2 Trees and Their Applications

Transcription

1 Trs n Tr Appltons. Proprts o trs.. Crtrzton o trs Dnton. A rp s ll yl (or orst) t ontns no yls. A onnt yl rp s ll tr. Quston. Cn n yl rp v loops or prlll s? Notton. I G = (V, E) s rp n E, tn G wll not t rp G wt t rmov; tt s, G = (V, E {}). Smlrly, = uv or u, v V n E, tn G + nots t rp G wt t ; tt s, G + = (V, E {}). I W = v 0 v... v n W = v + v +... m v m r (v 0, v )- n (v, v m )-wl, rsptvly, n rp G, tn W W wll not t ontnton o W n W, tt s, t (v 0, v m )-wl v 0 v... v + v +... m v m. Torm. I G s tr, tn or ll u, v V (G) tr s unqu (u, v)-pt n G. Proo. T ny vrts u, v n tr G. Sn G s onnt, tr xsts t lst on (u, v)-pt n G. Suppos P n Q r two stnt (u, v)-pts. Tn tr xsts n = xy o P tt s not n o Q (or v-vrs). Now G ontns n (x, y)-wl n n n (x, y)-pt R. But tn Rxy s yl n G, ontrton. Hn G must ontn unqu (u, v)-pt. Torm. Any tr wt n vrts s prsly n s. Proo. By nuton on n. I n =, tn G = K n E(G) = 0 = n s lm. Suppos t sttmnt s tru or ll trs wt wr tn n vrts. Lt G tr wt n vrts, wr n. Tn G ontns n uv. Sn uv s t unqu (u, v)- pt, t rp G uv s sonnt n s xtly two onnt omponnts G n G. Sn G n G r onnt n yl, ty r trs, sy wt n n n vrts, wr n = n +n n n, n < n. By t nuton ypotss, E(G ) = n or =,. Hn E(G) = E(G ) + E(G ) + = (n ) + (n ) + = n. Corollry. A tr wt n vrts s t lst two vrts o r. Proo. Lt G = (V, E) tr wt n vrts. Sn t s onnt, w v (u) or ll u V. Lt n t numr o vrts o r n G. Tn E = u V (u) n + (n n ). Sn E = n, w v w rsults n n. (n ) n + (n n ),

2 Torm. Lt G rp wt n vrts n n s. T ollown sttmnts r quvlnt: () G s onnt. () G s yl. () G s tr. Proo. () (): Assum G s onnt. Lt H mnml onnt spnnn surp o G (.. H s onnt spnnn surp o G su tt H s sonnt or ll E(H)). Tn H must yl (otrws H woul onnt or n o yl). Hn H s tr n so E(H) = V (H) = n = E(G). Sn H G, w v E(H) = E(G) n n H = G. It ollows tt G s yl. () (): Assum G s yl. Lt H mxml yl rp vn G s spnnn surp (.. H + s yl or ll E(H)). Tn H must onnt n so t s tr. Hn E(H) = V (H) = n = E(G). Sn G H, w v E(H) = E(G) n n H = G. It ollows tt G s tr. () (): Ts ollows rtly rom t nton o tr. Exrs.8 Sow tt ny two vrts n looplss rp G r onnt y unqu pt, tn G s tr. Exrs. Prov Corollry. y sown tt t orn n trmnus o lonst pt n non-trvl tr ot v r. Exrs.0 Sow tt () onnt omponnt o orst s tr; () G s orst n only E(G) = V (G) ω(g)... Cut s Dnton. A ut n rp G s n o G su tt ω(g ) > ω(g) (tt s, t rmovl o nrss t numr o onnt omponnts). Quston. I s ut o rp G, wt s t xt vlu o ω(g ) ω(g)? I G s onnt, wt n you sy out t rp G? Torm. E(G) s ut o rp G n only ls n no yl o G. Proo. Lt E(G) ut o G. Tn ω(g ) > ω(g) n tr xst u, v V (G) su tt u n v r onnt n G ut not n G. Tus, tr xsts (u, v)-pt P n G tt ontns t = xy, sy P xyq, wr P n Q r (u, x)- n (y, v)-pt, rsptvly. Now, ls n yl C, tn C s n (x, y)-pt n G, n n P (C )Q s (u, v)-wl n G, ontrton. Hn os not l n yl. Convrsly, lt E(G) n tt ls n no yl. Suppos = xy s not ut. Tn ω(g ) = ω(g), n ny two vrts onnt n G r onnt n G. In prtulr, sn x G y, tr xsts n (x, y)-pt P n G. But tn P + s yl o G ontnn, ontrton. Hn s ut.

