Mutually Independent Hamiltonian Cycles of Pancake Networks

Transcription

1 Mutually Indpndnt Hamiltonian Cycls of Pancak Ntworks Chng-Kuan Lin Dpartmnt of Mathmatics National Cntral Univrsity, Chung-Li, Taiwan 00, R O C discipl@ms0urlcomtw Hua-Min Huang Dpartmnt of Mathmatics National Cntral Univrsity, Chung-Li, Taiwan 00, R O C huang@mathncudutw Jimmy J M Tan Dpartmnt of Computr Scinc National Chiao Tung Univrsity, Hsinchu, Taiwan 00, R O C jmtan@cisnctudutw Lih-Hsing Hsu Information Enginring Dpartmnt Ta Hwa Institut of Tchnology, Hsinchu, Taiwan 07, R O C lhhsu@cisnctudutw Abstract A hamiltonian cycl C of G is dscribd as u, u,,u n(g), u to mphasiz th ordr of nods in C Thus, u is th bginning nod and u i is th i-th nod in C Two hamiltonian cycls of G bginning at a nod x, C = u, u,, u n(g), u and C = v, v,, v n(g), v, ar indpndnt if x = u = v, and u i v i for vry i n(g) A st of hamiltonian cycls {C, C,,C k } of G ar mutually indpndnt if any two diffrnt hamiltonian cycls ar indpndnt Th mutually indpndnt hamiltonicity of graph G, IHC(G), is th maximum intgr k such that for any nod u of G thr xist k-mutually indpndnt hamiltonian cycls of G starting at u In this papr, w ar going to study IHC(G) for th n-dimnsional pancak graph W will prov that IHC(P ) = and IHC( ) = n if n Introduction An intrconnction ntwork conncts th procssors of paralll computrs Its architctur can b rprsntd as a graph in which th nods corrspond to procssors and th dgs corrspond to connctions Hnc, w us graphs and ntworks intrchangably Thr ar many mutually conflicting rquirmnts in dsigning th topology for computr ntworks Akrs and Krishnamurthy [] proposd a family of intrsting intrconnction ntworks, th n-dimnsional pancak graph Thy showd that th pancak graphs ar nod transitiv Hung t al [7] studid th hamiltonian connctivity on th faulty pancak graphs Th mbdding of cycls and trs into th pancak graphs whr discussd in [,, 7, 8] Gats and Papadimitriou [5] studid th diamtr of th pancak graphs Up to now, w do not know th xact valu of th diamtr of th pancak graphs [6] For a graph dfinitions and notation w follow [] G = (V, E) is a graph if V is a finit st and E is a subst of {(u, v) (u, v) is an unordrd pair of V } W say that V is th nod st and E is th dg st W us n(g) to dnot V Lt S b a subst of V Th subgraph of G inducd by S, G[S], is th graph with V (G[S]) = S and E(G[S]) = {(x, y) (x, y) E(G) and x, y S} W us G S to dnot th subgraph of G inducd by V S Two nods u and v ar adjacnt if (u, v) is an dg of G Th st of nighbors of u, dnot by N G (u), is {v (u, v) E} Th dgr d G (u) of a nod u of G is th numbr of dgs incidnt with u Th minimum dgr of G, writtn δ(g),

2 is min{d G (x) x V } A path is a squnc of nods rprsntd by v 0, v,,v k with no rpatd nod and (v i, v i+ ) is an dg of G for all 0 i k W us Q(i) to dnot th i-th nod v i of Q = v, v,,v k W also writ th path v 0, v,, v k as v 0,,v i, Q, v j,, v k, whr Q is a path form v i to v j A path is a hamiltonian path if it contains all nods of G A graph G is hamiltonian connctd if thr xists a hamiltonian path joining any two distinct nods of G A cycl is a path with at last thr nods such that th first nod is th sam as th last on A hamiltonian cycl of G is a cycl that travrss vry nod of G A graph is hamiltonian if it has a hamiltonian cycl A hamiltonian cycl C of graph G is dscribd as u, u,,u n(g), u to mphasiz th ordr of nods in C Thus, u is th bginning