂, … , Q_h−1, Q_h+1, … , Q_k, (new) Q_{k +1} because z is not contained in Q₁, Q₂, … , Q_h−1, Q_{h +1}, … , Q_k, (new) Q_{k +1} and |λ|≤|dk +1|+1=k/2=|z|<|ρ|.

3.2.2.3 Case 3: |d_k+1|=k/2

Q'₁, Q'₂, … , Q'_r, Q_r+1, … , Q_k were first obtained by executing Paths1(Ω, k, k, {d ,₁ d , … ,₂ d , d_{r r+1}, … , d_k}, {1, 2, … , k}), where Q'_i ends at d_i for all 1≤i≤r. That Ω is minimal was assured by Lemma 12 (substituting r, d_i , and Ω for h, d'i, and Ψ, respectively). By Lemma 8, Q'1, Q'₂, … , Q'_r, Q_r+1, … , Q_k are disjoint to each other. Besides, each Q'_i is shortest with length |d_i|=k/2 because _{, ( )} = _{, ( )} =1

i

i i d

i d d

d _{Ω Φ} .

For all r+1≤j≤k, if |dj|≤k/2−1, then Qj has length at most |d_j|+2≤k/2+1. If |dj|=k/2, then dj_,Ω(d_j) =1 (by our assumption), which implies by Lemma 8 that Q_j is shortest with length |d_j|=k/2. We note

|d_j|≤|dk +1|= k/2.

For all 1≤i≤r, Qi was constructed as the combination of the subpath of Q'_i from s to f_i and two links (f_i, fi ) and ( f_i , d_i) if d_i =d_h for some r+1≤h≤k+1, and the combination of Q'i and a link (d_i, d_i) else (refer to Figure 3), where f_i is the immediate predecessor of d_i in Q'_i. Q_i has length k/2+1. Q1, Q₂, … , Q_k are disjoint to each other, provided that Q_i is disjoint with Q_r+1, Q_r+2, … , Q_k and Q₁, Q₂, … , Q_r are disjoint to each other. We first show that Q_i is disjoint with Q_r+1, Q_r+2, … , Q_k. It suffices to show that d_i is not contained internally in Q_r+1, Q_r+2, … , Q_k and f_i is not contained in Q_r+1, Q_r+2, … , Q_k. The former is assured by Lemma 13 (substituting r, d_i , and Ω for h, d'i, and Ψ, respectively). The latter can be shown as follows. Suppose that α is a node in Q_j, where r+1≤j≤k. When |dj|=k/2, we have |α|≤k/2 because Qj

is shortest with length k/2. When |dj|≤k/2−1, we have |α|≤|dj|+1≤k/2. Since | f_i |=k−(|d_i|−1)=k/2+1>

|α|, f_i is not contained in Q_r+1, Q_r+2, … , Q_k.

We then show that Q₁, Q₂, … , Q_r are disjoint to each other. It suffices to show that f ,₁ f , … ,₂ f_r are all distinct, {f ,₁ f , … ,₂ f }∩{d_r 1, d₂, … , d_r} is empty, and each of f ,₁ f , … ,₂ f , d_{r 1}, d₂, … , d_r is not contained in Q'₁, Q'₂, … , Q'_r. Clearly, f ,₁ f , … ,₂ f_r are all distinct. Since | f_i |=k/2+1 and

|d_i|=k/2 for all 1≤i≤r, we have {f ,₁ f , … ,₂ f }∩{d_r 1, d₂, … , d_r} empty. Besides, each of f ,₁ f , … ,₂ fr is not contained in Q'₁, Q'₂, … , Q'_r because Q'₁, Q'₂, … , Q'_r are all shortest with length k/2. Since Lemma 13 assures that each of d₁, d₂, … , d_r is not contained internally in Q'₁, Q'₂, … , Q'_r, we only need to

show {d₁, d₂, … , d_r}∩{d ,₁ d , … ,₂ d } empty. Suppose conversely d_{r i}=d , where 1≤i≤r, 1≤j≤r, and i≠j. _j It follows by Lemma 3 (d_i,Φ(di) =0, dj_,Φ(d_j) =0, and 1

) , ( ) (

, = =

j j dj

d

i d

d _{Φ Φ} ) that Φ is not minimal, which is a contradiction.

Q_{k +1} was constructed as the combination of a link (s, s) and a shortest path from s to d_{k +1} in a k-cube whose length is (k−|dk +1|)+1≤k/2+1. Qk +1 is not necessarily disjoint with Q₁, Q₂, … , Q_k. As stated below, there are two situations in which some of Q₁, Q₂, … , Q_{k +1} need to be changed, in order to satisfy the disjoint property. The two situations have to be treated sequentially.

