23 Minimum Spanning Trees

Transcription

1 3 Mnmum Spnnn Trs Eltron rut sns otn n to mk t pns o svrl omponnts ltrlly quvlnt y wrn tm totr. To ntronnt st o n pns, w n us n rrnmnt o n wrs, onntn two pns. O ll su rrnmnts, t on tt uss t lst mount o wr s usully t most srl. W n mol ts wrn prolm wt onnt, unrt rp G D.V; E/,wrV s t st o pns, E s t st o possl ntronntons twn prs o pns, n or.u; / E, wvwtw.u;/ spyn t ost (mount o wr n) to onnt u n. W tn ws to n n yl sust T E tt onnts ll o t vrts n wos totl wt w.t / D X w.u;/.u;/t s mnmz. Sn T s yl n onnts ll o t vrts, t must orm tr, w wll spnnn tr sn t spns t rp G. Wll t prolm o trmnn t tr T t mnmum-spnnn-tr prolm. Fur 3. sows n xmpl o onnt rp n mnmum spnnn tr. In ts ptr, w sll xmn two lortms or solvn t mnmumspnnn-tr prolm: Kruskl s lortm n Prm s lortm. W n sly mk o tm run n tm O.E l V/usn ornry nry ps. By usn Fon ps, Prm s lortm runs n tm O.E C V l V/,wmprovs ovr t nry-p mplmntton jv j s mu smllr tn jej. T two lortms r ry lortms, s sr n Cptr 6. E stp o ry lortm must mk on o svrl possl os. T ry strty vots mkn t o tt s t st t t momnt. Su strty os not nrlly urnt tt t wll lwys n lolly optml solutons T prs mnmum spnnn tr s sortn orm o t prs mnmum-wt spnnn tr. W r not, or xmpl, mnmzn t numr o s n T,snllspnnntrsv xtly jv j s y Torm B..

2 3. Grown mnmum spnnn tr Fur 3. Amnmumspnnntroronntrp. Twtsonsrsown, n t s n mnmum spnnn tr r s. T totl wt o t tr sown s 37. Ts mnmum spnnn tr s not unqu: rmovn t.; / n rpln t wt t.; / yls notr spnnn tr wt wt 37. to prolms. For t mnmum-spnnn-tr prolm, owvr, w n prov tt rtn ry strts o yl spnnn tr wt mnmum wt. Altou you n r ts ptr npnntly o Cptr 6, t ry mtos prsnt r r lss pplton o t tortl notons ntrou tr. Ston 3. ntrous nr mnmum-spnnn-tr mto tt rows spnnntrynonttm. Ston3.vstwolortms tt mplmnt t nr mto. T rst lortm, u to Kruskl, s smlr to t onnt-omponnts lortm rom Ston.. T son, u to Prm, rsmls Djkstr s sortst-pts lortm (Ston.3). Bus tr s typ o rp, n orr to prs w must n tr n trms o not just ts s, ut ts vrts s wll. Altou ts ptr ouss on trs n trms o tr s, w sll oprt wt t unrstnn tt t vrts o tr T r tos tt som o T s nnt on. 3. Grown mnmum spnnn tr Assum tt w v onnt, unrt rp G D.V; E/ wt wt unton w W E! R, nwwstonmnmumspnnntrorg. T two lortms w onsr n ts ptr us ry ppro to t prolm, ltou ty r n ow ty pply ts ppro. Ts ry strty s ptur y t ollown nr mto, w rows t mnmum spnnn tr on t tm. T nr mto mns st o s A,mntnntollownloopnvrnt: Pror to trton, A s sust o som mnmum spnnn tr. At stp, w trmn n.u; / tt w n to A wtout voltn ts nvrnt, n t sns tt A [.u; / s lso sust o mnmum spnnn