nod and u i is th i-th nod in C Two hamiltonian cycls of G bginning at a nod x, C = u, u,, u n(g), u and C = v, v,, v n(g), v, ar indpndnt if x = u = v, and u i v i for vry i n(g) A st of hamiltonian cycls {C, C,,C k } of G ar mutually indpndnt if any two diffrnt hamiltonian cycls ar indpndnt Th mutually indpndnt hamiltonianicity of graph G, IHC(G), is th maximum intgr k such that for any nod u of G thr xist k-mutually indpndnt hamiltonian cycls of G starting at u Obviously, IHC(G) δ(g) if G is a hamiltonian graph Th concpt of mutually indpndnt hamiltonian cycls can b applid in a lot of aras Considr th following scnario In Christmas, w hav a holiday of 0 days A tour agncy will organiz a 0-day tour to Italy Suppos that thr will b a lot of popl joining this tour Howvr, th maximum numbr of popl stay in ach local ara is limitd, say 00 popl, for th sak of hotl contract On trivial solution is on th First-Com-First-Srv basis So only 00 popl can attnd this tour (Not that w can not schdul th tour in a piplind mannr bcaus th holiday priod is fixd) Nonthlss, w obsrv that a tour is lik a hamiltonian cycl basd on a graph, in which a nod is dnotd as a hotl and any two nods ar joind with an dg if th associatd two hotls can b travld in a rasonabl tim Thrfor, w can organiz all th diffrnt subgraphs, i ach subgraph has its own tour In this way, w do not allow two subgroups stay in th sam ara during th sam tim priod In othr words, any two diffrnt tours ar indd indpndnt hamiltonian cycls Suppos that thr ar 0-mutually indpndnt hamiltonian cycls Thn w may allow 000 popl to visit Italy on Christmas vacation For this rason, w would lik to find th maximum numbr of mutually indpndnt hamiltonian cycls Such applications ar usful for task schduling and rsourc placmnt, which ar also important for complir optimization to xploit paralllism In this papr, w study mutually indpndnt hamiltonian cycls of pancak graph In th following sction, w giv th dfinition of th pancak graphs and rviw som of th prvious work usd in this papr In sction, w prov that IHC(P ) = and IHC( ) = n if n Th pancak graphs Lt n b a positiv intgr W us n to dnot th st {,,, n} Th n-dimnsional pancak graph, dnotd by, is a graph with th nod st V ( ) = {u u u n u i n and u i u j for i j} Th adjacncy is dfind as follows: u u u i u n is adjacnt to v v v i v n through an dg of dimnsion i with i n if v j = u i j+ for all j i and v j = u j for all i < j n W will us boldfac to dnot a nod of Hnc, u,u,,u n dnot a squnc of nods in In particular, dnots th nod n Th pancak graphs P, P, and P ar illustratd in Figur P P a b P b Figur : Th pancak graphs P, P, and P By dfinition, is an (n )-rgular graph with n! nods Morovr, it is nod transitiv [] Lt u = u u u n b an arbitrary nod of W us (u) i to dnot th i-th componnt u i of u, and us {i} to dnot th i-th subgraph of inducd by thos nods u with (u) n = i Thn can b dcomposd into n nod disjoint subgraphs {i} for all i n such that ach {i} is isomorphic to Thus, th pancak graph a

