MinSpanning

MinSpanning - Minimum Spanning Trees Given an undirected...

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Subhash Suri UC Santa Barbara Minimum Spanning Trees Given an undirected graph G = ( V,E ) , with edge costs c ij . A spanning tree T of G is a cycle-free subgraph that spans all the nodes. The cost of T is the sum of the costs of the edges in T . MST is the smallest cost spanning tree. a b c e d f g 5 3 2 12 16 4 a b c e d f g 8 5 3 14 10 2 18 12 30 16 26 4 Graph G MST
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Subhash Suri UC Santa Barbara Applications of MST a b c e d f g 5 3 2 12 16 4 a b c e d f g 8 5 3 14 10 2 18 12 30 16 26 4 Graph G MST Direct applications: interconnection of entities. 1. electrical devices (circuit boards) 2. utilities (gas, oil) 3. computers or communication devices by high speed lines. 4. cable service customers Indirect applications. 1. Optimal message passing. 2. Data storage. 3. Cluster analysis
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Subhash Suri UC Santa Barbara Optimality Conditions Greedy incremental flavor: add one edge at a time. Each step colors an edge of G blue (accept) or red (reject). Co lor Invariant (CI): MST containing all blue edges, and no red edges. Recall that a cut in G = ( V,E ) is a partition of it vertices ( X,V - X ) .
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Subhash Suri UC Santa Barbara Blue and Red Rules x y x y Cut Rule (Blue) Red Rule (Cycle) Blue (Cut) Rule: Select a cut not crossed by any blue edge. Among the uncolored edges crossing the cut, make the minimum cost edge blue. Red (Cycle) Rule: Select a simple cycle with no red edges. Among all uncolored edges of the cycle, make the maximum cost one red.
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Subhash Suri UC Santa Barbara Generic MST Algorithm Theorem: Apply red and blue rules in arbitrary order until neither rule applies. The resulting set of blue edges forms a MST. a b c e d f g 5 3 2 12 16 4 a b c e d f g 8 5 3 14 10 2 18 12 30 16 26 4 Graph G MST Proof has two parts: 1. The Color Invariant (CI) holds. 2. All edges are ultimately colored.
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Subhash Suri UC Santa Barbara Correctness of Blue Rule Let T * be the MST guaranteed by the CI before the last coloring step. e e e’ MST T* MST satisfying color invariant Cut Suppose the last step was to color e blue. Consider the cut ( X,V - X ) to which blue rule applied. Some edge e 0 of T * must cross this cut. The graph T * + e contains a cycle containing both e and e 0 , and cost ( e ) cost ( e 0 ) . (Why?) So T * + e - e 0 is also a MST.
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UC Santa Barbara Correctness of Red Rule MST T* MST satisfying color invariant e’ Cycle e’ e Suppose edge e colored red. If e 6∈ T * , then T * still satisfies CI. Otherwise, consider T * - e . It has two components. The cycle used in coloring e has some edge e 0 with one end in each of these components. By choice,
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MinSpanning - Minimum Spanning Trees Given an undirected...

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