The first situation is that Q_{k +1} intersects with Q_t for some 1≤t≤r at a node x (≠s). Qt and Q_m need to be changed as follows: Q_t was reconstructed as the combination of the subpath of Q_m from s to g and two links (g, g ) and ( g , d_t), and Q_m was reconstructed as the combination of the subpath of (old) Q_t from s to

x and a shortest path from x to d_m, where d_t =d_m for some r+1≤m≤k and g is the immediate predecessor of d_m in (old) Q_m. That x (=f_t) is contained in (old) Q_t and d_m surely exists is explained below.

According to the construction of (old)Q_t, we have x≠dk +1 and f_t∈(old)Q_t. Besides, |x|≤k/2+1=| f_t | if dt=d_h for some r+1≤h≤k+1, and |x|≤k/2 else. On the other hand, since x ∈Qk +1, we have |x|≥|dk +1|+1=

k/2+1. Hence |x|=k/2+1, which implies that x= f_t (hence x=f_t) and x is the immediate predecessor of d_{k +1} in Q_{k +1}. Since f_t =x∈(old)Q_t, we have d_t=d_h. We have d_t≠dk +1, for otherwise (d_t=d_{k +1}) both f_t and f_t are adjacent to d_{k +1}, which is impossible because k≥4. Hence d_t =d_m for some r+1≤m≤k.

Since x=f_t and d_m=d_t , (new) Q_m is identical with Q'_t. Hence (new) Q_m is shortest with length k/2.

(New) Q_t has length greater by one than that of (old) Q_m. (Old) Q_m is shortest with length k/2, as explained below. By Lemma 9 (substituting d_t and d_m for d_h and d_l, respectively), we have d_m,Φ(dm) =1. Hence, d_m,Ω(dm) = d_m,Φ(dm) =1 and by Lemma 8 (old) Q_m is shortest with length |d_m|=k/2. Besides, |g|=

|d_m|−1=k/2−1. That Q1, … , Q_t−1, (new) Q_t, Q_t+1, … , Q_m−1, (new) Q_m, Q_m+1, … , Q_k are disjoint to each other is mainly due to the construction method. More reasons are supplemented as follows. (New) Q_m is disjoint with Q₁, … , Q_t−1, Q_t+1, … , Q_m−1, Q_m+1, … , Q_k, as a consequence of (new) Q_m identical with Q'_t. Since (old) Q_m is disjoint with Q'₁, Q'₂, … , Q'_r, we have g≠fi (hence g≠ fi ) for all 1≤i≤r. (New) Qt is disjoint with Q₁, … , Q_t−1, Q_t+1, … , Q_m−1, Q_m+1, … , Q_k, as a consequence of g≠ f_i and |g |=k/2+1>|α|.

(New) Q_t and (new) Q_m are disjoint because g∉(new) Q_m.

Q_{k +1} is disjoint with Q₁, … , Q_t−1, (new) Q_t, Q_t+1, … , Q_r, as explained below. Suppose that β ∉{s, x, d_{k +1}} is a node in Q_{k +1} and γ is a node in Q₁, … , Q_t−1, (new) Q_t, Q_t+1, … , Q_r. The construction method assures |β|>|x|=k/2+1 (recall that x is the immediate predecessor of dk +1 in Q_{k +1}) and |γ|≤k/2+1. Hence we have β≠γ. The construction method also assures that x and d_{k +1} are not contained in Q₁, … , Q_t−1, (new) Q_t, Q_t+1, … , Q_r (recall x= f_t ≠g ).

To sum up, Q₁, Q₂, … , Q_k that we obtained with the first situation taken into consideration are disjoint to each other. Besides, Q_{k +1} is disjoint with Q₁, Q₂, … , Q_r. It should be noted that Q_{k +1} intersects with at most one of (old)Q₁, (old)Q₂, … , (old)Q_r. If Q_{k +1} intersects with another (old)Q_w (1≤w≤r and w≠t) at a node x' (∉{s, x}), in addition to (old)Q_t, then we have |x'|=k/2+1=|x|. Since both x and x' are contained in Q_{k +1}, we have x=x', which is a contradiction.

The second situation is that Q_{k +1} intersects with Q_l for some r+1≤l≤k. Since |α|≤k/2=|d_{k +1}|, the intersection of Q_{k +1} and Q_l must be d_{k +1}. Q_{k +1} was reconstructed as the subpath of Q_l from s to d_{k +1} whose length is at most k/2. (New) Qk +1 is disjoint with Q₁, Q₂, … , Q_l−1, Q_l+1, … , Q_k. Q_l needs to be reconstructed, depending on whether k is even or odd. If k is even, then Q_l was reconstructed as the combination of a link (s, s), a shortest path from s to d_l in a k-cube, and a link (d_l , d_l), which has length (k−|d_l |)+2=|d_l|+2. We have |d_l|=k/2−1, as explained below. It was shown that (old) Ql has length at most k/2+1. Also, (old) Ql has length at least |d_{k +1}|+1=k/2+1 because dk +1∈(old) Q_l. Hence (old) Q_l has length exactly k/2+1. Since |dl|≤k/2, we have |dl|=k/2−1 as a consequence of Lemma 8.

