Weighted Graphs. Weighted graphs may be either directed or undirected.

Size: px

Start display at page:

Download "Weighted Graphs. Weighted graphs may be either directed or undirected."

Tracy Patterson
5 years ago
Views:

1 1 In mny ppltons, o rp s n ssot numrl vlu, ll wt. Usully, t wts r nonntv ntrs. Wt rps my tr rt or unrt. T wt o n s otn rrr to s t "ost" o t. In ppltons, t wt my msur o t lnt o rout, t pty o ln, t nry rqur to mov twn lotons lon rout, t. t Struturs & Alortms MQun

2 Sortst Pts (SSA) 2 Gvn wt rp, n snt no S, w woul lk to n pt o lst totl wt rom S to o t otr vrts n t rp. T totl wt o pt s t sum o t wts o ts s. W v sn tt prormn FS or BFS on t rp wll prou spnnn tr, ut ntr o tos lortms tks wts nto ount. Tr s smpl, ry lortm tt wll solv ts prolm. t Struturs & Alortms MQun

3 jkstr's SSA Alortm* 3 Lt S t st o vrts o Gor w w v trmn t sortst pt stn (v)rom sto v. Intlly, S= {s} n (s)= 0. s Sussvly, oos t vrtx vw stss: p(v) = mnmum( (u)+ wt(u,v) ) (u,v):u S Tn vto S, n st (v)to p(v). Trmnt wn S ontns ll t vrts tt r rl rom s. t Struturs & Alortms * MQun

4 jkstr's Alortm Tr 4 Lt t sour vrtx. S = {} 0 S = {,} 0 40 t Struturs & Alortms MQun

5 jkstr's Alortm Tr Contnun: S = {,, } S = {,,, } S = {,,,, } S = {,,,,, } S = {,,,,,, } t Struturs & Alortms MQun

6 jkstr's Alortm Tr Contnun: 6 S = {,,,,,,, } S = {,,,,,,,, } T orrsponn tr s sown t lt. As sr, t lortm os not mntn ror o t s tt wr us, ut tt n sly rm. t Struturs & Alortms MQun

7 Lmttons 7 jkstr's SSA Alortm only works or rps wt non-ntv wts. S t Bllmn-For Alortm, w works vn t wts r ntv, prov tr s no ntv yl( yl wos totl wt s ntv). t Struturs & Alortms MQun

8 Mnml Spnnn Tr 8 Gvn wt rp, w woul lk to n spnnn tr or t rp tt s mnml totl wt. T totl wt o spnnn tr s t sum o t wts o ts s. W wnt to n spnnn tr T, su tt T' s ny otr spnnn tr or t rp tn t totl wt o T s lss tn or qul to tt o T'. t Struturs & Alortms MQun

9 Jrnk-Prm MST Alortm By moyn jkstr s SSA Alortm to ul lst o t s tt r us s vrts r, n storn t stn rom nos to t urrnt tr (rtr tn rom nos to t sour) w otn Prm s Alortm (V Jrnk, 1930 n R C Prm, 197). It turns out tt ts lortm os, n t, rt spnnn tr o mnml wt t rp to w t s ppl s onnt. Sn t omplx stps n Prm s lortm r t sm s jkstr s, no tl xmpl s tr r. 9 QTP: wy os jkstr's SSA Alortm not nssrly n mnmum-wt spnnn tr? t Struturs & Alortms MQun

10 Jrnk-Prm MST Alortm Tr A B C E F G H I n n n n n n A n n n n n n B n B n E n E n H E C t Struturs & Alortms MQun

11 Corrtnss o jkstr s Alortm 11 Lt S t st o xplor nos;.., nos or w w v ssn (lm) nl mnmum stn. Lt s t strt vrtx. For ny vrtx un S, lt (u) t (tul) sortst pt stn rom sto u. For ny vrtx vn t rp ut not n S, lt p(v) t sortst stn rom s to vtt w v oun so r. Tn: u s v t Struturs & Alortms MQun

12 Corrtnss o jkstr s Alortm 12 Tm: or no un S, (u)s t lnt o t sortst pt rom sto u. proo y nuton on S : I S = 1, tn S = {} s n ( s) = 0, w s lrly mnml. Suppos S = K 1, n tt t rsult ols or K. Lt v t nxt vrtx to S, n lt ( u, v) t us, wr us n S. u s v t Struturs & Alortms MQun

13 Corrtnss o jkstr s Alortm 13 Suppos tr s notr pt n Grom sto v. Lt ( x, y) t rst n tt pt tt lvs S. But tn ( x)+ wt( x, y) u ( ) + wt( uv, ), otrws ywoul v n to S nst o v. So, tr nnot sortr pt rom sto v. u s v x y t Struturs & Alortms MQun

Lecture 20: Minimum Spanning Trees (CLRS 23)

Lecture 20: Minimum Spanning Trees (CLRS 23) 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