3 Corollry. Lt G onnt rp. Tn G s tr n only vry o G s ut. Proo. Exrs. Exrs. Sow tt vry vrtx o rp G s vn r, tn G s no ut. (Hnt: Sow tt E(G) s n -sont unon o yls.).. Spnnn trs Dnton. A spnnn tr o rp G s spnnn surp o G tt s tr. Torm. Evry onnt rp s spnnn tr. Proo. Lt G onnt rp n T mnml onnt spnnn surp o G (tt s, onnt spnnn surp o G wt t proprty tt T s sonnt or ll E(T )). But tn vry o T s ut n y Corollry., T s tr. Hn T s spnnn tr o G. Corollry.8 I G s onnt rp, tn E(G) V (G). Proo. Exrs. Corollry. sows tt tr s mnml onnt rp: rmovn ny wll sonnt t. T nxt torm sows tt tr s mxml yl rp: n ny rts yl. Torm. Lt G onnt rp, T ts spnnn tr, n E(G) E(T ). Tn T + ontns unqu yl. Proo. Lt E(G) E(T ), = uv. Sn T s onnt, t ontns (u, v)-pt P, w os not ontn t. Hn P uv s yl n T + tt ontns. Lt C ny yl o T +. Sn T s yl, C nssrly ontns. Furtrmor, C s (u, v)-pt n T. But sn T ontns unqu (u, v)-pt, T + ontns unqu yl. Exrs.0 Dtrmn t numr o prws non-somorp spnnn trs n o t rps n Fur. Rrns or Ston.: Bony. Grp trvrsls In ts ston w onsr two mtos or onstrutn spnnn trs n rp, n, t t sm tm, or trvrsn (vstn ll vrts o) rp. Ts two mtos r lso us n mny otr rp lortms, or xmpl, or nn onnt omponnts o rp. 8

4 Fur : Exrs.... Dpt-rst sr (or trn) Dpt-rst sr s ntrl to lortms or nn ut-vrts n los o rp, stron omponnts o rt rp, n or trmnn wtr or not rp s plnr. Dpt-rst sr n son tr trn n us to solv prolms n w n xustv sr o ll possl solutons s rqur. Assum G s smpl onnt rp. W onstrut pt-rst spnnn tr T n G s ollows. Strtn t vn vrtx v 0, onstrut lonst pt T = v 0 v... v n G. So v s no nours tt r not lry n T. I T ontns ll t vrts o G, w r on. I not, tr to t prvous vrtx v on t pt T n oos nour v or v su tt v s not lry n T. A vrtx v n v v to t tr T. Contnu uln pt rom v s lon s possl, lwys n only vrts not lry n T. Wn ts s no lonr possl, tr to t prvous vrtx n t tr T. Sn no wt ot nponts n T s vr to T, t n rsult s spnnn tr. To urnt unqunss o t rsult, t vrts o G r orr n t stp, t rst vll vrtx s osn to to T. Alortm. Dpt-Frst Sr (rursv vrson) prour DF S(G: onnt rp wt vrts u, u,..., u n ) T := tr onsstn only o vrtx u vst(u ) {T s t pt-rst spnnn tr o G} prour vst(v: vrtx o G) {t rp G n ts tr T r lol vrls r} wl v s nours not yt n T n w := rst nour o v not yt n T T := T + vw vst(w) n Exmpl. Construt pt-rst spnnn tr n t rp n Fur.