(New) Q_l is disjoint with Q₁, … , Q_l−1, Q_{l +1}, … , Q_k, (new) Q_{k +1}, as explained below. Suppose that η

∉{s, dl , d_l}is a node in (new) Q_l and τ is a node in Q1, Q₂, … , Q_r. We have |η|>|d_l |=k/2+1=k/2+1.

According to the construction method, we have |τ|=k/2+1 if τ∈{ f ,₁ f , … ,₂ f ,_r g }, and |τ|<k/2+1 else. Besides, we have d_l∉{f1, f₂, … , f_r, g} (equivalently, d_l ∉{ f ,₁ f , … ,₂ f ,_r g }). Hence (new) Q_l is disjoint with Q₁, Q₂, … , Q_r. Since |α|≤k/2, (new) Ql is disjoint with Q_r+1, … , Q_l−1, (old) Q_l, Q_l+1,… , Q_k, which also implies that (new) Q_l and (new) Q_{k +1} are disjoint.

If k is odd, then Q_l was reconstructed as the combination of a link (s, s), a shortest path from s to y in a k-cube, and a link (y, d_l), where y is an adjacent node of d_l so that |y|=|d_l|+1 and y∉(old) Q_i for all 1≤i≤k. We have |dl|=k/2−1 similarly. (New) Ql has length (k−|y|)+2=k/2+1. Suppose that qc's,

1≤c≤k−|dl|+1, are k−|dl|+1 distinct adjacent nodes of d_l in a k-fcube with |q_c|=|d_l|+1 (one of q_c's is d_l ). In the following, we show the existence of y by showing the existence of a q_w∉(old) Q_i for all 1≤i≤k, where 1≤w≤k−|dl|+1.

We first show that each (old) Q_u with dl_,Φ(d_u) =1 does not contain any q_c, where 1≤u≤k. Since dk +1∈ (old) Q_l and |d_{k +1}|=|d_l|+1, we have d_l,Ω(dl) =0 by Lemma 8. By Lemma 11 (d_l,Φ(dl)=d_l,Ω(dl)=0), we have

|d_u|≤|dl| and d_u,Φ(du) =1. By our assumption, we have r+1≤u≤k, and hence d_u,Ω(du) =d_u,Φ(du) =1. By Lemma 8, (old) Q_u is shortest with length |d_u|. Since |d_u|≤|dl|<|d_l|+1=|q_c|, (old) Q_u does not contain q_c. Then we show that each (old) Q_v with dl_,Φ(d_v) =0 contains at most one q_c, where 1≤v≤k. It suffices to show that each (old) Q_v with d_l,Φ(dv) =0 contains at most one node, say ρ, with |ρ|=k/2=|qc|. When 1≤v≤r, each (old) Q_v contains exactly one ρ according to the construction method. When r+1≤v≤k, it was shown that if

|d_v|= k/2, then (old) Qv is shortest with length k/2. Hence (old) Qv contains exactly one ρ. If

|d_v|≤k/2−1, then Lemma 8 assures that (old) Qv contains at most one ρ. Since there are k−|dl|+1 distinct q_c's but only k−|dl| (old) Q_v's with d_l,Φ(dv) =0, a q_w as required can be found.

Suppose that λ ∉{s, y, dl} is a node in (new) Q_l. Since |λ|>|y|=k/2≥|α| and y is not contained in Q₁, … , Q_l−1, (old) Q_l, Q_l+1,… , Q_k, we have (new) Q_l disjoint with Q_r+1, … , Q_l−1, Q_l+1,… , Q_k, (new) Q_{k +1} (recall that (new)Q_{k +1} is a subpath of (old)Q_l). If (new)Q_l intersects with Q_v for some 1≤v≤r at a node z (≠s), thenQ_v and Q_w need to be reconstructed in order that the final Q₁, Q₂,… , Q_{k +1} are disjoint, where d_v =d_w for some r+1≤w≤k and w≠l. We have z≠dl because (old)Q_l is disjoint with Q₁, Q₂, … , Q_r. That is, z is contained in the subpath of (new)Q_l from s to y. Since |y|=k/2=|dk +1|, the reconstruction of Q_v and Q_w can be made all the same as the reconstruction of Q_t and Q_m in the first situation that Q_{k +1} intersects with Q_t for some 1≤t≤r (substituting y, v, w, p, z, and the subpath of (new)Q_l from s to y for d_{k +1}, t, m, g, x, and Q_{k +1}, respectively).

