Spanning Trees. BFS, DFS spanning tree Minimum spanning tree. March 28, 2018 Cinda Heeren / Geoffrey Tien 1

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1 Spnnn Trs BFS, DFS spnnn tr Mnmum spnnn tr Mr 28, 2018 Cn Hrn / Gory Tn 1

2 Dpt-rst sr Vsts vrts lon snl pt s r s t n o, n tn ktrks to t rst junton n rsums own notr pt Mr 28, 2018 Cn Hrn / Gory Tn 2

3 Dpt-rst sr Alortm Alortm DFS G, v { mrk v s vst or ll vrts u jnt to v, o { (u s not n vst) { DFS G, u } } mrk v s ns output v to ns orr lst } Mr 28, 2018 Cn Hrn / Gory Tn 3

4 Dpt-rst sr , 3 1 0, 2, 4, 6, , 3, 8, 9 3 0, , 5, 6, , 4, 7 Vst: T T T T T T T T T T 7 1, 4, Vst orr: Fns orr: Runnn tm? Cn lso mplmnt lk BFS usn stk Mr 28, 2018 Cn Hrn / Gory Tn 4

5 Appltons o rp trvrsl? Otr ppltons: Topolol sort (DFS) Moln tr low Cyl tton Sortst pt n mz (BFS) Mr 28, 2018 Cn Hrn / Gory Tn 5

6 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 Mr 28, 2018 Cn Hrn / Gory Tn 6

7 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 Mr 28, 2018 Cn Hrn / Gory Tn 7

8 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 Mr 28, 2018 Cn Hrn / Gory Tn 8

9 DFS spnnn tr Root t DFS spnnn tr t ny vrtx,.. Mrk vrts n s us n DFS trvrsl,,,,,,,, E,,,, Rturn G = V, E Mr 28, 2018 Cn Hrn / Gory Tn 9

10 BFS spnnn tr Root t BFS spnnn tr t ny vrtx,.. Mrk vrts n s us n BFS trvrsl,,,,,,,, E,,,, Rturn G = V, E Mr 28, 2018 Cn Hrn / Gory Tn 10

11 Mnmum spnnn trs T suprm rulr o Boolsvll s pskt Prolm: nstlln powr lns n Boolsvll to supply ll t rsntl/ommrl/t. strts ut t osts $5 pr unt stn to onstrut t powr ln Fn onurton o mnml ost tt onnts ll t strts Ts s mnml spnnn tr Mr 28, 2018 Cn Hrn / Gory Tn 11

12 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 B 7 C 1 D B C 1 D B C 1 D E F E F E F Wt = 15 Wt = 13 Mr 28, 2018 Cn Hrn / Gory Tn 12

13 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 Mr 28, 2018 Cn Hrn / Gory Tn 13

14 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! Mr 28, 2018 Cn Hrn / Gory Tn 14

15 Kruskl's lortm Exmpl D 16 9 A 8 12 G C 13 F 4 B H 7 16 E MST wt: 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 Mr 28, 2018 Cn Hrn / Gory Tn 15

16 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 Mr 28, 2018 Cn Hrn / Gory Tn 16

17 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 Mr 28, 2018 Cn Hrn / Gory Tn 17

18 Rns or ts lsson Crrno & Hnry Cptrs (Spnnn trs, mnmum spnnn trs) Nxt lss: Crrno & Hnry, Cptr (Sortst pts) Mr 28, 2018 Cn Hrn / Gory Tn 18

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