3 can b constructd rcursivly Lt H n, w us Pn H to dnot th subgraph of inducd by i H V ( {i} ) For i j n, w us {i,j} to dnot th subgraph of inducd by thos nods u with (u) n = i and (u) n = j Obviously, {i,j} {j,i} and {i,j} is isomorphic to By dfinition, thr is xactly on nighbor v of u such that u and v ar adjacnt through an i-dimnsional dg with i n W us (u) i to dnot th uniqu i-nighbor of u W hav ((u) i ) i = u and (u) n {(u)} For i, j n and i j, w us E i,j to dnot th st of dgs btwn {i} and {j} Lmma Lt i, j n with i j and n Thn E i,j n = (n )! Lmma Lt u and v b two distinct nods of with d(u,v) Thn (u) (v) Thorm [7] Suppos that F is a subst of V ( ) with F n and n Thn F is hamiltonian connctd Thorm Lt {a, a,,a r } b a subst of n for som positiv intgr r n with n 5 Assum that u and v ar two distinct nods of with u and v {ar} Thn thr is a hamiltonian path H = u = x, H,y,x, H,y,,x r, H r,y r = v of r i= {ai} joining u to v such that x = u, P {a} n y r = v, and H i is a hamiltonian path of P {ai} n joining x i to y i for vry i r Proof W st x as u and st that y r as v W know that {ai} is isomorphic to for vry i r By Thorm, this statmnt holds for r = Thus, w assum that r By Lmma, En ai,ai+ = (n )! 6 for vry i r W choos (y i,x i+ ) E ai,ai+ for vry i r with y x and x r y r By Thorm, thr is a hamiltonian path H i of {ai} joining x i to y i for vry i r Thn H = u = x, H,y,x, H,y,,x r, H r,y r = v forms a dsird path S Figur for illustration on { a} u=x { a } x { ar} x r H H y H y r y r =v Figur : Illustration for Thorm on IHC( ) Lmma Lt k n with n, and lt x b a nod of Thr is a hamiltonian path P of {x} joining th nod (x) n to som nod v with (v) = k Proof Suppos that n = Sinc P is nod transitiv, w may assum that x = Th rquird paths of P {} ar listd blow: k = ()()()()()()()() ()()()()()()()() ()()()()()()() k = ()()()()()()()() ()()()()()()()() ()()()()()()() k = ()()()()()()()() ()()()()()()()() ()()()()()()() k = ()()()()()()()() ()()()()()()()() ()()()()()()() With Thorm, w can find th rquird hamiltonian path in for vry n 5 Lmma Lt a, b n with a b and n, and lt x b a nod of Thr is a hamiltonian path P of {x} joining a nod u with (u) = a to a nod v with (v) = b Proof Suppos that n = Sinc P is nod transitiv, w may assum that x = Without loss of gnrality, w may assum that a < b W hav a and b Th rquird paths of P {} ar listd blow: ()()()()()()()() ()()()()()()()() ()()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()()() With Thorm, w can find th rquird hamiltonian path on for vry n 5 Lmma 5 Lt n, and a, b n with a b Assum that x and y ar two adjacnt nods of Thr is a hamiltonian path P of {x,y} joining a nod u with (u) = a to a nod v with (v) = b

4 Proof Sinc is nod transitiv, w may assum that x = and y = () i for som i {,,, n} Without loss of gnrality, w assum that a < b Thus, a n and b W prov this lmma by induction on n For n =, th rquird paths of P {, () i } ar listd blow: y = ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() y = ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() y = ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() ()()()()()()()() ()()()()()()()() ()()()()()() Suppos that this lmma is tru for P k for vry k < n W hav th following cass: Cas y = () i for som i and i n, i y {n} Lt c n {a} By induction, thr is a hamiltonian path R of {n} {, () i } joining a nod u with (u) = a to a nod z with (z) = c W choos a nod v in n {c} with (v) = b By Thorm, thr is a hamiltonian path H of n joining th nod (z) n to v Thn u, R,z, (z) n, H,v forms th dsird path Cas y = () n, i y {} Lt c n {, a} and d n {, b, c} By Lmma, thr is a hamiltonian path R of {n} {} joining a nod u with (u) = a to a nod w with (w) = c Again, thr is a hamiltonian path H of {} {() n } joining a nod z with (z) = d to a nod v with (v) = b By Thorm, thr is a hamiltonian path Q