5 Fur : Exmpl. Consr t tm omplxty o t lortm. Assumn tt t nours o ny vrtx r ssl n t rown tr T n umnt n onstnt numr o oprtons, w only n to ount t numr o tms t wl loop s ntr n prour vst. Prour vst s ll or vrtx v o t rp, n or ts vrtx, t wl loop s ntr t most (v) tms. So t wl loop s ntr t most v V (G) (v) = tms ltotr, wr = E(G). Ts vs tm omplxty O(), or O(n ) sn n(n ). Exrs. Construt pt-rst spnnn tr n t rp n Fur wt t vrts orr n rvrs orr (.. strtn t vrtx ). Wt out strtn t vrtx wt vrts n snn orr? Exrs. Dsr n lortm usn DFS tt ns t onnt omponnts o rp. Exrs. Suppos you n yoursl n mz. Dsr n lortm usn DFS tt ts you out o t mz. Assum tt you v p o l to mr t psss n/or ntrstons. (You my lso ssum tt l mrs n rs.).. Brt-Frst Sr Brt-rst sr n us to n low-umntn pts n lortms or mxmum lows n ntwors. An, ssum G s smpl onnt rp. T rt-rst sr lortm onstruts spnnn tr T o G y rst nnn out rom vn vrtx v. At stp, n unpross vrtx v s tn o quu, mr pross, n or nour w o v not lry n T, t vw s to t tr T. T vrtx w s tn to t quu o unpross vrts. Vrts r tus to T lvl y lvl: vrts t stn rom v rst, tn vrts t stn, stn n so on. An, or t purpos o unqunss o output, t vrts o G r orr n nours o vn vrtx nsrt nto t quu n tr prorty orr. 0

6 Alortm. Brt-Frst Sr prour BF S(G: onnt rp wt vrts v, v,..., v n ) T := tr onsstn only o vrtx v Q := (v ) {Q s t quu o unpross vrts} wl Q s not mpty n v := rst vrtx n Q rmov v rom Q {now pross vrtx v} wl v s nour not n T n w := rst nour o v not n T put w to t n o Q T := T + vw n n {T s t rt-rst spnnn tr o G} Exmpl. Construt rt-rst spnnn tr n t rp n Fur Fur : Exmpl. Exrs.8 Sow tt t tm omplxty o rt-rst sr s O(), or O(n ), wr n = V (G) n = E(G). Exrs. Construt rt-rst spnnn tr n t rp n Fur wt t vrts orr n rvrs orr(.. strtn t vrtx ). Wt out strtn t vrtx wt vrts n snn orr? Exrs.0 Dsr n lortm usn BFS tt, or vn vrtx u n onnt rp G, trmns t stns o ll otr vrts rom u.

7 .. Appltons o trn Som prolms n solv only y n xustv sr o ll possl solutons. On wy to sr systmtlly s to us son tr, n w ll ntrnl vrts (vrts o r ) rprsnt sons n ll lvs (vrts o r ) rprsnt tr soluton or n. To n soluton, w strt wt squn o sons n n ttmpt to r soluton. Ts s rprsnt y pt rom t root o t son tr (rst vrtx n pt-rst sr) to l. On t s nown tt vn squn o sons os not l to soluton, w tr n t son tr n strt notr squn o sons. T prour s ontnu untl soluton (or ll solutons) s oun, or ls, untl s s stls tt no solutons xst. For t rst xmpl, w n t noton o vrtx-olourn o rp. Dnton. Lt G smpl rp. A -vrtx-olourn o G s mppn : V (G) {,,..., } su tt or ll u, v V (G), uv E(G) (u) (v). T rp G s s to -vrtx-olourl tr xsts -vrtx olourn o G. Spn o olourn, r s notr mportnt rtrzton o prtt rps; ts on uss -vrtx-olourn nst o o yls. Torm. A rp s prtt n only t mts -vrtx-olourn Proo. Exrs. Exmpl. Construt son tr n us trn to n -vrtx-olourn o t rp n Fur. Fur : Exmpl. Exrs. Construt son tr n us trn to trmn wtr or not t rp n Fur n t Ptrsn rp (Fur ) r -vrtx-olourl.