Consequently, (new) Q_v and (new) Q_w have lengths k/2+1 and k/2, respectively. Besides, Q1, … , (new)Q_v, … , Q_l−1, Q_l+1, … , (new)Q_w, … , Q_k, (new)Q_{k +1} are disjoint (l<w is assumed), (new)Q_l is disjoint with Q₁, … , (new)Q_v, … , Q_r, and (new)Q_l is disjoint with (new) Q_w (because (new) Q_w has length

k/2<k/2=|y|<|λ|). We have d_v=d_w for some r+1≤w≤k with the same arguments as dt =d_m for some r+1≤m≤k in the first situation that Qk +1 intersects with Q_t for some 1≤t≤r. Further, d_v ≠dl (i.e., w≠l) was shown below.

Suppose conversely d =d. Since y is adjacent to d, either y=d or y is adjacent to d in a k-cube. If

y=d_l , then y=d_v, which contradicts to y ∉Qv. If y is adjacent to d_l in a k-cube, then d_l and d_l are at a distance of three at most in a k-cube, as explained below, which contradicts to our assumption of k≥4. It suffices to show a path (d_l, y, z, d_l ) in a k-cube. We have shown x= f_t above (for the situation that Q_{k +1} that were obtained after the mth modification, where 0≤m≤v−u and D⁽⁰⁾ and Ψ⁽⁰⁾ denote the initial D and Ψ, respectively. By Lemma 12 (substituting u, d_i , and Ψ⁽⁰⁾ for h, d'_i, and Ψ, respectively), Ψ⁽⁰⁾ is minimal.

Ψ^(m+1)(d_h)=Ψ^(m)(d_h)=Φ(dh). On the other hand, if Ψ^(m)(d_l) and Ψ^(m)(d_h) were swapped, then Ψ^(m+1)(p_l)=Φ(dl) and Ψ^(m+1)(d_h)=Φ(dh) remain valid after a slight modification of Φ, as explained below. We simply swap Φ(dl) and Φ(dh), and then we have Ψ^(m+1)(p_l)=(new)Ψ^(m)(d_l)=(old)Ψ^(m)(d_h)=(old)Φ(dh)=(new)Φ(dl)

Q_v+2, … , Q_k. Since |α|>|d_{k +1}|=k/2+1, |β|≤k/2, and |γ|<k/2+1, we have α≠β, α≠γ, d_{k +1}≠β, and d_{k +1}≠γ. For all 1≤i≤r, Qi was constructed, depending on whether d_i =d_w for some v+1≤w≤k or not. If di =d_w, then Q_i was constructed as the combination of the subpath of Q'_i from s to f_i and two links (f_i, f_i ) and ( f_i , d_i) (refer to Figure 4(a)), where f_i denotes the immediate predecessor of d_i in Q'_i. If f_i ∈Qk +1, then Q'_i and Q_w need to be swapped prior to the construction of Q_i. Q'_i is shortest with length k/2−1. By Lemma 9 (d_i,Φ(di) =0 and d_i =d_w), we have d_w,Φ(dw) =1. Since d_w,Ψ(dw) =d_w,Φ(dw) =1, Q_w is shortest with length

|d_w|=|d_i |=k/2−1 by Lemma 8. Hence Qi has length k/2. Qi is disjoint with Q'_r+1, … , Q'_v, Q_{v +1}, … , Q_{k +1}, as explained below.

We first assume f_i∉Qk +1. For all u+1≤t≤v, since Φ(dt)=Ψ(pt) and pt_,Ψ(p_t) =1, we have pt_,Φ(d_t) =1. By Lemma 13 (substituting v and d_i (or p_i) for h and d'_i, respectively), each of d₁, d₂, … , d_v is not contained internally in Q'₁, … , Q'_v, Q_{v +1}, … , Q_k. Since |f_i|=k/2+2 is greater than |β|, |γ|, and |d_c| for all v+1≤c≤k, f_i is not contained in Q'_r+1, … , Q'_v, Q_v+1, … , Q_k. Hence Q_i is disjoint with Q'_r+1, … , Q'_v, Q_v+1, … , Q_k. Suppose that τ∉{ f_i, d_i} is a node in Q_i. Since |τ|≤|f_i|=k/2−2<|α| and |τ|<|dk +1|, each node in Q_i−{ f_i, d_i} is not contained in Q_{k +1}. We have d_i ∉Qk +1 because |d_i|≤|dk +1|<|α|. Hence Q_i and Q_{k +1} are disjoint. If f_i∈ Qk +1, then Q'_i and Q_w were swapped and f_i was changed to be the immediate predecessor of d_i in (new) Q'_i ( f_i was changed accordingly). We have (new) f_i∉Qk +1, because (new)f_i ≠(old) f_i,

|(new) f_i|=|(old) f_i|, and the subpath of Q_{k +1} from s to d_{k +1} is shortest. With the same arguments, Q_i is disjoint with Q'_r+1, … , Q'_v, Q_v+1, … , Q_{k +1}.

