cs.IT 2011-04-29 0

Mengers Paths with Minimum Mergings

For an acyclic directed graph with multiple sources and multiple sinks, we prove that one can choose the Merger's paths between the sources and the sinks such that the number of mergings between these paths is upper bounded by a constant depending on…

Authors: Guangyue Han

Mengers Paths with Minimum Mergings
Menger’s P aths with Minim um Mergings ∗ Guangyue Han Departmen t of M a thematics Univ ersit y of Hong Kong P okfulam Road, Hong Kong e-mail: ghan@hku.hk August 18, 202 1 Abstract F or an acyclic directed graph with m ultiple sour ces and m ultiple sink s, w e p ro ve that one can c ho ose the Menger’s paths b etw een the sources and the sinks suc h that the n u m b er of mergings b etw een these paths is upp er b ound ed b y a constant dep en ding only on the min-cuts b etw een the sour ces and the sinks, regardless of the size and top ology of the graph. W e also g ive b ounds on the min im um n u m b er of mergings b et ween these paths, and discuss ho w it d ep ends on the min-cuts. 1 In tro ductio n Let G ( V , E ) denote an acyclic directed gra ph, where V denotes the set of all the vertice s (p oints ) in G and E denotes the se t of all the edges in G . Using these not a tions, the edge-connectivit y v ersion of Me nger’s t heorem [7] states: Theorem 1.1 (Menger, 1927) . F or any u , v ∈ V , the m a ximum n umb er of p airwise e dge- disjoint dir e cte d p aths fr om u to v in G e quals the mi n -cut b etwe en u an d v , name ly the minimum numb er of e dges in E who s e de letion destr oys al l dir e cte d p a ths fr om u to v . W e call a ny set consisting of the maxim um n um b er of pairwise edge-disjoin t directed paths from u t o v a set of Menger’s p aths from u and v . Apparen tly , for fixed u , v ∈ V , there ma y exist m ultiple sets of Menger’s paths. F or m paths β 1 , β 2 , · · · , β m in G ( V , E ), we say these paths mer ge at e ∈ E if 1. e ∈ ∩ m i =1 β i ; 2. there are at least tw o distinct f , g ∈ E suc h that f , g are immediately ahead of e on some β i , β j , j 6 = i , respective ly . ∗ This work is partially supp orted by a grant from the Univ ersity Gr ants Committee of the Hong Ko ng Spec ial Administrative Region, China (Pro ject No. AoE /E-02 /08). S (source) A B C D (b) (a) (c) T P S f r a g r e p la c e m e n t s β 1 β 1 β 1 β 2 β 2 β 2 β 3 β 4 Figure 1 : examples of mergings and non-mergings Roughly speaking, condition 1 sa ys that β 1 , β 2 , · · · , β m internal ly interse ct at e (namely , all β i ’s share a common edge e ), condition 2 says immediately b efore all β i ’s in ternally inte rsect at e , at least tw o of them are differen t. W e call e together with all the subseq uent edges shared b y all β i ’s (un til they branc h off ) mer ge d subp ath b y β i ( i = 1 , 2 , · · · , m ) at e ; and w e often sa y all β i ’s merge at the ab o v e-mentioned merged subpath. In this pap er we will coun t n um b er of mergings wit hout mu lt iplicities: all the mergings at t he same edge e will be coun ted as one merging at e . Example 1.2. In Fig ure 1(a), paths β 1 and β 2 share some ve rtex, ho w ev er n o t edges/subpaths, so β 1 and β 2 do not merge. In Figure 1(b), paths β 1 and β 2 do share edge S → T , whe re S is a source, ho wev er condition 2 is not satisfied, therefore β 1 and β 2 do not merge, although they in ternally in t ersect at S → T . In Figure 1(c), β 1 and