(Mnmum) Spnnn Trs Spnnn trs Kruskl's lortm Novmr 23, 2017 Cn Hrn / Gory Tn 1
Spnnn trs Gvn G = V, E, spnnn tr o G s onnt surp o G wt xtly V 1 s mnml sust o s tt onnts ll t vrts o G G = Spnnn trs Novmr 23, 2017 Cn Hrn / Gory Tn 2
Spnnn tr proprts A spnnn tr G = V, E must: ontn ll t vrts o G onnt not ontn yls A tr, ut not spnnn tr G = Contns ll t vrts n s t orrt numr o s, ut s yl n s not onnt Novmr 23, 2017 Cn Hrn / Gory Tn 3
Construtn spnnn trs Bk to trvrsls Spnnn trs n onstrut y prormn trvrsl strtn rom ny vrtx, mrkn trvl s n vst vrts.. Brt-rst sr, Dpt-rst sr Novmr 23, 2017 Cn Hrn / Gory Tn 4
DFS spnnn tr Root t DFS spnnn tr t ny vrtx,.. Mrk vrts n s us n DFS trvrsl,,,,,,,, E,,,, Rturn G = V, E Novmr 23, 2017 Cn Hrn / Gory Tn 5
BFS spnnn tr Root t DFS spnnn tr t ny vrtx,.. Mrk vrts n s us n BFS trvrsl,,,,,,,, E,,,, Rturn G = V, E Novmr 23, 2017 Cn Hrn / Gory Tn 6
Mnmum spnnn trs T suprm rulr o Grpln s pskt Prolm: nstlln wtrwys n Grpln to supply ll t rsntl/ommrl/t. strts ut t osts $5 pr unt stn to onstrut t wtrwy Fn onurton o mnml ost tt onnts ll t strts Ts s mnml spnnn tr Novmr 23, 2017 Cn Hrn / Gory Tn 7
Mnml spnnn tr Gvn onnt rp G = V, E wt unonstrn wts Output rp G = V, E wt t ollown rtrsts G s spnnn surp o G Alortm rqurmnts G s onnt n yl (.. tr) T sum o t wts o E s mnml mon ll su spnnn trs A A A 8 2 4 4 2 B 7 C 1 D B C 1 D B C 1 D 2 3 9 5 2 3 5 2 3 5 E F E F E F Wt = 15 Wt = 13 Novmr 23, 2017 Cn Hrn / Gory Tn 8
Kruskl's lortm or mnmum spnnn trs Gry lortm, puls n 1956 Buls spnnn tr rom svrl onnt omponnts Rptly ooss t mnmum-wt jonn two onnt omponnts, w os not orm yl, untl st s V 1 s KrusklsAlortm() { st E = ø wl ( E V 1) { Fn mnmum wt E su tt E A to E } } os not ontn yls Novmr 23, 2017 Cn Hrn / Gory Tn 9
Kruskl's lortm Dt typs or mplmntton KrusklsAlortm() { st E = ø wl ( E V 1) { Fn mnmum wt E su tt E A to E } } os not ontn yls W n ADTs tt support our rqur oprtons ntly How o w n t mnmum wt? Prorty quu! How n w k or yls n prorm unon? Dsjont sts! Novmr 23, 2017 Cn Hrn / Gory Tn 10
Kruskl's lortm Exmpl D 16 9 A 8 12 G 6 10 2 C 13 F 4 B 3 17 5 11 H 7 16 E MST wt: 38 13 A prq A, B 2 B, C 3 G, H 4 E, F 5 F, G 10 F, H 11 D, F 12 B, G 13 E B C D G A, C 6 C, F 13 F H B, E 7 A, D 16 prq ost p orr rry ul rmovmn Not tt no nsrtons prorm tr ul C, D 8 E, H 16 Ovrll ost? D, G 9 C, E 17 Not: only on rton lst n prq or omptnss n ts sl Novmr 23, 2017 Cn Hrn / Gory Tn 11
A slt tnnt mz onstruton Wt mks oo mz? A un o jnt rooms E room s vrtx Opn wll twn rooms orm n Unprtl, not sly solv Hly rnn, mny ns Just nou wlls to t rom ny room to ny otr room Esplly strt n ns out Novmr 23, 2017 Cn Hrn / Gory Tn 12
Mz unr onstruton So r, numr o wlls v n knok own, wl otrs rmn Now w onsr t wll twn rooms A n B Soul w knok t own? I A n B r otrws onnt I A n B r not otrws onnt A B Alortm: Wl s rmn n E Rmov rnom = u, v rom E I u n v v not n onnt A to E Mrk u n v s onnt Ts s lot lk Kruskl's lortm! Solv t usn sjont sts n rnom slton Novmr 23, 2017 Cn Hrn / Gory Tn 13
Rns or ts lsson Wkp ttps://n.wkp.or/wk/kruskl%27s_lortm Nxt lss: Komn Cptr 12.6 (Prm's lortm, sortst pts) Novmr 23, 2017 Cn Hrn / Gory Tn 14