3 66 Cptr 3 Mnmum Spnnn Trs tr. W ll su n s or A,snwntslytoA wl mntnn t nvrnt. GENERIC-MST.G; w/ A D; wl A os not orm spnnn tr 3 nn.u; / tt s s or A A D A [.u; / 5 rturn A W us t loop nvrnt s ollows: Intlzton: Atr ln, tst A trvlly stss t loop nvrnt. Mntnn: T loop n lns mntns t nvrnt y n only s s. Trmnton: All s to A r n mnmum spnnn tr, n so t st A rturn n ln 5 must mnmum spnnn tr. T trky prt s, o ours, nn s n ln 3. On must xst, sn wn ln 3 s xut, t nvrnt tts tt tr s spnnn tr T su tt A T. Wtn t wl loop oy, A must propr sust o T,n tror tr must n.u; / T su tt.u; / 6 A n.u; / s s or A. In t rmnr o ts ston, w prov rul (Torm 3.) or ronzn s s. T nxt ston srs two lortms tt us ts rul to n s s ntly. W rst n som ntons. A ut.s; V S/ o n unrt rp G D.V; E/ s prtton o V.Fur3.llustrtstsnoton.Wsyttn.u; / E rosss t ut.s; V S/ on o ts nponts s n S n t otr s n V S. Wsyttutrspts sta o s no n A rosss t ut. An s lt rossn ut ts wt s t mnmum o ny rossn t ut. Not tt tr n mor tn on lt rossn ut n t s o ts. Mor nrlly, w sy tt n s lt stsyn vn proprty ts wt s t mnmum o ny stsyn t proprty. Our rul or ronzn s s s vn y t ollown torm. Torm 3. Lt G D.V; E/ onnt, unrt rp wt rl-vlu wt unton w n on E. LtA sust o E tt s nlu n som mnmum spnnn tr or G, lt.s; V S/ ny ut o G tt rspts A, nlt.u; / lt rossn.s; V S/.Tn,.u; / s s or A.

4 3. Grown mnmum spnnn tr S V S 7 () 0 S V S S 7 0 V S () Fur 3. Two wys o vwn ut.s; V S/ o t rp rom Fur 3.. () Blk vrts r n t st S, nwtvrtsrnv S. T s rossn t ut r tos onntn wt vrts wt lk vrts. T.; / s t unqu lt rossn t ut. A sust A o t s s s; not tt t ut.s; V S/ rspts A, snnooa rosss t ut. () T sm rp wt t vrts n t st S on t lt n t vrts n t st V S on t rt. An rosss t ut t onnts vrtx on t lt wt vrtx on t rt. Proo Lt T mnmum spnnn tr tt nlus A, nssumttt os not ontn t lt.u; /, sntos,wron. Wsll onstrut notr mnmum spnnn tr T 0 tt nlus A [.u; / y usn ut-n-pst tnqu, try sown tt.u; / s s or A. T.u; / orms yl wt t s on t smpl pt p rom u to n T,sFur3.3llustrts. Snu n r on oppost ss o t ut.s; V S/,tlstonnT ls on t smpl pt p n lso rosss t ut. Lt.x; y/ ny su. T.x; y/ s not n A, ustut rspts A. Sn.x; y/ s on t unqu smpl pt rom u to n T,rmovn.x; y/ rks T nto two omponnts. An.u; / ronnts tm to orm nwspnnntrt 0 D T.x; y/ [.u; /. W nxt sow tt T 0 s mnmum spnnn tr. Sn.u; / s lt rossn.s; V S/n.x; y/ lso rosss ts ut, w.u;/ w.x;y/. Tror, w.t 0 / D w.t / w.x;y/ C w.u;/ w.t / :

5 6 Cptr 3 Mnmum Spnnn Trs x u p y v Fur 3.3 T proo o Torm 3.. Blk vrts r n S, nwtvrtsrnv S. T s n t mnmum spnnn tr T r sown, ut t s n t rp G r not. T s n A r s, n.u; / s lt rossn t ut.s; V S/. T.x; y/ s n on t unqu smpl pt p rom u to n T.ToormmnmumspnnntrT 0 tt ontns.u; /,rmov t.x; y/ rom T n t.u; /. But T s mnmum spnnn tr, so tt w.t / w.t 0 /;tus,t 0 must mnmum spnnn tr lso. It rmns to sow tt.u; / s tully s or A. WvA T 0, sn A T n.x; y/ 6 A; tus,a [.u; / T 0.Consquntly,snT 0 s mnmum spnnn tr,.u; / s s or A. Torm 3. vs us ttr unrstnn o t workns o t GENERIC- MST mto on onnt rp G D.V; E/. As t mto pros, t st A s lwys yl; otrws, mnmum spnnn tr nlun A woul ontn yl, w s ontrton. At ny pont n t xuton, t rp G A D.V; A/ s orst, n o t onnt omponnts o G A s tr. (Som o t trs my ontn just on vrtx, s s t s, or xmpl, wn t mto ns: A s mpty n t orst ontns jv j trs, on or vrtx.) Morovr, ny s.u; / or A onnts stnt omponnts o G A, sn A [.u; / must yl. T wl loop n lns o GENERIC-MST xuts jv j tms us t ns on o t jv j s o mnmum spnnn tr n trton. Intlly, wn A D;,trrjV j trs n G A,ntrtonrustt numr y. Wntorstontnsonlysnltr,tmtotrmnts. T two lortms n Ston 3. us t ollown orollry to Torm 3..