of n {} joining th nod (w) n to th nod (z) n Thn u, R,w, (w) n, Q, (z) n,z, H,v forms th dsird path Lmma 6 Lt a, b n with n Assum that x is a nod of, and x and x ar two distinct nighbors of x Thr is a hamiltonian path P of {x,x,x } joining a nod u with (u) = a to a nod v with (v) = b Proof Sinc is nod transitiv, w may assum that x = Morovr, w assum that x = () i and x = () j for som i, j n {} with i < j Without loss of gnrality, w assum that a < b Thus, a n and b W prov this lmma by induction on n For n =, th rquird paths of P {, () i, () j } ar listd blow: x = and x = ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() x = and x = ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()()

5 x = and x = ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() x = and x = ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() ()()()()()()() Suppos that this lmma is tru for P k for vry k < n W hav th following cass: Cas j n, i x {n} and x {n} Lt c n {, a} By induction, thr is a hamiltonian path R of {n} {,x,x } joining a nod u with (u) = a to a nod z with (z) = c W choos a nod v in {} with (v) = b By Thorm, thr is a hamiltonian path H of n joining th nod (z) n to v W st P = u, R,z, (z) n, H,v, Thn P forms th dsird path Cas j = n, i x {n} and x {} Lt c n {, a} and d n {, b, c} By Lmma 5, thr is a hamiltonian path R of {n} {,x } joining a nod u with (u) = a to a nod z with (z) = c By Lmma, thr is a hamiltonian path H of {} {x } joining a nod w with (w) = d to a nod v with (v) = b By Thorm, thr is a hamiltonian Q of n {} joining th nod (z) n to th nod (w) n W st P = u, R,z, (z) n, Q, (w) n,w, H,v, Thn P forms th dsird path Our main rsult for th pancak graph is statd in th following thorm Thorm IHC(P ) = and IHC( ) = n if n Proof It is asy to s that P is isomorphic to a cycl with six nods, so IHC(P ) = is (n )-rgular graph, it is clar that IHC( ) n Sinc is nod transitiv, w only nd to show that thr xist (n )-mutually indpndnt hamiltonian cycls of starting form th nod For n =, w prov that IHC(P ) by listing th rquird hamiltonian cycls as following: ()()()()()()()()() C ()()()()()()()()() ()()()()()()() ()()()()()()()()() C ()()()()()()()()() ()()()()()()() ()()()()()()()()() C ()()()()()()()()() ()()()()()()() Lt n 5 Lt B b th (n ) n matrix with { i + j if i + j n, b i,j = i + j n + if n < i + j Mor prcisly, n n 5 n B = n n n n It is not hard to s that (b i,, b i,,, b i,n ) forms a prmutation of {,,, n} for vry i with i n Morovr, b i,j b i,j for any i < i n and j n In othr words, B forms a latin rctangl with ntris in {,,, n} W construct {C, C,, C n } as follows: () k = By Lmma, thr is a hamiltonian path H of {b,n} {} joining a nod x with x () n and (x) = n to th nod () n By Thorm, thr is a hamiltonian path H of n t= {b,t} joining th nod () n to th nod (x) n with H (i+(j )(n )!) {b,j } for vry i (n )! and for vry j n W st C =, () n, H, (x) n,x, H, () n, () k = By Lmma 5, thr is a hamiltonian path Q of {b,n } {, () } joining a nod y with (y) = n to a nod z with (z) = By Thorm, thr is a hamiltonian Q of n t= {b,t} joining th nod (() ) n to th nod (y) n such that Q (i + (j )(n )!) {b,j } for vry i (n )! and for vry j n By Thorm, thr is a hamiltonian path Q of {b,n} joining th nod (z) n to th nod () n W st C =, (), (() ) n, Q, (y) n,y, Q,z, (z) n, Q, () n, () k n By Lmma 6, thr is a hamiltonian path R k of P {b k,n k+} n {, () k, () k } joining a nod w k with (w k ) = n to a nod v k with (v k ) = By Thorm, thr is a hamiltonian path R k of n k t= P {b k,t} n joining th

