225_32_MST_Correctness

225_32_MST_Correctness -...

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1 Start at vertex 0. Number the minimum weight  spanning tree edges with 1, 2, 3, … according to the  order that they would be added to the MST by the  Dijkstra/Prim MST algorithm.
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2 Grades for Assignment 3B: If you go into connex to the place where you had submitted assignment 3B,  your mark (out of 40) should be there at the top of the page. There is detailed  feedback on your program (assuming you submitted something) at the bottom  of the page. Look for something like this: Instructor's attachments to this submission marking_output.txt  ( 10 KB; Nov 29, 2011 5:26 pm )  Click on the marking_output.txt to get the feedback on your program that  contributed your grade. Final exam tutorial: Wednesday Dec. 7 at 12:00pm,  Elliott 061. Assignment 5:  due Friday, beginning of class.
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3 Kruskal example:
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4 Proof of Correctness for the Kruskal and  Dijkstra/Prim MST algorithms. Why do these algorithms correctly compute a minimum  weight spanning tree?
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5 Dijkstra/Prim example:
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6 What's a Cycle? c yc le   o f a  g ra ph is  a n a lte rna ting  s e q ue nc e  o f  ve rtic e s  a nd e dg e s  o f the  fo rm   v 0 , (v 0 , v 1 ), v 1 , (v 1 , v 2 ),  v 2 , (v 2 , v 3 ),  … ,v k-1 , (v k-1 , v k ), v k   where except for v 0   = v k   the  vertices are distinct
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7 What's a Cut? Le t (X,Y) be a partition of the nodes of a graph. The  cut  induced by that partition is the set of all edges  {x,y} with x in X and y in Y. 
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8 The  Red / Blue  Rules for MST Author- Frank Ruskey. C o l o r Invariant: The re  is  a  MS T c o nta ining  a ll the  re d e dg e s   a nd no ne  o f the  b lue  e dg e s .  Red Rule: Le t C  be a cut without red edges. Color red the smallest  uncolored edge of C.  Blue Rule: Le t C  be a cycle without blue edges. Color blue the  largest uncolored edge of C. 
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9 Red Rule: Dijkstra/Prim Le t C  be a cut without red edges. Color red the smallest  uncolored edge of C. 
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10 Blue Rule: Kruskal’s Le t C  be a cycle without blue edges. Color blue the  largest uncolored edge of C. 
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