6 3. Grown mnmum spnnn tr 6 Corollry 3. Lt G D.V; E/ onnt, unrt rp wt rl-vlu wt unton w n on E. Lt A sust o E tt s nlu n som mnmum spnnn tr or G,nltC D.V C ;E C / onnt omponnt (tr) n t orst G A D.V; A/. I.u; / s lt onntn C to som otr omponnt n G A,tn.u; / s s or A. Proo T ut.v C ;V V C / rspts A, n.u; / s lt or ts ut. Tror,.u; / s s or A. Exrss 3.- Lt.u; / mnmum-wt n onnt rp G. Sow tt.u; / lons to som mnmum spnnn tr o G. 3.- Prossor Str onjturs t ollown onvrs o Torm 3.. Lt G D.V; E/ onnt, unrt rp wt rl-vlu wt unton w n on E. LtA sust o E tt s nlu n som mnmum spnnn tr or G, lt.s; V S/ ny ut o G tt rspts A, nlt.u; / s or A rossn.s; V S/.Tn,.u; / s lt or t ut. Sow tt t prossor s onjtur s norrt y vn ountrxmpl Sow tt n.u; / s ontn n som mnmum spnnn tr, tn t s ltrossnsomutotrp. 3.- Gv smpl xmpl o onnt rp su tt t st o s.u; / W tr xsts ut.s; V S/ su tt.u; / s lt rossn.s; V S/ os not orm mnmum spnnn tr Lt mxmum-wt on som yl o onnt rp G D.V; E/. Prov tt tr s mnmum spnnn tr o G 0 D.V; E / tt s lso mnmum spnnn tr o G. Tts,trsmnmumspnnntroG tt os not nlu.

7 630 Cptr 3 Mnmum Spnnn Trs 3.-6 Sow tt rp s unqu mnmum spnnn tr, or vry ut o t rp, tr s unqu lt rossn t ut. Sow tt t onvrs s not tru y vn ountrxmpl Aru tt ll wts o rp r postv, tn ny sust o s tt onnts ll vrts n s mnmum totl wt must tr. Gv n xmpl to sow tt t sm onluson os not ollow w llow som wts to nonpostv. 3.- Lt T mnmum spnnn tr o rp G, nltl t sort lst o t wts o T.SowttornyotrmnmumspnnntrT 0 o G, t lst L s lso t sort lst o wts o T Lt T mnmum spnnn tr o rp G D.V; E/, nltv 0 sust o V.LtT 0 t surp o T nu y V 0,nltG 0 t surp o G nu y V 0.SowttT 0 s onnt, tn T 0 s mnmum spnnn tr o G Gvn rp G n mnmum spnnn tr T,supposttwrst wt o on o t s n T. Sow tt T s stll mnmum spnnn tr or G. Morormlly,ltT mnmum spnnn tr or G wt wts vn y wt unton w. Cooson.x; y/ T n postv numr k, n n t wt unton w 0 y w 0.u; / D ( w.u;/.u; /.x; y/ ; w.x;y/ k.u; / D.x; y/ : Sow tt T s mnmum spnnn tr or G wt wts vn y w ? Gvn rp G n mnmum spnnn tr T,supposttwrst wt o on o t s not n T.Gvnlortmornntmnmum spnnn tr n t mo rp.