8 0 8 Fur : Exrs. Exrs. T n-quns prolm ss ow n quns n pl on n n n ssor so tt no two quns tt otr (tt s, no two quns l n t sm row, olumn, or ny onl). Construt son tr or t -quns prolm. Exrs. Suppos you v n ons w loo ntl xpt tt on o t ons s ountrt n tror ltr tn otrs. Usn pn ln, you ws to n t ountrt on wt smllst numr o wns. Construt son tr or o t ss n = 8 n n =. Exrs. Usn son tr, nrt ll susts S o t st {,,,, } wt t proprty tt t sum o lmnts n S s n t ntrvl [0, 0]. Rrns or Ston.: Rosn. T Conntor Prolm mnmum spnnn trs n t ry lortm Imn t ollown prolm: A ompny plns to ul ommuntons ntwor onntn numr o ts. E two o t ts must ln wt tlpon ln, ut not nssrly rtly. Tr s vn ost ssot wt rt ln twn two ts. W lns soul ult so tt t totl ost o t ntwor s mnmz? (S Fur 8.) Ts prolm s solv y nn mnmum spnnn tr n wt rp. Dnton.8 A wt rp s rp G totr wt unton w : E(G) R ssnn to numrl wt. (Not: Otn w sll ssum tt ll wts r non-ntv.) W sll prsnt two lortms or onstrutn mnmum spnnn trs: Prm s lortm n Krusl s lortm. Ty pro y sussvly n s o smllst wt tt o not rt yl. T rn s tt n Prm s lortm, t st o s lry osn t stp orms tr n t rp, wl n Krusl s lortm t orms orst. Bot r xmpls o ry lortms.

9 Dnton. A ry lortm s n ppro to solvn n optmzton prolm y mn wt sms to t st o t vry stp (rtr tn onsrn ll squns o stps tt my l to soluton). A ry ppro my or my not l to sl soluton, n t os, t my or my not n n optml soluton. For vry ry lortm w n to prov tt t ns n optml soluton... Prm s lortm Lt G onnt wt rp wt n vrts. Prm s lortm strts y oosn n o smllst wt n puttn t nto t tr T. At sussv stp, w to T n o smllst wt su tt s xtly on npont n T. Ts urnts tt T + wll tr. W stop wn n s v n osn. By Torm., T s spnnn tr o G. In Torm. w sow tt T s n t mnmum spnnn tr. Alortm.0 Prm s Alortm prour P rm(g: wt onnt rp wt n vrts) T := tr onsstn o mnmum-wt only or := to n n := o mnmum wt tt s xtly on npont n T T := T + n {T s mnmum spnnn tr o G} Not tt to m t lortm trmnst, t s o G must prorr, n t stp, mssl s onsr n tr prorty orr. Exmpl. Construt mnmum spnnn tr n t rp n Fur 8 usn Prm s lortm. (Us t lxorp orrn o t s.) 8 8 Fur 8: Exmpl.

10 Torm. Lt G onnt wt rp wt n vrts. Any spnnn tr T onstrut y Prm s lortm ppl to G s mnmum spnnn tr o G. Proo. Lt,,..., n t sussv s osn y Prm s lortm, so E(T ) = {,,..., n }. In ton, lt n ny o G not n T. (I G = T, tn tr s notn to prov.) For =,,..., n, lt T = T [{,,..., }]. Now, or ny spnnn tr T o G, lt (T ) t smllst vlu o su tt E(T ). Lt T mnmum spnnn tr o G wt (T ) s lr s possl. Suppos T s not mnmum spnnn tr o G. Tn (T ) = < n. Ts mns tt,,..., l n ot T n T wl E(T ). By Torm., T + ontns unqu yl C. Sn T s yl, tr s n o C tt os not l n T. In prtulr, n osn so tt prsly on o ts nponts ls n T (ust ollow lon C strtn rom t npont o tt ls n T ; su n xsts sn otrws C woul yl n T = T + ). By Torm., s not ut o T + us t ls n yl. Hn T = (T + ) s onnt rp wt n s, n tror notr spnnn tr o G. Clrly, w(t ) = w(t ) w( ) + w( ). Now, n Prm s lortm, ws osn s n o smllst wt su tt xtly on npont o ls n T. Sn xtly on npont o ls n T, w now tt w( ) w( ). But tn w(t ) w(t ). Ts mns tt T s lso mnmum spnnn tr o G. Howvr, (T ) > = (T ), ontrtn t o o T. Tror, T = T n T s mnmum spnnn tr o G. It n sown tt wt pproprt mplmntton, Prm s lortm s o omplxty O( lo n), wr s t numr o s n n s t numr o vrts n onnt wt rp... Krusl s lortm Lt G onnt wt rp wt n vrts. Krusl s lortm strts y oosn n o smllst wt n puttn t nto t orst F. At sussv stp, w to F n o smllst wt su tt s t most on npont n F. Ts urnts tt F + wll orst. W stop wn n s v n osn. By Torm., F s spnnn tr o G. In Exrs. you wll sow tt F s n t mnmum spnnn tr. Alortm. Krusl s Alortm prour Krusl(G: wt onnt rp wt n vrts) F := trvl rp