β 2 merge at edge A → B , at subpath A → B → C ; β 2 and β 3 merge a t edge A → B , a t subpath A → B → C → D ; β 1 , β 2 and β 3 merge at edge A → B , at subpath A → B → C ; β 4 merges with β 3 at edge B → C , at subpath B → C → D ; there are t wo mergings in F ig ure 1(c), a t edge A → B , and at edge B → C , respective ly . In this pap er, w e will consider an acyclic directed graph G ( E , V ) with n sources and n sinks. Unless sp ecified otherwise, w e will use S 1 , S 2 , · · · , S n to denote the sources and R 1 , R 2 , · · · , R n to denote the sinks; c i will b e used to denote the min-cut b et w een S i and R i , and α i = { α i, 1 , α i, 2 , · · · , α i,c i } will b e used to denote a set of Menger’s paths from S i and R i . W e will study how α i ’s merge with each other; more specifically , we sho w that appro pria tely c hosen Menger’s paths will o nly merge with each ot her finitely man y times. In par t icular, w e deal with the case when all sources and sinks are distinct in Section 2 , and the case when the sources are iden tical and the sinks are distinct in Section 3. F or b oth of cases, w e will study ho w the minim um merging n um b er dep ends on the min-cuts. W e remark tha t when n = 1, F ord-F ulk erson algorithm [2] can find the min-cut and a set of Menger’s path b etw een S 1 and R 1 in p olynomial time. The LDP (Link Disjoint Problem) 2 asks if there are tw o edge-disjoin t paths from S 1 , S 2 to R 1 , R 2 , resp ectiv ely . The fact that the LDP problem is NP-complete [3] su g gests the intricac y of the problem when n ≥ 2. Notation and Con ven tion. F o r a path γ in an acyclic direct gr a ph G , let a ( γ ) , b ( γ ) denote the starting p oint and the ending p oin t of γ , resp ectiv ely; let γ [ s, t ] denote the subpath of γ with the starting p oin t s and the ending point t . F or t wo distinct paths γ , π in G , w e sa y γ is smal ler than π if there is a directed path from b ( γ ) t o a ( π ); if γ , π and the connecting path from b ( γ ) to a ( π ) are subpaths of path β , w e say γ is smal ler than π on β . Note that this definition also a pplies to the cas e when paths degenerate to v ertices/edges ; in other w ords, in the definition, γ , π or the connecting path f rom b ( γ ) to a ( π ) can b e v ertices / edges in G , whic h can b e vie wed as degenerated paths. If b ( γ ) = a ( π ), w e use γ ◦ π to denote the path obtained b y concatenating γ and π subseque ntly . F or a set of vertice s v 1 , v 2 , · · · , v j in G , define G | v 1 , · · · , v j ) to b e subgraph of G consisting of the set of ve rt ices (denoted b y V 0 ), eac h of whic h is smaller than some v j , and the set of all t he edges, eac h of whic h is inciden t with s o me v ertex in V 0 . 2 Minim um Mergings M In this section, w e consider any acyclic directed graph G with n distinct sources and n distinct sinks. Let M ( G ) denote the minimum n umber of mergings ov er all p ossible Menger’s path sets α i ’s, i = 1 , 2 , · · · , n , and let M ( c 1 , c 2 , · · · , c n ) denote the suprem um of M ( G ) o v er all p ossible c hoices of suc h G . In the followin g , w e shall prov e that Theorem 2.1. F or any c 1 , c 2 , · · · , c n , M ( c 1 , c 2 , · · · , c n ) < ∞ , and furthermor e, we have M ( c 1 , c 2 , · · · , c n ) ≤ X i 0 (w e shall choose K large enough lat er) and eac h j = 1 , 2 , · · · , k , pic k merged subpaths γ 0 ,j suc h that γ 0 ,j b elongs to path