8 3. T lortms o Kruskl n Prm T lortms o Kruskl n Prm T two mnmum-spnnn-tr lortms sr n ts ston lort on t nr mto. Ty us sp rul to trmn s n ln 3 o GENERIC-MST. In Kruskl s lortm, t st A s orst wos vrts r ll tos o t vn rp. T s to A s lwys lst-wt n t rp tt onnts two stnt omponnts. In Prm s lortm, t st A orms snl tr. T s to A s lwys lst-wt onntn t tr to vrtx not n t tr. Kruskl s lortm Kruskl s lortm ns s to to t rown orst y nn, o ll t s tt onnt ny two trs n t orst, n.u; / o lst wt. Lt C n C not t two trs tt r onnt y.u; /. Sn.u; / must lt onntn C to som otr tr, Corollry 3. mpls tt.u; / s s or C.Kruskl slortmqulssrylortmus t stp t s to t orst n o lst possl wt. Our mplmntton o Kruskl s lortm s lk t lortm to omput onnt omponnts rom Ston.. It uss sjont-st t strutur to mntn svrl sjont sts o lmnts. E st ontns t vrts n on tr o t urrnt orst. T oprton FIND-SET.u/ rturns rprsnttv lmnt rom t st tt ontns u. Tus,wntrmnwtrtwovrtsu n lon to t sm tr y tstn wtr FIND-SET.u/ quls FIND-SET./. To omn trs, Kruskl s lortm lls t UNION prour. MST-KRUSKAL.G; w/ A D; or vrtx G:V 3 MAKE-SET./ sort t s o G: E nto nonrsn orr y wt w 5 or.u; / G: E,tkn n nonrsn orr y wt 6 FIND-SET.u/ FIND-SET./ 7 A D A [.u; / UNION.u; / rturn A Fur 3. sows ow Kruskl s lortm works. Lns 3 ntlz t st A to t mpty st n rt jv j trs, on ontnn vrtx. T or loop n lns 5 xmns s n orr o wt, rom lowst to st. T loop

9 63 Cptr 3 Mnmum Spnnn Trs () 7 0 () 7 0 () 7 0 () 7 0 () 7 0 () 7 0 () 7 0 () 7 0 Fur 3. T xuton o Kruskl s lortm on t rp rom Fur 3.. S s lon to t orst A n rown. T lortm onsrs n sort orr y wt. An rrow ponts to t unr onsrton t stp o t lortm. I t jons two stnt trs n t orst, t s to t orst, try mrn t two trs. ks, or.u; /, wtrtnpontsu n lon to t sm tr. I ty o, tn t.u; / nnot to t orst wtout rtn yl,ntssr. Otrws,ttwovrtslontornt trs. In ts s, ln 7 s t.u; / to A, nlnmrstvrts n t two trs.

10 3. T lortms o Kruskl n Prm 633 () 7 0 (j) 7 0 (k) 7 0 (l) 7 0 (m) 7 0 (n) 7 0 Fur 3., ontnu Furtr stps n t xuton o Kruskl s lortm. T runnn tm o Kruskl s lortm or rp G D.V; E/ pns on ow w mplmnt t sjont-st t strutur. W ssum tt w us t sjont-st-orst mplmntton o Ston.3 wt t unon-y-rnk n pt-omprsson ursts, sn t s t symptotlly stst mplmntton known. Intlzn t st A n ln tks O./ tm, n t tm to sort t s n ln s O.E l E/. (WwllountortostotjV j MAKE-SET oprtons n t or loop o lns 3 n momnt.) T or loop o lns 5 prorms O.E/ FIND-SET n UNION oprtons on t sjont-st orst. Alon wt t jv j MAKE-SET oprtons, ts tk totl o O..V C E/.V // tm, wr s t vry slowly rown unton n n Ston.. Bus w ssum tt G s onnt, w v jej jv j, nsotsjont-stoprtons tk O.E.V // tm. Morovr, sn.jv j/ D O.l V/D O.l E/,ttotl runnn tm o Kruskl s lortm s O.E l E/. OsrvnttjEj < jv j, w v l jej D O.l V/,nsownrstttrunnntmoKruskl s lortm s O.E l V/.

