L16-MST

L16-MST - L16 4102 abhi shelat 10.22.2013 Greedy Alg Min...

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L16 4102 10.22.2013 abhi shelat Greedy Alg: Min Span Trees
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MST
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connecting houses image:www.princegeorgeva.org , thefranciscofamily.org, www.rightdriveacademy.co.uk, www.ccscambridge.org, www.drawingcoach.com, www.pastoral.org.uk , www.daasgallery.com
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connecting houses image:www.princegeorgeva.org , thefranciscofamily.org, www.rightdriveacademy.co.uk, www.ccscambridge.org, www.drawingcoach.com, www.pastoral.org.uk , www.daasgallery.com
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connecting houses image:www.princegeorgeva.org , thefranciscofamily.org, www.rightdriveacademy.co.uk, www.ccscambridge.org, www.drawingcoach.com, www.pastoral.org.uk , www.daasgallery.com
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connecting houses image:www.princegeorgeva.org , thefranciscofamily.org, www.rightdriveacademy.co.uk, www.ccscambridge.org, www.drawingcoach.com, www.pastoral.org.uk , www.daasgallery.com
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connecting houses image:www.princegeorgeva.org , thefranciscofamily.org, www.rightdriveacademy.co.uk, www.ccscambridge.org, www.drawingcoach.com, www.pastoral.org.uk , www.daasgallery.com
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e d f c b a 10 8 8 h g i 2 11 9 5 6 1 3 3 9 12
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graphs clrs [ch 22]
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defnition: path a sequence oF nodes with the property that simple path: cycle:
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defnition:tree a tree is connected graph:
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what we want: e d f c b a 10 8 8 h g i 2 11 9 5 6 1 3 3 9 12
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minimum spanning tree min ± ( u,v ) T w ( u, v ) T E looking for a set of edges that (a) connects all vertices (b) has the least cost
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facts min ± ( u,v ) T w ( u, v ) T E looking for a set of edges that (a) connects all vertices (b) has the least cost how many edges does solution have ? does solution have a cycle?
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strategy add lightest edge that does not create a cycle start with an empty set of edges A repeat for v-1 times:
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example 6 8 h g i 2 11 9 5 12 3 7 10 9 3 1 8 a b c f d e
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kruskal e d f c b a 10 8 1 3 3 9 12 6 5 9 11 2 i g h 8 7 e d f c b a 10 8 1 3 3 9 12 6 5 9 11 2 i g h 8 7 e d f c b a 10 8 1 3 3 9 12 6 5 9 11 2 i g h 8 7 e d f c b a 10 8 1 3 3 9 12 6 5 9 11 2 i g h 8 7 e d f c b a 10 8 1 3 3 9 12 6 5 9 11 2 i g h 8 7 e d f c b a 10 8 1 3 3 9 12 6 5 9 11 2 i g h 8 7 e d f c b a 10 8 1 3 3 9 12 6 5 9 11 2 i g h 8 7 e d f c b a 10 8 1 3 3 9 12 6 5 9 11 2 i g h 8 7
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1 T ⇥⌅ 2 repeat V - 1 times: 3 add to T the lightest edge e E that does not create a cycle why does this work?
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defnition: cut
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example of a cut 6 8 h g i 2 11 9 5 12 3 7 10 9 3 1 8 a b c f d e
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defnition: crossing a cut
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u S v V - S defnition: crossing a cut an edge crosses a graph cut (S,V-S) iF e =( u, v ) 6 8 h g i 2 11 9 5 12 3 7 10 9 3 1 8 a b c f d e
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example of a crossing 6 8 h g i 2 11 9 5 12 3 7 10 9 3 1 8 a b c f d e
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defnition: respect
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cut theorem
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L16-MST - L16 4102 abhi shelat 10.22.2013 Greedy Alg Min...

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