ψ j and | R (0) | b ( γ 0 , 1 ) , b ( γ 0 , 2 ) , · · · , b ( γ 0 ,k )) | M = K ; note that, without loss of gene r a lit y , w e can assume that γ 0 ,j is t he largest merged subpath from R (0) | b ( γ 0 , 1 ) , b ( γ 0 , 2 ) , · · · , b ( γ 0 ,k )) on ψ j (one can c ho ose γ 0 ,j to b e S 1 if suc h merged subpath does not e xist on ψ j ). Now set S (1) = S (0) ∪ R (0) | b ( γ 0 , 1 ) , b ( γ 0 , 2 ) , · · · , b ( γ 0 ,k )) and R (1) = R (0) \ R (0) | b ( γ 0 , 1 ) , b ( γ 0 , 2 ) , · · · , b ( γ 0 ,k )) . If a merged subpath is the smallest or the largest one on a φ -path, we say it is a terminal merged subpath on the φ -path, or simply a φ -terminal merged subpath. No w supp o se that w e already obtain S ( i ) = S ( i − 1) ∪ R ( i − 1) | b ( γ i − 1 , 1 ) , b ( γ i − 1 , 2 ) , · · · , b ( γ i − 1 ,k )) and R ( i ) = R ( i − 1) \ R ( i − 1) | b ( γ i − 1 , 1 ) , b ( γ i − 1 , 2 ) , · · · , b ( γ i − 1 ,k )) , where R ( i − 1) | b ( γ i − 1 , 1 ) , b ( γ i − 1 , 2 ) , · · · , b ( γ i − 1 ,k )) contains exactly K me rg ings and at least one φ -terminal merged subpath, w e then contin ue to pic k merged subpath γ i,j on ψ j from R ( i ) suc h that | R ( i ) | b ( γ i, 1 ) , b ( γ i, 2 ) , · · · , b ( γ i,k )) | M = K , here, a gain, eac h γ i − 1 ,j , j = 1 , 2 , · · · , k , is c hosen to b e largest merged subpath on ψ j ; and w e set S ( i +1) = S ( i ) ∪ R ( i ) | b ( γ i, 1 ) , b ( γ i, 2 ) , · · · , b ( γ i,k )) , 17 and R ( i +1) = R ( i ) \ R ( i ) | b ( γ i, 1 ) , b ( γ i, 2 ) , · · · , b ( γ i,k )) . W e will further con tinue in this fashion, if necessary , to obta in S (2) , R (2) , S (3) , R (3) , · · · un til w e obtain S ( i 0 ) , R ( i 0 ) suc h that S ( i 0 ) \ S ( i 0 − 1) do es not con tain a ny φ -terminal merged subpaths. Note that each S ( j ) \ S ( j − 1) ( j = 1 , 2 , · · · , i ) has K mergings, w e th us call eac h of them a K - trunk . The first i 0 − 1 K -trunk, S ( j ) \ S ( j − 1) ( j = 1 , 2 , · · · , i 0 − 1) con ta ins some φ - terminal merged subpaths, w e thus call these K -t r unks singular ; on t he other hand, the i 0 -th K - trunk do es not con tain an y terminal merged subpaths o n an y φ -path, w e then call this K - trunk normal . By Theorem 2.1 and the fact that G is non- r ero ut a ble, w e now c ho ose K so larg e that the num b er of critical merged subpaths within S ( i 0 ) \ S ( i 0 − 1) is la r ger than k (thus the num b er of critical merged subpaths within S ( i 0 ) is larger t ha n k ), here w e say a merged subpath is critic al within a subgraph of G if the asso ciated φ -path, after merging a t t his merged subpath, do es not merge an ymore within this subgraph (note that since S ( i 0 ) \ S ( i 0 − 1) is normal, the φ -path will con tinue to merge within R ( i 0 ) ). No w, let T i 0 denote the se t of all the merged subpaths w it hin R ( i 0 ) whic h can semi-reach some critical merged su bpa t h within S ( i 0 ) through g roup ψ from below. One c hec ks at least one o f those ψ -paths, eac h of whic h con tains at least one critical merged subpath within S ( i 0 ) , do es not con tain an y merged subpath within T i 0 (since, otherwise, by a usual bac k- tracing argumen t, one c hec ks that there exists a merging reduc ing rerouting) . Assume that ξ i 0 , 1 , ξ i 0 , 2 , · · · , ξ i 0 ,m i 0 (1 ≤ m i 0 ≤ k − 1) a r e the largest merged subpaths from T i 0 , and they b elong to paths ψ j i 0 , 1 , ψ j i 0 , 2 , · · · , ψ j