Rückblick: Starke Zusammenhangskomponenten

Transkript

1 Rückblick: Starke Zusammenhangskomponenten Der Algorithmus von Kosaraju bestimmt die starken Zusammenhangskomponenten eines gerichteten Graphen wie folgt: Schritt 1: Bestimme den transponierten Graphen G T (V,E T ) und führe ganzheitliche Tiefensuche auf diesem durch Schritt 2: Ordne die Knoten des Graphen in absteigender Reihenfolge ihrer Endzeitpunkte aus Schritt 2 Schritt : Führe eine ganzheitliche Tiefensuche auf dem ursprünglichen Graphen durch, betrachte hierbei die Knoten gemäß der Reihenfolge aus Schritt 2 als Startknoten 64

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3 Algorithmus von Kosaraju <latexit sha1_base64="hnhjp68jly8zkyrv0qot78qqps=">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</latexit> <latexit sha1_base64="hnhjp68jly8zkyrv0qot78qqps=">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</latexit> <latexit sha1_base64="hnhjp68jly8zkyrv0qot78qqps=">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</latexit> <latexit sha1_base64="khbf1edlb1egv2gxvm5ovw40hr4=">aaagzxiclvrnb9naehvlccv8txdkslbvtjjjbcebvlwqsly4vbsjtjwsqfqvj8kqa6+1uya4jlz5q/wzumipyfrqiluafeopzn5nl2djxuykhulvvzaflo5w71sr92oohjx4/wv17eir5jah0cwdcnlhyaqmbdbvvde5cadhgry7kcpfvwzhkq8+ktieay+hgv0salw2nw6vjlafrdgnegunzyflkhiwkwnx1wocnbexw6vixckuikc7ghyde44inbpet9eevmcizwmzdjaetxswz+vkqcs6zgqanslihmgdkrkgifqivqqbibybj6aykcb9smfsxqmncgfzey1xbzdlzgkjhvsxgunfvnbad5ipu24hdxiaaslbtpklsrjyixpf2vmnfuiugytyf8kqsk1qnvdplcxu+zae8bdcv48speat+ikfxuugpr0cx2a+lovjsazkxsfuplptjam9ywwgrpp+lsizrys90surmgtxjss5tkfktxjqlfe6euzbekjimsu5tuunhdrmqvktof6u1oamtssdu0rpl8o9jopxtpb58t6mv5rg7chzkyfugclmubp7drvk4tti1c7it4t4etzgof1dt+rozrbveh/gnbdyta6uazd07wlud/jjpihuirhkxu2fapbgtngytcr9smjoe5lpiietkldtgkqjid7oeq8d2szwd7iqzlnxrna0b4pdbnqv0chzhs5isg+llhvaqap1vhexvjnkezyplhylxpvzwqre1kluvdgeeudcd6vvzjvw+1bqomw0kow5ikoi4yur+mq1knwafes1gymgupsidlgaholrmtnwumja7u/xotf4s+aomgbpyg2qhlkrvzd9jwdtsxdwoavpozuuprjeonlpt1e2pzvr+dnfik8zz46wxadjgw2pfeg8cgl2dvh5uflv+v/5up1sj6tfqt5y6vftepdpmvvx7x2q1ijg=</latexit> Schritt 1: Bestimmen des transponierten Graphen

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Algorithmus von Kosaraju Schritt 1: Erweiterte ganzheitliche Tiefensuche auf transponiertem Graphen col = pre = beg = end = rep = ord = 67

5 Algorithmus von Kosaraju Schritt 2: Ordnen der Knoten in absteigender Reihenfolge ihrer Endzeitpunkte aus Schritt 1 Die Knoten werden also in der Reihenfolge als Startknoten im folgenden Schritt betrachtet 68

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Algorithmus von Kosaraju Schritt : Erweiterte ganzheitliche Tiefensuche auf dem ursprünglichen Graphen mit neu bestimmter Reihenfolge col = pre = beg = end = rep = ord = Starke Zusammenhangskomponenten: 69

