Minimum Spanning Tree. EECS 214, Fall 2017
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1 Minimum Spnning Tr EECS 4, Fll 0
2 Dinition: spnning tr For onnt omponnt o grph, spnning tr is yl-r sust o gs tht touh vry non-isolt vrtx:
3 Dinition: spnning tr For onnt omponnt o grph, spnning tr is yl-r sust o gs tht touh vry non-isolt vrtx:
4 Dinition: spnning tr For onnt omponnt o grph, spnning tr is yl-r sust o gs tht touh vry non-isolt vrtx:
5 Dinition: spnning orst I grph hs multipl omponnts thn h will hv its own spnning tr, orming spnning orst: g h i
6 Dinition: miniml spnning tr In wight grph, spnning tr (or orst) is miniml i th sum o its wights is miniml ovr ll possil spnning trs: 4
7 Computing n MST It s surprisingly sy thr r two simpl, gry lgorithms: Prim s Kruskl s
8 6 Prim s lgorithm Buil tr g-y-g, s ollows:. Strt th tr t ny vrtx. Fin th smllst g onnting tr vrtx to non-tr vrtx, n it to th tr. Rpt until ll vrtis r in th tr
9 6 Prim s lgorithm Buil tr g-y-g, s ollows:. Strt th tr t ny vrtx. Fin th smllst g onnting tr vrtx to non-tr vrtx, n it to th tr. Rpt until ll vrtis r in th tr
10 6 Prim s lgorithm Buil tr g-y-g, s ollows:. Strt th tr t ny vrtx. Fin th smllst g onnting tr vrtx to non-tr vrtx, n it to th tr. Rpt until ll vrtis r in th tr
11 6 Prim s lgorithm Buil tr g-y-g, s ollows:. Strt th tr t ny vrtx. Fin th smllst g onnting tr vrtx to non-tr vrtx, n it to th tr. Rpt until ll vrtis r in th tr
12 6 Prim s lgorithm Buil tr g-y-g, s ollows:. Strt th tr t ny vrtx. Fin th smllst g onnting tr vrtx to non-tr vrtx, n it to th tr. Rpt until ll vrtis r in th tr
13 6 Prim s lgorithm Buil tr g-y-g, s ollows:. Strt th tr t ny vrtx. Fin th smllst g onnting tr vrtx to non-tr vrtx, n it to th tr. Rpt until ll vrtis r in th tr
14 Kruskl s lgorithm Buil svrl trs n join thm, s ollows:. Strt with trivil tr t vry vrtx. Consir th gs in orr rom smllst to lrgst. Whn n g woul join two sprt trs, us it omin thm into on tr
15 Kruskl s lgorithm Buil svrl trs n join thm, s ollows:. Strt with trivil tr t vry vrtx. Consir th gs in orr rom smllst to lrgst. Whn n g woul join two sprt trs, us it omin thm into on tr
16 Kruskl s lgorithm Buil svrl trs n join thm, s ollows:. Strt with trivil tr t vry vrtx. Consir th gs in orr rom smllst to lrgst. Whn n g woul join two sprt trs, us it omin thm into on tr
17 Kruskl s lgorithm Buil svrl trs n join thm, s ollows:. Strt with trivil tr t vry vrtx. Consir th gs in orr rom smllst to lrgst. Whn n g woul join two sprt trs, us it omin thm into on tr
18 Kruskl s lgorithm Buil svrl trs n join thm, s ollows:. Strt with trivil tr t vry vrtx. Consir th gs in orr rom smllst to lrgst. Whn n g woul join two sprt trs, us it omin thm into on tr
19 Kruskl s lgorithm Buil svrl trs n join thm, s ollows:. Strt with trivil tr t vry vrtx. Consir th gs in orr rom smllst to lrgst. Whn n g woul join two sprt trs, us it omin thm into on tr
20 Implmnting Kruskl s lgorithm W n wy to kp trk o th isjoint trs
21 9 Disjoint Sts ADT (k Union-Fin) Looks lik: 0 { } { } 4 6 Signturs: union(disjointsts, Nt, Nt): Voi in(disjointsts, Nt): Nt Bhvior: union(, p, q) uss p n q s sts to join togthr in(, p) rturns rprsnttiv lmnt tht will th sm or vry lmnt o p s st
22 0 Kruskl s lgorithm using isjoint sts Input: A wight grph grph o n vrtis Output: A minimum spnning orst orst (rprsnt s grph) u nw union-in univrs with n ojts; orst grph o n vrtis n 0 gs; or h g (u, v) in inrsing orr o wight w o i in(u, u) in(u, v) thn union(u, u, v); Eg(orst, u, v) n n
23 Implmnting union-in
24 Gol Eiint t strutur or union-in: in n union ommns n intrlv numr o oprtions m n hug numr o ojts n n hug Lt s lso think out iiny in trms o Kruskl s lgorithm: O(E log E + ET in + VT union )
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