i 0 ,m i 0 , resp ectiv ely . L et ¯ T i 0 = m i 0 [ j =1 ψ j i 0 ,j [ b ( γ i 0 − 1 ,j i 0 ,j ) , b ( ξ i 0 ,j )] , here, one can c hec k that ψ j i 0 ,j [ b ( γ i 0 − 1 ,j i 0 ,j ) , b ( ξ i 0 ,j )] is the “segmen t” of ψ j i 0 ,j that is within R ( i 0 ) and before b ( ξ i 0 ,j ), or more formally , ψ j i 0 ,j [ b ( γ i 0 − 1 ,j i 0 ,j ) , b ( ξ i 0 ,j )] = ψ j i 0 ,j [ S 1 , b ( ξ i 0 ,j )] ∩ R ( i 0 ) . Note that for an y ξ i 0 ,j , j = 1 , 2 , · · · , m i 0 , the aso ciated φ -path, from ξ i 0 ,j , ma y merge outside ¯ T i 0 next t ime; if this φ -path merge within ¯ T i 0 again after a n um b er of mergings outside ¯ T i 0 , w e call it an excursiv e φ -path (with resp ect to ξ i 0 ,j ). One c hec ks that there are at most k − 2 excursiv e φ -pa t hs (since, otherwise, we can find a cycle in G , whic h is a con tradiction). So, letting L i 0 denote the num b er of φ - paths that con t a ins at least one merged subpath within R ( i 0 ) | b ( ξ i 0 , 1 ) , b ( ξ i 0 , 2 ) , · · · , b ( ξ i 0 ,m i 0 )), the num b er of connected φ -pat hs is upp er bounded b y L i 0 + ( k − 2). Then, b y induction ass umptions, | R ( i 0 ) | b ( ξ i 0 , 1 ) , b ( ξ i 0 , 2 ) , · · · , b ( ξ i 0 ,m i 0 )) ∩ ¯ T i 0 | M ≤ C m i 0 ( L i 0 + ( k − 2)) ≤ C k − 1 L i 0 + C k − 1 ( k − 2) . On the other hand, fo r any merged subpath, sa y η , from ¯ T i 0 other than ξ i 0 ,j , j = 1 , 2 , · · · , m i 0 , the associat ed φ -path, from η , can only merge within ¯ T i 0 (note that this implies that, from R ( i 0 ) | b ( ξ i 0 , 1 ) , b ( ξ i 0 , 2 ) , · · · , b ( ξ i 0 ,m i 0 )), at most k − 1 φ - pa ths can merge further; this f act will b e used la t er in the pro of ). One che cks that there exists at least o ne ψ j i 0 ,j , j = 1 , 2 , · · · , m i 0 , 18 whic h do es not merge with an y φ - paths within R ( i 0 ) | b ( ξ i 0 , 1 ) , b ( ξ i 0 , 2 ) , · · · , b ( ξ i 0 ,m i 0 )) \ ¯ T i 0 (again, since, otherwise, w e can find a cycle in G , whic h is a con tradiction). Th us, b y induction assumptions, | R ( i 0 ) | b ( ξ i 0 , 1 ) , b ( ξ i 0 , 2 ) , · · · , b ( ξ i 0 ,m i 0 )) \ ¯ T i 0 | M ≤ C k − 1 L i 0 , It then immediately follo ws that | R ( i 0 ) | b ( ξ i 0 , 1 ) , b ( ξ i 0 , 2 ) , · · · , b ( ξ i 0 ,m i 0 )) | M ≤ 2 C k − 1 L i 0 + C k − 1 ( k − 2) . No w set S ( i 0 +1) = S ( i 0 ) ∪ R ( i 0 ) | b ( ξ i 0 , 1 ) , b ( ξ i 0 , 2 ) , · · · , b ( ξ i 0 ,m i 0 )) and R ( i 0 +1) = R ( i 0 ) \ R ( i 0 ) | b ( ξ i 0 , 1 ) , b ( ξ i 0 , 2 ) , · · · , b ( ξ i 0 ,m i 0 )) . W e roughly summarize what w e ha ve done so far. Roug hly sp eaking, from the source side of graph G , w e k eep “cutting” K -tr unks, S ( i ) \ S ( i − 1) , i = 1 , 2 , · · · , i 0 − 1 , from G , un til we obtain a normal K -trunk, S ( i 0 ) \ S ( i 0 − 1) , then, b y cutting all the merged subpaths smaller than some of those merged subpaths that can semi-reac h some critical merged subpaths within S ( i 0 ) through ψ f rom b elo w, w e obtain a ˜ K - trunk, S ( i 0 +1) \ S ( i 0 ) . Similar operat ions can be done to R ( i 0 +1) . More prec isely , we kee p cutting K -trunks from R ( i 0 +1) un til w e obta in a norma l K - t r unk S ( i 1 ) \ S ( i 1 − 1) , then w e cut all the merged subpaths smaller than some of those m erg ed subpaths that can se mi- reac h some critical merged subpaths within S ( i 1 ) to obtain a ˜ K -trunk, S ( i 1 +1) \ S ( i 1 ) with | S ( i 1 +1) \ S ( i 1 ) | M ≤ 2 C k − 1 L i 1 + C k − 1 ( k − 2) , where L i 1 denotes the num b er of φ -paths that contains at least one merged subpath within S ( i 1 +1) \ S ( i 1 ) . W e con tinue these op erations in a n iterat ive fashion to further obtain normal K - trunks and ˜ K - trunks (here, again, w e a r e fo llo wing the same notational conv en tion as b efore), S ( i 1 +1) \ S ( i 1 ) , · · · , S ( i 2 ) \ S ( i 2 − 1) , S ( i 2 +1) \ S ( i 2 ) , · · · , S ( i 3 ) \ S ( i 3 − 1) , S ( i 3 +1) \ S ( i 3 ) , · · · un til there are no merged subpaths left in the graph. As stated b efore, from each ˜ K -trunk, at most k − 1 φ -paths can merge further, so at least one o f the φ -paths from this trunk will not go for w ard to merge any mo r e, implying there exist at most n ˜ K - trunks. Note also that the n um b er o f mergings within all K -trunks will b e upp er b ounded b y 3 K n , since w e can only hav e at most 2 n singular K -trunks and n normal K -t r unks. Summing up a ll the merged subpaths contained in all ˜ K - trunks, w e conclude that the n um b er of merged s ubpaths within all ˜ K - trunks is upp er b o unded b y 2 C k − 1 ( L i 0 + L i 1 + · · · + L i i + · · · ) + nC k − 1 ( k − 2) ≤ 2 C k − 1 ( n + ( k − 1)( n − 1)) + nC k − 1 ( k − 2) , (here ( k − 1)( n − 1) is the upp er b ound on the num ber of φ - pa ths that ma y b elong to more than one ˜ K - trunk) whic h implies that the n um b er of mergings in G can b e upp er b ounded b y | G | M ≤ C k n, for s o me constant C k . 19 3 Minim um Mergings M ∗ In this section, w e consider any acyclic dire cted graph G with one s o urce and n distinct sinks. Let M ∗ ( G ) denote the minim um n um b er of mergings ov er all po ssible Me nger’s path sets α i ’s, i = 1 , 2 , · · · , n , a nd let M ∗ ( c 1 , c 2 , · · · , c n ) denote the suprem um of M ∗ ( G ) ov er all p ossible c hoices of suc h G . W e also hav e t he follo wing “finitenes s” theorem for M ∗ : Theorem 3.1. F or any c 1 , c 2 , · · · , c n , M ∗ ( c 1 , c 2 , · · · , c n ) < ∞ , and furthermor e, we have M ∗ ( c 1 , c 2 , · · · , c n ) ≤ X i 1 = M (1 , 1), whic h implies M do es not satisfy the equalit y in Proposition 3.6; • through Propo sition 3.6, w e see that M ∗ (2 c, c ) = M ∗ ( c, c ) ≤ M ∗ ( c, c ) + M ∗ ( c, c ) , and strict inequalit y in the ab o v e expression holds as long a s M ∗ ( c, c ) > 0, thus M ∗ do es not satisfy the inequality in Proposition 2.11; namely , not lik e M , M ∗ is not sup-linear in its parameters; • Prop osition 3.7 implies that M ∗ (1 , n ) = 0, while from Example 2 .15, w e ha v e M (1 , n ) = n , whic h implie s M do es not satisfy the equalit y in Prop osition 3.7. The follo wing prop osition reve a ls a r elationship b et w een M and M ∗ . Prop osition 3.9. F or any n , we hav e M ∗ ( n + 1 , n + 1) ≤ M ( n, n ) − n + 1 . Pr o of. Consider the case when G has o ne source S and tw o sinks R 1 , R 2 , and the min-cut b et we en the source S a nd ev ery sink is equal to n + 1 . F or each s ink R i , pic k a se t of Menger’s paths α i = { α i, 1 , α i, 2 , · · · , α i,n +1 } . By Prop osition 3.5, we can assume ev ery α 1 - path merges with certain α 2 -path and vice v ersa. As sho wn in the pro of of Prop osition 3.6, w e can further a ssume α 1 ,i shares subpath starting fr o m S with α 2 ,i , i = 1 , 2 , · · · , n + 1, after p ossible reroutings. No w, if ev ery α 1 -path merges with some α 2 -path, for instance, α 1 ,i first me r g es with α 2 ,δ ( i ) at