7 Motivation <latexit sha1_base64="6sya/fjqg4mavbs2jr/zkhy8d0=">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</latexit> <latexit sha1_base64="6sya/fjqg4mavbs2jr/zkhy8d0=">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</latexit> <latexit sha1_base64="6sya/fjqg4mavbs2jr/zkhy8d0=">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</latexit> <latexit sha1_base64="2lnsb5vialabb2vowwbkvvqdnq=">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</latexit> Wir lernen nun zwei gierige Algorithmen kennen, um für einen ungerichteten gewichteten zusammenhängenden Graphen einen minimalen Spannbaum zu bestimmen Ein minimaler Spannbaum ist eine Teilmenge T der Kantenmenge E, so dass der Graph G(V, T) ein Baum ist und zudem das Gewicht w(t) = ÿ w(u, v) (u,v)œt unter allen Spannbäumen minimal sein muss 70

8 <latexit sha1_base64="hhlwksuvku4olvgeeputfpqqgki=">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</latexit> <latexit sha1_base64="rcvcu1bswnbds7tiq2we9wzkmta=">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</latexit> <latexit sha1_base64="rcvcu1bswnbds7tiq2we9wzkmta=">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</latexit> <latexit sha1_base64="bdiq761qaemxqfmropt/jxqpxfm=">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</latexit> Motivation Da der Graph G(V, T) ein Baum sein muss, wissen wir, dass T = V - 1 gelten muss Minimale Spannbäume haben praktische Anwendungen z.b. beim Entwurf von Kommunikationsnetzen und Schaltungen WND 55 0 MZG HOM SLS ZW SB 71

9 5. Algorithmus von Kruskal Der Algorithmus von Kruskal ähnelt dem Algorithmus zum Bestimmen von (schwachen) Zusammenhangskomponenten und verwendet ebenfalls eine Union-Find-Datenstruktur Ein wichtiger Unterschied ist, dass der Algorithmus von Kruskal die Kanten des Graphen in aufsteigender Reihenfolge ihres Gewichts betrachtet 72

10 Algorithmus von Kruskal Wird eine Kante betrachtet und befinden sich die dadurch verbundenen Knoten noch in verschiedenen disjunkten Teilmengen, so werden diese Teilmengen vereinigt und die Kante ist Teil des minimalen Spannbaums Zu jedem Zeitpunkt bilden die disjunkten Teilmengen in der Union-Find-Datenstruktur einen Wald aller Knoten Da der Graph zusammenhängend ist, bleibt am Ende nur eine disjunkte Teilmenge, nämlich die gesamte Knotenmenge, übrig 7

11 Algorithmus von Kruskal 1 public class Kruskal { 2 // Knoten u der Kanten (u,v) im minimalen Spannbaum 4 private int[] tu; 5 6 // Knoten v der Kanten (u,v) im minimalen Spannbaum 7 private int[] tv; 8 9 public int[] gettu() { 10 return tu; 11 } 12 1 public int[] gettv() { 14 return tv; 15 } public void computeminimumspanningtree(graph g, Weighting w) { Die Arrays tu und tv enthalten bei Beendung des Algorithmus die im minimalen Spannbaum enthaltenen Kanten 74

12 Algorithmus von Kruskal 17 public void computeminimumspanningtree(graph g, Weighting w) { // Kanten initialisieren und sortieren 20 int[] gu = new int[g.getedgecount() / 2]; 21 int[] gv = new int[g.getedgecount() / 2]; 22 int[] gw = new int[g.getedgecount() / 2]; 2 int i = 0; for ( int u = 0; u < g.getnodecount(); u++) { 26 for ( int v : g.getlist(u)) { 27 // Kante (u,v) bzw. (v,u) nur einmal betrachten 28 if (u < v) { 29 gu[i] = u; 0 gv[i] = v; 1 gw[i] = w.getweight(u, v); 2 i++; } 4 } 5 } 6 sort(gu, gv, gw); 7 8 // Union -Find- Datenstruktur initialisieren 75