On the other hand, if d_i ≠dj for all v+1≤j≤k, then Qi was constructed as the combination of Q'_i and a link (d_i, d_i) (refer to Figure 4(b)), which has length k/2. Qi is disjoint with Q'_r+1, … , Q'_v, Q_v+1, … , Q_{k +1}, provided d_i ∉{d_r+1, … , d_u , p_u+1, … , p_v, d_{k +1}} and d_i is not contained internally in Q'_r+1, … , Q'_v, Q_v+1, … , Q_{k +1}. We have d_i ∉{d_r+1, d_r+2 , … , d_u } because d_i∉{d_r+1, d_r+2, … , d_u}, d_i ∉{p_u+1, p_u+2, … , p_v} because {d ,₁ d , … ,₂ d_u}∩{pu+1, p_u+2, … , p_v} is empty, and d_i ≠dk +1 because |d |=k/2−1≠|d_i k +1|. We have shown above that d_i is not contained internally in Q'_r+1, … , Q'_v, Q_v+1, … , Q_k. Since |d_i|=k/2+1=|dk +1|<|α|, d_i is not contained internally in Q_{k +1}.

Next we show that Q₁, Q₂, … , Q_r are disjoint to each other. Without loss of generality, we assume

dc∈{dv+1, d_v+2, … , d_k} for all 1≤c≤m and d_c∉{dv+1, d_v+2, … , d_k} for all m+1≤c≤r, where 0≤m≤r. It suffices to show that f ,₁ f , … ,₂ f_m are all distinct, {f ,₁ f , … ,₂ f_m}∩{d1, d₂, … , d_r} is empty, and each of f ,₁ f , … ,₂ f_m, d₁, d₂, … , d_r is not contained in Q'₁, Q'₂, … , Q'_r. The first two hold because f₁, f₂, … , f_m are all distinct, | f_c|=k/2+2 for all 1≤c≤m, and |di|=k/2+1 for all 1≤i≤r. The last holds because Q'₁, Q'₂, … , Q'_r are all shortest with length k/2−1.

For all r+1≤j≤u, Qj was constructed as the combination of Q'_j and a link (d , d_{j j}), which has length

k/2+1. Qj is disjoint with Q₁, Q₂, … , Q_r, Q'_u+1, … , Q'_v, Q_v+1, … , Q_{k +1}, provided d_j ∉{d1, d₂, … , d_r, p_u+1, … , p_v, d_v+1, … , d_{k +1}} and d_j is not contained internally in Q₁, Q₂, … , Q_r, Q'_u+1, … , Q'_v, Q_{v +1}, … , Q_{k +1}. By our assumption (refer to the beginning paragraph enclosed by /* and */ in Case 4), we have d_j ∉{d1, d₂, … , d_k}. Since |d |=k/2≠k/2−1=|p_j t| for all u+1≤t≤v, we have d_j ∉{pu+1, p_u+2, … , p_v}. Also we have

dj ≠dk +1 because |d |≠|d_j k +1|. On the other hand, since d_j is not contained internally in Q'₁, … , Q'_v, Q_v+1, … , Q_k, we only need to show d_j∉{ f ,₁ f , … ,₂ f ,_r d ,₁ d , … ,₂ d } and d_{r j} is not contained internally in Q_{k +1}. The former holds because |d_j|=k/2≠k/2+2=|f_i | and |d_j|= k/2≠k/2−1=|d_i| for all 1≤i≤r. The latter holds because |d_j|<|α|. Further, Q_r+1, Q_r+2, … , Q_u are disjoint to each other because d_j ∉{d1, d₂, … , d_k} and d_j is not contained internally in Q'_r+1, Q'_r+2, … , Q'_u.

For all u+1≤t≤v, Qt was constructed as the combination of Q'_t and two links (p_t, p_t) and (p_t, d_t), which has length k/2+1 (refer to Figure 5). Qt is disjoint with Q₁, Q₂, … , Q_u, Q_v+1, Q_v+2, … , Q_{k +1}, provided p_t∉{d1, d₂, … , d_u, d_{v +1}, d_{v +2}, … , d_{k +1}}, p_t is not contained in Q₁, Q₂, … , Q_u, Q_v+1, Q_v+2, … , Q_{k +1}, and d_t is not contained internally in Q₁, Q₂, … , Q_u, Q_{v +1}, Q_v+2, … , Q_{k +1}. By our assumption, we have p_t∉ {d ,₁ d , … ,₂ d_u , d_v+1, d_v+2, … , d_k} and p_t∉{d ,₁ d , … ,₂ d_u , d_v+1, d_v+2, … , d_k, d_{k +1}}, from which p_t∉ {d₁, d₂, … , d_u} and p_t ∉{d1, d₂, … , d_u} are implied. Since |p_t|=k/2−1≠|dk +1|, we have p_t≠dk +1. Suppose that η ∉{d1, d₂, … , d_u, d_{v +1}, d_v+2, … , d_{k +1}} is a node in Q₁, Q₂, … , Q_u, Q_{v +1}, Q_{v +2}, … , Q_{k +1}. We have p_t≠η