m erg ed subpath γ i , here δ denotes certain mapping from { 1 , 2 , · · · , n + 1 } to { 1 , 2 , · · · , n + 1 } . One then c heck s that there exist i ( i ≤ n +1) and m ( m ≤ n + 1) suc h that δ m ( i ) = i , and one can further c ho ose m to b e the smallest suc h “p erio d”. Ho w eve r , in this case certain reroutings of α 2 can b e done b y replacing α 2 ,δ j ( i ) [ S, b ( γ δ j − 1 ( i ) )] b y α 1 ,δ j − 1 ( i ) [ S, b ( γ δ j − 1 ( i ) )], j = 1 , · · · , m (here δ 0 ( i ) △ = i ), to redu ce the merging n um b er. So, without loss of generalit y , w e can assume, after further p ossible reroutings, α 1 ,n +1 do es 22 not merge with any other paths, and α 2 , 1 do esn’t merge with an y other paths either b y similar a rgumen t; in other w o r ds, all mergings are b y paths α 1 , 1 , α 1 , 2 , · · · , α 1 ,n and paths α 2 , 2 , α 2 , 3 , · · · , α 2 ,n +1 . With the fact that for j = 2 , 3 , · · · , n , α 1 ,j shares a subpath (whic h will not b e coun ted when computing M ∗ ( G )) with α 2 ,j , w e es t a blish the theorem. Prop osition 3.10. F or any n , we hav e M ∗ (2 , 2 , · · · , 2 | {z } n ) = M ∗ (2 , · · · , 2 , 2 | {z } n − 1 ) + 1 . Pr o of. Giv en any acyclic directed graph G with one source S a nd n sinks R 1 , R 2 , · · · , R n , where the min-cut b et wee n S and R i is 2, pic k a set of Menger’s paths α i = { α i, 1 , α i, 2 } from S to R i for a ll feasible i . Again b y a new merging, we mean a merging among α 1 , α 2 , · · · , α n , ho w ever not a mo ng α 1 , α 2 , · · · , α n − 1 . Assume that α 1 , α 2 , · · · , α n − 1 are c hosen suc h that the mergings among themselv es is no more than M ∗ (2 , 2 , · · · , 2 | {z } n − 1 ), w e shall prov e that whe neve r α n newly merges with α 1 , α 2 , · · · , α n − 1 more than 2 times, one can alw ays reroute certain paths to decrease the t o tal n umber of mergings within α 1 , α 2 , · · · , α n . Apparently this will b e sufficie nt to imply: M ∗ (2 , 2 , · · · , 2 | {z } n ) ≤ M ∗ (2 , · · · , 2 , 2 | {z } n − 1 ) + 1 . In the followin g , for an y j , if w e use p to refer to one of the t w o paths in α j , w e will use ¯ p to refer to the other path in α j . Consider the fo llowing tw o s cenarios: 1. for t w o certain Me nger’s paths p, q , p merges with q and ¯ p merges with ¯ q ; 2. for a Menger’s path p ∈ α n whic h newly merges with q 1 , q 2 , · · · , q l at subpath γ (here w e ha ve listed a ll the paths merging with p at γ ), p shares a subpath with ev ery q j b efore the new merging. F or scenario 1, supp o se p merges with q a t γ , and ¯ p merges with ¯ q at ε . Then one can alw ay s rerout e p [ S, a ( γ )] using q [ S, a ( γ )], reroute ¯ p [ S, a ( ε )] using ¯ q [ S, a ( ε )]; or alternativ ely reroute q [ S, a ( γ )] using p [ S, a ( γ )], reroute ¯ q [ S, a ( ε )] using ¯ p [ S, a ( ε )]. So in the follow ing w e assume that scenario 1 nev er o ccurs. F or scenario 2, supp o se that b efor e p newly merges with q 1 , q 2 , · · · , q l at γ , p shares a subpath ε j with ev ery q j . W e can assume ¯ p merges with eve ry q j [ b ( ε j ) , a ( γ )], o therwise one can reroute p [ b ( ψ j ) , a ( φ )] using q j [ b ( ψ j ) , a ( φ )] (and thus the new merging at γ disap- p ear); w e can also assume f or some path i , ¯ q i merges with p [ b ( ε i ) , a ( γ )], otherwise one can reroute ev ery q j [ b ( ε j ) , a ( γ )] using p [ b ( ε j ) , a ( γ )] and consequen tly all paths q 1 , q 