11 63 Cptr 3 Mnmum Spnnn Trs Prm s lortm Lk Kruskl s lortm, Prm s lortm s spl s o t nr mnmum-spnnn-tr mto rom Ston 3.. Prm s lortm oprts mu lk Djkstr s lortm or nn sortst pts n rp, w w sll s n Ston.3. Prm s lortm s t proprty tt t s n t st A lwys orm snl tr. As Fur 3.5 sows, t tr strts rom n rtrry root vrtx r n rows untl t tr spns ll t vrts n V.Estpstot tr A ltttonntsa to n solt vrtx on on w no o A s nnt. By Corollry 3., ts rul s only s tt r s or A; tror, wn t lortm trmnts, t s n A orm mnmum spnnn tr. Ts strty quls s ry sn t stp t s to t tr n tt ontruts t mnmum mount possl to t tr s wt. In orr to mplmnt Prm s lortm ntly, w n st wy to slt nwtotottrormytsna. Intpsuoolow, t onnt rp G n t root r o t mnmum spnnn tr to rown r nputs to t lortm. Durn xuton o t lortm, ll vrts tt r not n t tr rs n mn-prorty quu Q s on ky ttrut. For vrtx,tttrut:ky s t mnmum wt o ny onntn to vrtx n t tr; y onvnton, :ky D tr s no su. T ttrut : nms t prnt o n t tr. T lortm mpltly mntns t st A rom GENERIC-MST s A D.; :/ W V r Q : Wn t lortm trmnts, t mn-prorty quu Q s mpty; t mnmum spnnn tr A or G s tus A D.; :/ W V r : MST-PRIM.G; w; r/ or u G:V u:ky D 3 u: D NIL r:ky D 0 5 Q D G:V 6 wl Q ; 7 u D EXTRACT-MIN.Q/ or G:AjŒu Q n w.u;/ < :ky 0 : D u :ky D w.u;/

12 3. T lortms o Kruskl n Prm 635 () 7 0 () 7 0 () 7 0 () 7 0 () 7 0 () 7 0 () 7 0 () 7 0 () 7 0 Fur 3.5 T xuton o Prm s lortm on t rp rom Fur 3.. T root vrtx s. Ssrnttrnrown,nlkvrtsrnttr.Atstpo t lortm, t vrts n t tr trmn ut o t rp, n lt rossn t ut s to t tr. In t son stp, or xmpl, t lortm s o o n tr.; / or.; / to t tr sn ot r lt s rossn t ut.

13 636 Cptr 3 Mnmum Spnnn Trs Fur 3.5 sows ow Prm s lortm works. Lns 5 st t ky o vrtx to (xpt or t root r, woskysstto0 so tt t wll t rst vrtx pross), st t prnt o vrtx to NIL, nntlztmnprorty quu Q to ontn ll t vrts. T lortm mntns t ollown tr-prt loop nvrnt: Pror to trton o t wl loop o lns 6,. A D.; :/ W V r Q.. T vrts lry pl nto t mnmum spnnn tr r tos n V Q. 3. For ll vrts Q, : NIL, tn:ky < n :ky s t wt o lt.; :/ onntn to som vrtx lry pl nto t mnmum spnnn tr. Ln 7 nts vrtx u Q nnt on lt tt rosss t ut.v Q; Q/ (wt t xpton o t rst trton, n w u D r u to ln ). Rmovn u rom t st Q s t to t st V Q o vrts n t tr, tus n.u; u:/ to A. Tor loop o lns upts t ky n ttruts o vry vrtx jnt to u ut not n t tr, try mntnn t tr prt o t loop nvrnt. T runnn tm o Prm s lortm pns on ow w mplmnt t mnprorty quu Q. I w mplmnt Q s nry mn-p (s Cptr 6), w n us t BUILD-MIN-HEAP prour to prorm lns 5 n O.V / tm. T oy o t wl loop xuts jv j tms, n sn EXTRACT-MIN oprton tks O.l V/tm, t totl tm or ll lls to EXTRACT-MIN s O.V l V/. T or loop n lns xuts O.E/ tms ltotr, sn t sum o t lnts o ll jny lsts s jej. Wtntor loop, w n mplmnt t tst or mmrsp n Q n ln n onstnt tm y kpn t or vrtx tt tlls wtr or not t s n Q,nuptnttwntvrtxsrmov rom Q. T ssnmnt n ln nvolvs n mplt DECREASE-KEY oprton on t mn-p, w nry mn-p supports n O.l V/tm. Tus, t totl tm or Prm s lortm s O.V l V C E l V/D O.E l V/,ws symptotlly t sm s or our mplmntton o Kruskl s lortm. W n mprov t symptot runnn tm o Prm s lortm y usn Fon ps. Cptr sows tt Fon p ols jv j lmnts, n EXTRACT-MIN oprton tks O.l V/ mortz tm n DECREASE-KEY oprton (to mplmnt ln ) tks O./ mortz tm. Tror, w us Fon p to mplmnt t mn-prorty quu Q,trunnntmoPrm s lortm mprovs to O.E C V l V/.