because |p_t|=k/2+1≠|η| (as a consequence of our construction method). Since d_t is not contained internally in Q'₁, Q'₂, … , Q'_u, Q_v+1, Q_v+2, … , Q_k, d_t is not contained internally in Q₁, Q₂,… , Q_u, Q_v+1, … , Q_{k +1}, provided d_t ∉ { f ,₁ f , … ,₂ f ,_r d ,₁ d , … ,₂ d_u} and d_t is not contained internally in Q_{k +1}. The

former holds because |d_t|=k/2, | f_i |=k/2+2 and |d_i|=k/2−1 for all 1≤i≤r, and d_j ∉{d1, d₂, … , d_k} for all r+1≤j≤u (by our assumption). The latter holds because |dt|<k/2+1<|α|.

Q_u+1, Q_u+2, … , Q_v are disjoint to each other for the following reason. By our assumption, we have p_u+1, p_u+2, … , p_v all distinct (hence, p_u+1, p_u+2, … , p_v all distinct). For all u+1≤t≤v, since |pt|=k/2+1,

|d_t|=k/2, and Q't is shortest with length k/2−1, neither of p_t and d_t is contained in Q'_u+1, Q'_u+2, … , Q'_v. Besides, since |p_t|=k/2+1 and |dt|=k/2 for all u+1≤t≤v, we have {p_u+1, p_u+2, … , p_v}∩ {du+1, d_u+2, … , d_v} empty.

3.2.2.5 Case 5: |d_k+1|≥k/2+2

By carefully checking Section 3.2.2.4, it can be found that the execution of Case 4 remains correct even if there are the following two extensions: (1) |d_m|≤k/2+1 for all 1≤m≤k and |dk +1|≥k/2+1; (2) di=d_j is allowed if |d_i|=|d_j|<k/2, where 1≤i≤k+1 and 1≤j≤k+1. Since |qi|=k/2−1 for all 1≤i≤r', |d_j|≤k/2−2 for all r'+1≤j≤u', and |dc|≤k/2+1 for all u'+1≤c≤k, Q'1, … , Q'_u', Q_u'+1, … , Q_{k +1} can be obtained by the construction method of Case 4, where Q'_i ends at q_i for all 1≤i≤r' and Q'j ends at d_j for all r'+1≤j≤u'.

Besides, Q'₁, … , Q'_u', Q_u'+1, … , Q_{k +1} are disjoint to each other and their maximal length is not greater than

k/2+1. We also note that there is no complement link in Q'1, Q'₂, … , Q'_u'. Actually, Q'₁, Q'₂, … , Q'_u' have lengths smaller than k/2+1. Since Ω is minimal, Lemma 16 assures q_i,Ω(qi) =1 for all 1≤i≤r'. According to Lemma 8, Q'_i with 1≤i≤r' is shortest with length |qi|=k/2−1. Besides, Q'j with r'+1≤j≤u' has length not greater than |d_j|+2≤k/2.

For all 1≤i≤r', Qi was constructed as the combination of Q'_i and two links (q_i, q ) and (_i q , d_i i), which has length k/2+1 (refer to Figure 6). Qi is disjoint with Q'_r'+1, … , Q'_u', Q_u'+1, … , Q_{k +1}, provided q_i∉ {d_r'+1, … , d_u', d_u'+1, … , d_k, d_{k +1}}, q_i is not contained in Q'_r'+1, … , Q'_u', Q_u'+1, … , Q_{k +1}, and d_i is not contained internally in Q'_r'+1, … , Q'_u', Q_u'+1, … , Q_{k +1}. By our assumption (refer to the beginning paragraph enclosed by /* and */ in Case 5), we have q_i∉{d_r'+1, … , d_u', d_u'+1, … , d_k}. Also, q_i≠dk +1 because|q_i|<|d_{k +1}|.

Since |q_i |=k/2+1 and Q'r'+1, Q'_r'+2, … , Q'_u' have lengths at most k/2, qi is not contained in Q'_r'+1, Q'_r'+2, … , Q'_u'. q_i is also not contained in Q_u'+1, Q_u'+2, … , Q_k, as explained below.

We assume, without loss of generality, that Q_m+1, Q_m+2, … , Q_m+c contain p_m+1, p_m+2, … , p_m+c,

respectively (refer to Case 4 where Q_t was constructed as the combination of Q'_t and two links (p_t, p_t) and (p_t, d_t) for all u+1≤t≤v), where u'+1≤m+1≤m+c≤k. Suppose that α ∉{du'+1, d_u'+2, … , d_k}∪{p_m+1,

+2

pm , … , p_m+c} is a node in Q_u'+1, Q_u'+2, … , Q_k. We have q_i ≠α because |q_i |=k/2+1≠|α| (as a consequence of Case 4). By our assumption (refer to the beginning paragraphs enclosed by /* and */ in Case 4 and Case 5), we have q_i∉{pm+1, p_m+2, … , p_m+c}, i.e., q_i ∉{p_m+1, p_m+2 , … , p_m+c}, and q_i ∉{du'+1, d_u'+2, … , d_k}.