2 , · · · , q l can b e rerouted (and th us the new merging at γ disapp ear). But if f or some path i , ¯ q i merges with p [ b ( ε i ) , a ( γ )], sce na rio 1 o ccurs: p merges with q i , and ¯ p merges with ¯ q i . So in the follo wing w e assume sc enario 2 do es not o ccur either, i.e., there is alwa ys some q i suc h that b efore the new merging, p do es not internally inters ect with q i . W e s ay p newly merges with q i essential ly at γ if 1. b efore the new merging, p do es not internally intersec t (again meaning share subpath) with q i ; 23 2. ¯ p in ternally in tersects with q j [ S, a ( γ )]; 3. ¯ q i in ternally in tersects with p [ S, a ( γ )]. One c heck s tha t if p newly merges with some q i non-essen tially at γ , then either p [ S, a ( γ )] or q i [ S, a ( γ )] can b e rerouted. F urthermore if p newly merges with q i essen tially at γ , and ¯ p last merges with q i [ S, a ( γ )] at ε , then one can rero ute ¯ p b y replacing ¯ p [ S, a ( ε )] b y ¯ q i [ S, a ( ε )], so the new ¯ p shares subpath ¯ q i [ S, b ( ε )] staring f rom S ; in other words, af t er p ossible rero ut ing s, w e can further assume that ¯ p shares certain subpath with q i starting f r om S . No w supp ose p ∈ α n newly merges t wice a t γ 1 , γ 2 . F or i = 1 , 2, among all the Menger’s paths merging with p at γ i , let q i denote a n arbitra r ily c hosen path suc h that p newly merges with q i at γ i essen tially (note that q 1 6 = q 2 since b ot h of them merge with p essen tially). If ¯ q 2 merges with p [ b ( γ 1 ) , a ( γ 2 )] at subpath ε 1 , since ¯ q 2 do es not merge with ¯ p (scenario 1 do es not o ccur), one can reroute p [ S, a ( ε 1 )] using ¯ q 2 [ S, a ( ε 1 )] (then the new merging at γ 1 w ould dis- app ear). Consider the case when ¯ q 2 do es not merge with p [ b ( γ 1 ) , a ( γ 2 )]. If ¯ q 2 do es not merge with q 1 [ S, a ( γ 1 )] either, one can reroute q 2 [ S, a ( γ 2 )] using q 1 [ S, a ( γ 1 )] ◦ p [ a ( γ 1 ) , a ( γ 2 )]. Now consider the case when ¯ q 2 merges with q 1 [ S, a ( γ 1 )] a nd supp ose ¯ q 2 last merges with q 1 [ S, a ( γ 1 )] at ε 2 . If ¯ p do es not merge with q 1 [ b ( ε 2 ) , a ( γ 1 )], since ¯ q 2 w on’t merge with ¯ p , p [ S, a ( γ 1 )] can b e rerouted using ¯ q 2 [ S, b ( ε 2 )] ◦ q 1 [ b ( ε 2 ) , a ( γ 1 )] (then the new merging at γ 1 w ould disapp ear). No w consider the case when ¯ p do es merge with q 1 [ b ( ε 2 ) , a ( γ 1 )] at subpath ε 3 . But in this case, one can reroute q 2 [ S, a ( γ 2 )] using ¯ p [ S, a ( ε 3 )] ◦ q 1 [ a ( ε 3 ) , a ( γ 1 )] ◦ p [ a ( γ 1 ) , a ( γ 2 )]. Apply the argumen ts ab ov e to arbitrarily c hosen pair q 1 , q 2 essen tially merging with p , together with the fact that non-essen tial merging will disapp ear after appropriat e reroutings, w e conclude that ultimately certain r eroutings to reduce the n umber of mergings are alw ays p ossible when p ∈ α n newly me r g es t wice. No w supp ose p ∈ α n newly merges at γ 1 , and ¯ p ∈ α n newly merges at γ 2 . Let q 1 denote a n arbitrarily chose n path, among all the paths merging with p at γ 1 , suc h that p newly merges with q 1 at γ 1 essen tially; let q 2 denote an arbitrarily c hosen path, among all the paths merging with ¯ p a t γ 2 , suc h that ¯ p newly merges with q 2 at γ 2 essen tially (a gain one c heck s that q 1 6 = q 2 since t hey essen tially merge with p, ¯ p , resp ectiv ely). Apparen tly q 1 , q 2 m ust merge with each