11 or := to n n := o mnmum wt n E(G) E(F ) tt os not rt yl n F F := F + n {F s mnmum spnnn tr o G} Not tt to m t lortm trmnst, t s o G must prorr, n t stp, mssl s onsr n tr prorty orr. It n sown tt wt pproprt mplmntton, Krusl s lortm s o omplxty O( lo ), wr s t numr o s n onnt wt rp. Exmpl. Construt mnmum spnnn tr n t rp n Fur usn Krusl s lortm. (Us t lxorp orrn o t s.) 8 8 Fur : Exmpl. Exrs. Prov tt Krusl s lortm ppl to wt onnt rp prous mnmum spnnn tr. Exrs. Dsr n lortm tt onstruts: () mxmum spnnn tr n wt onnt rp; () mnmum spnnn orst n wt rp; () mnmum spnnn tr T n wt onnt rp su tt T ontns sp (yl) st o s. Rrns or Ston.: Rosn

12 . T Sortst Pts Prolm Gvn rp G n two vrts u n v n G, wt s stn rom u to v, tt s, t lnt o sortst (u, v)-pt, n G? In ts ston, w sll tully solv t wt vrson o ts prolm. But rst, wt o w mn y stn n lnt o pt n wt rp? Dnton. Lt G onnt wt smpl rp n u 0, u V (G). Furtrmor, lt P = u 0 u u... u (u 0, u )-pt n G. W n t wt (or lnt) o t pt P s w(p ) = w(u u ) n t (wt) stn rom u 0 to u s = st(u, v) = mn{w(p ) : P s (u 0, u ) pt}. Not tt pt lnt n stn n rp G orrspons to t pt wt (lnt) n stn n wt rp G wt ll wts qul to. On t otr n, P n Q r two pts n wt rp G, tn t s possl to v w(p ) < w(q) ltou P s mor s tn Q (v n xmpl!). T Sortst Pt Prolm ss to n t stn st(u 0, v 0 ) or two vrts u 0 n v 0 n onnt wt rp G. Dstr s Alortm, n nt lortm sr low, n t ns stns st(u 0, v) or ll vrts v V (G), owvr, t wors only or non-ntv wts. It strts y rown tr T rom t vrtx u 0. T tr T s t proprty tt t stp, or vry vrtx v n t tr, t sortst pt rom u 0 to v ontns only vrts lry n T. T nxt vrtx v to to t tr T s osn so tt mn{st(u 0, v) + w(vv ) : v V (T )} ( ) s s smll s possl ovr ll vrts not n T. (I v v, tn w n w(vv ) =.) An mportnt tur o t lortm s llln prour, w vos rptn lultons n (*). W not t vrtx st o t tr T y S. At t nnn, S =, t ll o vrtx u 0 s 0, n ll otr vrts v u 0 r ssn ll L(v) =. At sussv stp, vrtx v to to S s osn so tt L(v ) s s smll s possl ovr ll vrts not n S (ts s xtly t sm s Conton (*)). As v s to S, t lls o ll vrts r upt so tt t stp t ollown ols: n L(u) = st(u 0, u) or ll vrts u S (A) L(v) = mn{l(u) + w(uv) : u S} or ll vrts v S. (B) Not tt Clm (A) mpls tt t t n o t lortm L(v) s t stn rom u 0 to v or ll vrts v. Ts wll prov n Torm.0.