Then we show that d_i is not contained internally in Q'_r'+1, … , Q'_u', Q_u'+1, … , Q_{k +1}. Since |d_i|=k/2+2 and Q'_r'+1, Q'_r'+2, … , Q'_u' have lengths at most k/2, di is not contained internally in Q'_r'+1, Q'_r'+2, … , Q'_u'. We assume, without loss of generality, that Q_{m' +1}, Q_m'+2, … , Q_{m' +c'} contain f_m'+1, f_m'+2, … , f_m'+c' , respectively, where u'+1≤m'+1≤m'+c'≤k. Suppose that β ∉{ f_m'+1, f_m'+2, … , f_m'+c' } is a node in Q_u'+1, Q_u'+2, … , Q_k. We have |β|≤k/2+1 as a consequence of the execution of Case 4. Since |di|=k/2+2, we have d_i≠β. By our assumption, we have d_i =d_l(∉{fm'+1, f_{m' +2}, … , f_m'+c'}) for some u'+1≤l≤k, i.e., di∉{ f_m'+1,

+2 '

fm , … , f_m'+c' }. Since |d_i|≤|dk +1| and the subpath of Q_{k +1} from s to d_{k +1} is shortest, d_i is not contained internally in Q_{k +1}.

Q₁, Q₂, … , Q_r' are disjoint to each other, provided that q₁, q₂, … , q_r' are all distinct, q ,₁ q , … ,₂ q_r' are all distinct, each of q ,₁ q , … ,₂ q , d_{r' 1}, d₂, … , d_r' is not contained in Q'₁, Q'₂, … , Q'_r', and {q ,₁

q , … ,2 q }∩{d_r' 1, d₂, … , d_r'} is empty. By our assumption, we have q₁, q₂, … , q_r' all distinct (i.e., q ,₁ q , … ,2 q_r' all distinct). Since |q_i |=k/2+1 and |di|=k/2+2 for all 1≤i≤r', we have {q ,₁ q , … ,₂

'

q }∩{dr 1, d₂, … , d_r'} empty. Besides, since Q'_i is shortest with length |q_i|=k/2−1 for all 1≤i≤r', each of q ,1 q , … ,₂ q , d_{r' 1}, d₂, … , d_r' is not contained in Q'₁, Q'₂, … , Q'_r'.

For all r'+1≤j≤u', Qj was constructed depending on whether d =d_{j t} for some u'+1≤t≤k or not. If

dj =d_t, then Q_j was constructed as the combination of the subpath of Q'_j from s to g_j and two links (g_j, g ) _j and (g , d_{j j}), where g_j denotes the immediate predecessor of d_j in Q'_j. If g_j ∈Qk +1, then Q'_j and Q_t need to be swapped prior to the construction of Q_j. Since Q'_j has length at most k/2, Qj has length at most (k/2−1)+2=k/2+1. Qj is disjoint with Q₁, Q₂, … , Q_r', Q_u'+1, Q_u'+2, … , Q_{k +1}, as explained below.

If g ∉Qk +1, then it suffices to show that g is not contained in Q₁, Q₂, … , Q_r', Q_u'+1, Q_u'+2, … , Q_k

and d_j is not contained internally in Q₁, Q₂, … , Q_r', Q_u'+1, Q_u'+2, … , Q_{k +1}. We first show |g_j|≤k/2−2, i.e.,

|g |≥ _j k/2+2. Since there is no complement link in Q'j, we have |g_j|= |d |−1 or |_j d |+1. Since j

|d |≤k/2−2, we have |g_j j|≤k/2−1. If |gj|=k/2−1, then |d |=k/2−2. By Lemma 8 (|g_j j|=|d |+1), we have_j

) 0

, ( =

dj

dj

Ω . Consequently (|d |=k/2−2,_j d_j =d_t, and 0

)

, ( =

dj

dj

Ω ), a further modification of D and Ω is still required, which is a contradiction.

We show that g_j is not contained in Q₁, Q₂, … , Q_r', Q_u'+1, Q_u'+2, … , Q_k as follows. Since

|g |≥k/2+2 and Q'_j i is shortest with length |q_i|=k/2−1 for all 1≤i≤r', gj is not contained in Q'₁, Q'₂, … , Q'_r'. Further, g_j is not contained in Q₁, Q₂, … , Q_r', provided g_j ∉{q ,₁ q , … ,₂ q , d_{r' 1}, d₂, … , d_r'}.