other, otherwise one can reroute p [ S, a ( γ 1 )] using q 1 [ S, a ( γ 1 )] and reroute ¯ p [ S, a ( γ 2 )] using q 2 [ S, a ( γ 2 )] (then the t w o new mergings w ould disapp ear). Supp ose q 1 and q 2 last merge at ε 1 . W e claim that ¯ p m ust merge with q 1 [ b ( ε 1 ) , a ( γ 1 )], otherwise one can rero ute p [ S , a ( γ 1 )] using q 2 [ S, b ( ε 1 )] ◦ q 1 [ b ( ε 1 ) , a ( γ 1 )] ( p shares subpath with q 2 from S and does not merge with q 1 b efore γ 1 ). F urthermore ¯ p m ust merge with q 1 [ b ( ε 1 ) , a ( γ 1 )] at least once b efo r e a ( γ 2 ) (in other w o rds, ¯ p [ S, a ( γ 2 )] mus t merge with q 1 [ b ( ε 1 ) , a ( γ 1 )]), since o therwise, say ¯ p [ b ( γ 2 ) , R n ] merges with q 1 [ b ( ε 1 ) , a ( γ 1 )] at ε 2 , then one can reroute ¯ p [ S, a ( ε 2 )] with q 1 [ S, a ( ε 2 )] (thus the new merging at γ 2 w ould disapp ear). Similarly p [ S, a ( γ 1 )] m ust merge with q 2 [ b ( ε 1 ) , a ( γ 2 )]. No w supp ose ¯ p [ S, a ( γ 2 )] first merges with q 1 [ b ( ε 1 ) , a ( γ 1 )] at subpath ε 2 . Since scenario 1 do es not o ccur, ¯ q 1 w on’t merge with p , therefore it m ust share certain subpath with p staring from S (here w e remind the reader that p newly merges with q 1 essen tially , so ¯ q 1 will either merge with or share certain subpath with p fro m S ). Similarly supp ose p [ S, a ( γ 1 )] first merges with q 2 [ b ( ε 1 ) , a ( γ 2 )] at ε 3 , then ¯ q 2 m ust share certain subpath with ¯ p staring from S . No w since scenario 1 do es not o ccur, either ¯ q 2 w on’t merge with q 1 [ b ( ε 1 ) , a ( ε 2 )] or ¯ q 1 w on’t merge with q 2 [ b ( ε 1 ) , a ( ε 3 )]. If ¯ q 2 do es not merge with q 1 [ b ( ε 1 ) , a ( ε 2 )], then one can reroute q 2 [ b ( ε 1 ) , a ( γ 2 )] with q 1 [ b ( ε 1 ) , a ( ε 2 )] ◦ ¯ p [ a ( ε 2 ) , a ( γ 2 )]; if ¯ q 1 do es not merge with q 2 [ b ( ε 1 ) , a ( ε 3 )], then one can 24 reroute q 1 [ b ( ε 1 ) , a ( γ 1 )] with q 2 [ b ( ε 1 ) , a ( ε 3 )] ◦ p [ a ( ε 3 ) , a ( γ 1 )]. Apply the arg umen ts ab ov e to arbitrarily chos en pair q 1 , q 2 essen tially merging with p , together with the fact that non- essen tial merging will disapp ear after a ppro priate reroutings, w e conclude tha t ultimately certain reroutings t o reduce the n um b er of mergings are a lw ay s p ossible wh en when p ∈ α n newly me r g es and ¯ p ∈ α n newly me r g es. F or the other direction, a ssume that the subgraph consisting of α 1 , α 2 , · · · , α n − 1 ac hiev es M ∗ (2 , 2 , · · · , 2 | {z } n − 1 ), w e a dd α n suc h that for i = 1 , 2, α n,i share subpath with α n − 1 ,i , α n only merges with α n − 1 once, say α n, 1 merges with α n − 1 , 2 at γ , where γ is a largest merged subpath. One chec ks the graph consisting α 1 , α 2 , · · · , α n has M ∗ (2 , · · · , 2 , 2 | {z } n − 1 ) + 1 mergings, and the n um b er of mergings can’t b e redu ced, implying M ∗ (2 , 2 , · · · , 2 | {z } n ) ≥ M ∗ (2 , · · · , 2 , 2 | {z } n − 1 ) + 1 . W e th us pro v e the prop osition. Example 3.11. It immed ia tely follo ws from Prop osition 3.7 that M ∗ (1 , 1 , · · · , 1) = 0 . Example 3.12. It immed ia tely follo ws from Prop osition 3.10 that M ∗ (2 , 2 , · · · , 2 | {z } n ) = n − 1 , whic h w as first show n in [4]. In particular, M (2 , 2 ) = 1. F urther together with Prop osi- tion 3.6, w e ha ve M ∗ (2 , m ) = 1 for m ≥ 2. Note that M ∗ (2 , 2 , · · · , 2 | {z } n ) < X 1 ≤ i

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