13 Alortm.8 Dstr s Alortm prour Dstr(G: wt onnt rp; u 0 : vrtx) {to n t stns rom u 0 to ll otr vrts} {G s n vrts n non-ntv wt unton w, wr w(uv) = u v n uv E(G), n w(uu) = 0} or vry u V (G) L(u) := L(u 0 ) := 0 S := {} or := to n n v := vrtx n V (G) S su tt L(v ) s s smll s possl S := S {v } or vry v V (G) S L(v ) + w(vv ) < L(v) tn L(v) := L(v ) + w(vv ) n {L(v) s t stn rom u 0 to v } Exrs. Us Dstr s lortm to n stns rom vrtx n t wt rp n Fur.. W sll now prov orrtnss o Dstr s lortm. Not tt n ntrnl vrtx o pt s ny vrtx o t pt tt s not n npont; tt s, t ntrnl vrts o t pt v 0 v v... v v r t vrts v, v,..., v. Torm.0 Dstr s lortm ns t stns rom vn vrtx n onnt wt rp wt non-ntv wts. Proo. In ton to t notton ntrou n t lortm, lt S not t st S, n L (u) t ll o vrtx u tr t -t trton o t lortm (or = 0,,..., n). Tus S 0 =, L 0 (u 0 ) = 0, n L 0 (u) = or ll u u 0. By nuton, w sll prov tt t ollown ol or =,..., n: n L (u) = st(u 0, u) or ll vrts u S (A ) L (v) = mn{l (u) + w(uv) : u S } or ll vrts v S. (B ) Not tt (B ) sys tt or v S, L (v) s t lnt o sortst (u 0, v)-pt wt ll ntrnl vrts n S. It s not ult to s tt S = {u 0 }, n L (u 0 ) = 0, L (u) = w(u 0 u) or ll u u 0, n L (v) = or ll otr vrts v. Hn (A ) n (B ) ol or =. Ts provs t ss o nuton. Suppos (A ) n (B ) ol or som, < n (nuton ypotss). Lt v S + S, tt s, Dstr s lortm s osn to v to S urn t ( + )-st trton. W lm tt L (v ) s t lnt o sortst (u 0, v )-pt. Suppos not. Sn 8

14 Fur 0: Exmpl.

15 y nuton ypotss L (v ) s t lnt o sortst (u 0, v )-pt wt ll ntrnl vrts n S, tr xsts (u 0, v ) pt P = u 0... v... v, wr v s t rst vrtx not n S, su tt w(p ) < L (v ). But sn ll wts r non-ntv n L (v) s t lnt o sortst (u 0, v)-pt wt ll ntrnl vrts n S, w v w(p ) = w(u 0... v) + w(v... v ) w(u 0... v) L (v) L (v ), ontrton. Hn (A ) ols or S + = S {v }. Now t ny vrtx v S + n sortst (u 0, v)-pt P wt ll ntrnl vrts n S +. I P os not ontn v, tn w(p ) = L (v) y nuton ypotss n L + (v) = L (v). Otrws, P = Qv v, wr Q s sortst (u 0, v )-pt. In ts s, w(p ) = w(q)+w(v v) = L (v )+w(v v) = L + (v). W v tus prov t nuton stp or (B ). By nuton, (A ) n (B ) ol or ll =,..., n, wn Dstr s lortm orrtly omputs stns rom vrtx u 0. Lt us now stmt t orr o omplxty o Dstr s lortm (,..., r onstnts): n n O( + n + ( + (n + ))) = O( n + ) = O(n ). = Exrs. Fn stns rom vrtx n t rp rom Fur.. Exrs. Moy Dstr s lortm so tt t trmns sortst pts rtr tn mrly stns. Exrs. A ompny s rns n o sx ts C,..., C. T r or rt lt rom C to C s vn y t (, )-t ntry n t ollown mtrx ( nts tt tr s no rt lt): T ompny wss to prpr tl o pst routs twn prs o ts. Prpr su tl. Exrs. W v mnton ov tt Dstr s lortm wors only or rps wt ll wts non-ntv. Gv n xmpl o wt rp wt ntv wts or w Dstr s lortm norrtly omputs stns. Rrns or Ston.: Bony, Rosn = 0