Lecture 20: Minimum Spanning Trees (CLRS 23)

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1 Ltur 0: Mnmum Spnnn Trs (CLRS 3) Jun, 00 Grps Lst tm w n (wt) rps (unrt/rt) n ntrou s rp voulry (vrtx,, r, pt, onnt omponnts,... ) W lso suss jny lst n jny mtrx rprsntton W wll us jny lst rprsntton unlss stt otrws (O( V + E ) sp). W suss O( V + E ) rt-rst (BFS) n pt-rst sr (DFS) lortms n ow ty n us to omput.. onnt omponnts, sortst pt stns n unwt rps, n solv t topolol sortn prolm. W wll now strt sussn mor omplt prolms/lortms on wt rps. Mnmum Spnnn tr (MST) Prolm: Gvn onnt, unrt rp G =(V, E) wr (u, v) swt w(u, v). Fn yl st T E onntn ll vrts n V wt mnml wt w(t )= (u,v) T w(u, v) Not: Prolm s to n spnnn tr (yl st onntn ll vrts) o mnml wt. (w us mnmum spnnn tr s sort or mnmum wt spnnn tr). MST prolm s mny ppltons For xmpl, tnk out onntn ts wt mnml mount o wr (ts r vrts, wt o s r stns twn ty prs). Exmpl: Wt omsts =3 MST s not unqu:.. (, ) n xn wt (, )

2 . PRIM s lortm Gry lortm or omputn MST: Strt wt spnnn tr ontnn rtrry vrtx r n no s Grow spnnn tr y rptly n mnml wt onntn vrtx n urrnt spnnn tr wt vrtx not n t tr On t xmpl rp, t ry lortm woul work s ollows (strtn t vrtx ): ) ) ) ) ) ) ) ) Implmntton: To n mnml onnt to urrnt tr w mntn prorty quu on vrts not n t tr. T ky/prorty o vrtx s t wt o mnml wt onntn t to t tr. (W mntn pontr rom jny lst ntry o v to v n t prorty quu).

3 Anlyss: PRIM(r) For v V DO Insrt(Q, v, ) Cn(Q, r, 0) WHILE Q not mpty DO u = Dltmn(Q) For (u, v) E DO IF v Q n w(u, v) < ky(v) THEN vst[v] =u Cn(Q, v, w(u, v)) FI Wl loop runs V tms w prorm V Dltmn s W prorm t most on Cn or o t E s O(( V + E )lo V )=O( E lo V ) runnn tm. Corrtnss: As suss prvously, wn snn ry lortm t r prt s otn to prov tt t works orrtly. W wll prov Torm tt llows us to prov t orrtnss o nrl lss o ry MST lortms: Som ntons A ut S s prtton o V nto sts S n V \ S A (u, v) rosss ut S u S n v V \ S or v S n u V \ S A ut S rspts st T E no n T rosss t ut Exmpl: Cut S rspts T "ut" S = T V \ S Torm: I G =(V,E) s rp su tt T E s sust o som MST o G, nss ut rsptn T tn tr s MST or G ontnn T n t mnmum wt =(u, v) rossn S. 3

4 Not: Corrtnss o Prm s lortm ollows rom t Torm y nuton ut onsst o urrnt spnnn tr. Proo: Lt T MST ontnn T I T w r on I / T : Tr ot to (t lst) on otr (x, y) T rossn t ut S su tt tr s unqu pt rom u to v n T (T s spnnn tr) x Cut u y v = T Ts pt totr wt orms yl I w rmov (x, y) romt n nst, w stll v spnnn tr Nw spnnn tr must v sm wt s T sn w(u, v) w(x, y) Tr s MST ontnn T n. T Torm llows us to sr vry strt ry lortm or MST: T = Wl T V DO Fn ut S rsptn T Fn mnml rossn S T = T {} Prm s lortm ollows ts strt lortm. 3 Kruskl s Alortm Kruskl s lortm s notr mplmntton o t strt lortm. I n Kruskl s lortm: Strt wt V trs (on or vrtx) Consr s E n nrsn orr; t onnts two trs

5 Exmpl: ) ) ) ) ) ) ) ) Corrtnss o Kruskl s lortm ollows rom Torm: I mnml onnts two trs tn ut rsptn t urrnt st o s xsts (ut onsstn o vrts n on o t trs) 5

6 Implmntton: KRUSKAL T = FOR vrtx v V DO Mk-St(v) Sort s o E n nrsn orr y wt FOR =(u, v) E n orr DO IF Fn-St(u) Fn-St(v) THEN T=T {} Unon-St(u, v) FI Not: W n (Unon-Fn) t strutur tt supports: Mk-st(v): Crt st onsstn o v Unon-st(u, v): Unt st ontnn u n st ontnn v Fn-st(u): Rturn unqu rprsnttv or st ontnn u W us O( E lo E ) tmtosortsnwprorm V Mk-St, V Unonst, n E Fn-St oprtons. Nxt tm w wll suss smpl soluton to t Unon-Fn prolm (mntn st systm unr Fn-St n Unon-St) su tt Mk-St n Fn-St tk O() tm n Unon-St tks O(lo V ) tm mortz. Kruskl s lortm runs n tm O( E lo E + V lo V ) =O(( E + V )lo E )= O( E lo V ) lk Prm s lortm. Prm s lortm n mprov to O( V lo V + E ) usn notr p (Fon p) Vry rntly n O( V + E ) rnomz mnmum spnnn tr lortm s n vlop.

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