6 nod (() k ) n to th nod (w k ) n such that R k (i + (j )(n )!) P {b k,j} n for vry i (n )! and for vry j n k Again, thr is a hamiltonian path R k of n t=n k+ P {b k,t} n joining th nod (v k ) n to th nod (() k ) n such that R k(i+(j )(n )!) P {b k,n k+j+} n for vry i (n )! and for vry j k W st C k =, () k, (() k ) n, R k, (w k) n,w k, R k,v k, (v k ) n, R k, (() k ) n, () k, Thn {C, C,, C n } forms a st of (n )-mutually indpndnt hamiltonian cycls of starting from th nod Exampl: W illustrat th proof of Thorm with n = 5 as follows: W st 5 B = Thn w construct {C, C, C, C } as follows: () k = By Lmma, thr is a hamiltonian path H of P {b,5} 5 {} joining a nod x with x () and (x) = to th nod () By Thorm, thr is a hamiltonian path H of t= P {b,t} 5 joining th nod () 5 to th nod (x) 5 with H (i + (j )) P {b,j} 5 for vry i and for vry j W st C =, () 5, H, (x) 5,x, H, (), () k = By Lmma 5, thr is a hamiltonian path Q of P {b,} 5 {, () } joining a nod y with (y) = to a nod z with (z) = By Thorm, thr is a hamiltonian Q of t= P {b,t} 5 joining th nod (() ) 5 to th nod (y) 5 such that Q (i + (j )) P {b,j} 5 for vry i and for vry j By Thorm, thr is a hamiltonian path Q of P {b,5} 5 joining th nod (z) 5 to th nod () 5 W st C =, (), (() ) 5, Q, (y) 5,y, Q,z, (z) 5, Q, () 5, C C C C ( ) ( ) ( ) { } ( ) 5 H { } (() ) 5 { } (() ) 5 { } (() ) 5 Q R R 5 6 ( w) 5 6 { } { } { } { 5} { } { } 7 { 5} w - { (, ),( ) } R v 9 ( w) 5 { } ( v) { } - { (, ) } { 5} w 5 - {, ( ), () } R v 7 ( y ) 5 { } ( v) 5 { } 7 { 5} y Q z ( x ) 5 { } { } x { } () z 5 { } - H Q R R ( ( ) ) 5 ( ( ) ) 5 ( ) 0 ( ) Figur : Illustration for Thorm on ( ) ( ) () k By Lmma 6, thr is a hamiltonian path R k of P {b k,6 k} 5 {, () k, () k } joining a nod w k with (w k ) = to a nod v k with (v k ) = By Thorm, thr is a hamiltonian path R k of 5 k t= P {b k,t} 5 joining th nod (() k ) 5 to th nod (w k ) 5 such that R k(i + (j )) P {b k,j} 5 for vry i and for vry j 5 k Again, thr is a hamiltonian path R k of 5 t=7 k P {b k,t} 5 joining th nod (v k ) 5 to th nod (() k ) 5 such that R k(i + (j )) P {b k,6 k+j } 5 for vry i and for vry j k W st C k =, () k, (() k ) 5, R k, (w k) 5,w k, R k,v k, (v k ) 5, R k, (() k ) 5, () k, Thn {C, C, C, C } forms a st of -mutually indpndnt hamiltonian cycls of starting from th nod S Figur for illustration Rfrncs [] S B Akrs and B Krishnamurthy, A groupthortic modl for symmtric intrconnction ntworks, IEEE Transactions on Computrs, Vol 8, pp , 989 [] J A Bondy and U S R Murty, Graph Thory with Applications, North Holland, Nw York, 980 [] K Day and A Tripathi, A comparativ study of topological proprtis, IEEE Transactions on Paralll and Distributd Systms, Vol 5, pp 8, 99 [] W C Fang and C C Hsu, On th faulttolrant mbdding of complt binary tr in th pancak graph intrconnction ntwork, Information Scincs, Vol 6, pp 9 0, 000 [5] W H Gats and C H Papadimitriou, Bounds for sorting by prfix rvrsal, Discrt Mathmatics, Vol 7, pp 7 57, 979 [6] M H Hydari and I H Sudborough, On th diamtr of th pancak ntwork, Journal of Algorithms, Vol 5, pp 67 9, 997 [7] C N Hung, H C Hsu, K Y Liang, and LH Hsu, Ring mbdding in faulty pancak graphs, Information Procssing Lttrs, Vol 86, pp 7 75, 00

7 [8] A Kanvsky and C Fng, On th mbdding of cycls in pancak graphs, Paralll Computing, Vol, pp 9 96, 995