Clearly, we have g_j∉{q1, q₂, … , q_r'}, i.e., g_j ∉{q ,₁ q , … ,₂ q }. Since_r' d_i ∈{du'+1, d_{u' +2}, … , d_k} for all 1≤i≤r' and gj∉{du'+1, d_u'+2, … , d_k}, we have g_j∉{d ,₁ d , … ,₂ d }, i.e.,_r' g_j ∉{d1, d₂, … , d_r'}. On the other hand, g_j is not contained in Q_u'+1, Q_u'+2, … , Q_k because |g |>|_j β| and g_j ∉{fm'+1, f_m'+2, … , f_m'+c'} (i.e.,

gj ∉{ f_m'+1, f_m'+2, … , f_m'+c' }).

Then we show that d_j is not contained internally in Q₁, Q₂, … , Q_r', Q_u'+1, Q_u'+2, … , Q_{k +1}. Suppose that

γ ∉{d1, d₂, … , d_r'} is a node in Q₁, Q₂, … , Q_r'. Since |d_j|≥k/2+2 (by our assumption) and |γ|≤k/2+1 (refer to the construction method of Case 5), d_j is not contained internally in Q₁, Q₂, … , Q_r'. Besides, d_j is not contained internally in Q_u'+1, Q_u'+2, … , Q_k because |d_j|>|β| (i.e., d_j≠β) and d_j ∉{fm'+1, f_{m' +2}, … , f_m'+c'} (i.e., d_j∉{ f_m'+1, f_m'+2, … , f_m'+c' }). Since |d_j|≤|dk +1| and the subpath of Q_{k +1} from s to d_{k +1} is shortest, d_j is not contained internally in Q_{k +1}.

On the other hand, if g_j ∈Qk +1, then Q'_j and Q_t were swapped and g_j was changed to be the immediate predecessor of d_j in (new)Q'_j. It suffices to show (new)g_j ∉Qk +1, in order for Q_j to be disjoint with Q₁, Q₂, … , Q_r', Q_u'+1, Q_u'+2, … , Q_{k +1}. If dt_,Ω(d_t) =1 and 1

)

, ( =

dj

dj _Ω , then by Lemma 8 (old) Q_t and (old) Q'_j are shortest with lengths |d_t| (=|d |), which implies |(new)_j g_j|=|(old) g_j|, i.e.,

|(new)g |=|(old)_j g |. Since (old)_j g_j ∈Qk +1, |(new)g |=|(old)_j g |, (new)_j g_j ≠(old)g , and the subpath of _j Q_{k +1} from s to d_{k +1} is shortest, we have (new)g_j ∉Qk +1. If d_t,Ω(dt) =0, then Lemma 8 assures |(new)

g_j|=|d_t|+1=|d |+1, i.e., |(new)_j g |= |d_{j j}|−1<|dk +1|. It follows that we have (new)g_j ∉Qk +1. If 0

)

, ( =

dj

dj

Ω ,

then we have |(old) g_j|=| d |+1, i.e., |(old)_j g |=|d_{j j}|−1<|dk +1|. Hence, (old) g_j ∉Qk +1, which is a contradiction.

If d_j ≠dv for all u'+1≤v≤k, then Qj was constructed as the combination of Q'_j and a link (d , d_{j j}), which has length at most k/2+1. Qj is disjoint with Q₁, Q₂, … , Q_r', Q_u'+1, Q_u'+2, … , Q_{k +1}, provided d_j is not contained internally in Q₁, Q₂, … , Q_r', Q_u'+1, Q_u'+2, … , Q_{k +1} and d_j ∉{d1, d₂, … , d_r', d_{k +1}}. The former has been shown above. The latter holds because |d |≤k/2−2 and |d_j k +1|≥|di|≥k/2+2 for all 1≤i≤r'.

Finally we show that Q_r'+1, Q_r'+2, … , Q_u' are disjoint to each other. Without loss of generality, we assume d_c∈{du'+1, d_u'+2, … , d_k} for all r'+1≤c≤l' and dc∉{du'+1, d_u'+2, … , d_k} for all l'+1≤c≤u', where r'≤l'≤u'. It suffices to show that g_r'+1, g_r'+2, … , g_l' are all distinct, {g_r'+1 , g_r'+2, … , g_l' }∩{dr'+1, d_r'+2, … , d_u'} is empty, and each of g_r'+1, g_r'+2, … , g_l', d_r'+1, d_r'+2, … , d_u' is not contained in Q'_r'+1, Q'_r'+2, … , Q'_u'. The first two hold because g_r'+1, g_r'+2, … , g_l' are all distinct and {g_r'+1, g_r'+2, … , g_l'}∩{d_r'+1,

+2 '

dr , … , d_u'} is empty. The last holds because |g_c|≥k/2+2 for all r'+1≤c≤l', |dj|≥k/2+2 for all r'+1≤j≤u', and Q'r'+1, Q'_r'+2, … , Q'_u' have lengths at most k/2 (recall that none of Q'r'+1, Q'_r'+2, … , Q'_u' contain complement links).