13 Algorithmus von Kruskal 7 8 // Union -Find- Datenstruktur initialisieren 9 UnionFind uf = new UnionFind(g.getNodeCount()); 40 for ( int u = 0; u < g.getnodecount(); u++) { 41 uf.make(u); 42 } 4 44 // Baumkanten initialisieren 45 tu = new int[g.getnodecount() - 1]; 46 tv = new int[g.getnodecount() - 1]; 47 int j = 0; // Alle Kanten im Graphen betrachten 50 for ( int e = 0; e < g.getedgecount() / 2; e++) { 51 if (uf.find(gu[e])!= uf.find(gv[e])) { 52 uf.union(gu[e], gv[e]); 5 tu[j] = gu[e]; 54 tv[j] = gv[e]; 55 j++; 56 } 57 } 58 } 59 } 76

14 <latexit sha1_base64="+ssej7skcosxr2gupjri5oac7lw=">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</latexit> <latexit sha1_base64="+ssej7skcosxr2gupjri5oac7lw=">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</latexit> <latexit sha1_base64="+ssej7skcosxr2gupjri5oac7lw=">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</latexit> <latexit sha1_base64="v1sonzfrbuap57myuhbe/cpcdu=">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</latexit> Algorithmus von Kruskal <latexit 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<latexit 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<latexit sha1_base64="itsozftuvjizhli+idwhuf+6jv0=">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</latexit> <latexit sha1_base64="iidp8xd9jcmeqmncqehcnkrxlyw=">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</latexit> Beispiel: Algorithmus von Kruskal angewendet auf einen ungerichteten gewichteten Graphen gu = [ ] gv = [ ] gw = [ ] r = [ ] 77

15 <latexit sha1_base64="ggjmzehbfdhh/mepsqtt/jtkylu=">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</latexit> <latexit sha1_base64="ggjmzehbfdhh/mepsqtt/jtkylu=">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</latexit> <latexit sha1_base64="ggjmzehbfdhh/mepsqtt/jtkylu=">aaaddxicjvlnjtmwehbc1l+usajlhyfivottcc6b6rkxlgglrldxamjkseddq04cwrpdgljear4gm6ik8/ae/aaockiugssy1v6/m189njgcs6fqd//5rgxll66fgxnau/a9rsb/vbx8yvwgom66k0kcxmybfbjmukoeo18dswmjhnlxo/ycnoi1q2rssc4hsts7esncgllr0p7wpmzht1b0q4s0ivljq5q+rquwo75aa1y21k+70utnecfjkj6/wldxvt1ui/qcf2khv8owfqh/nc0n/hhe/q8ciz+z4ppxdlz/mlxwydlxysumussgtmp/byjimkuxeltcwsdoemjw8oc+tnlqufvglqkqmttt4uzs8fevfeohj6yzjkullyrq9qxfyoqlhptprgntbkem7o+lvyrl1yqqrabrfsrfmm7ddmikwlndglfotwpozmprizrjbk8qmj4jtfviskq2jazlougjrk0ghetbckop2aacbrford2wsqbgbvzkhi0ncj7vxqlbk2nmatzpbks61amagftc+25lf96b/bwwgy+mpg9wgwnwx6rhbiffkapcybeuam5cxzjzpcyxfnnjnwhrofu/uz/fljtr1fmrukc1zv/4asop9vg==</latexit> <latexit sha1_base64="tr0b20ew8iwx+bjdvsd995bp/yg=">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</latexit> Algorithmus von Kruskal <latexit 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<latexit 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<latexit 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<latexit 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Beispiel: Algorithmus von Kruskal angewendet auf einen ungerichteten gewichteten Graphen tu = [ ] tv = [ ] r = [ ] 78

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sha1_base64="afztqgfv9inruargstfa/9egjo=">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</latexit> Algorithmus von Kruskal <latexit 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<latexit 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Beispiel: Algorithmus von Kruskal angewendet auf einen ungerichteten gewichteten Graphen tu = [ ] tv = [ ] r = [ ] 79