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Minimum cost spanning tree and activity networks

Minimum cost spanning tree and activity networks. Data structure 2002/12/2. Minimum-cost spanning trees (MST) in a given graph (6.3). A minimum-cost spanning tree is a spanning tree of least cost Design strategy – greedy method Kruskal ’ s algorithm Edge by edge Prim ’ s algorithm

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Minimum cost spanning tree and activity networks

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  1. Minimum cost spanning tree and activity networks Data structure 2002/12/2

  2. Minimum-cost spanning trees (MST) in a given graph (6.3) • A minimum-cost spanning tree is a spanning tree of least cost • Design strategy – greedy method • Kruskal’s algorithm • Edge by edge • Prim’s algorithm • Span out from one vertex • Sollin’s algorithm • Hint: Construct from connected components • Leave as a homework

  3. An MST example 0 0 28 1 1 10 10 14 14 16 16 5 5 6 6 2 2 24 25 25 18 12 12 4 4 22 22 3 3

  4. Kruskal’s algorithm • Edge by edge • O(e log e) • Algorithm: T={}; While ((T contains less than n-1 edges) &&(E not empty)){ choose an edge (v,w) from E of lowest cost; delete (v,w) from E; if((v,w) does not create a cycle in T) add(v,w) to T; else discard(v,w); } If(T contains fewer than n – 1 edges) printf(“No spanning tree\n”);

  5. Prim’s algorithm • Begins from a vertex • O(n2) • Algorithm: T={}; TV = {0}; /* start with vertex 0 and no edges */ While (T contains fewer than n-1 edges){ let(u,v) be a least cost edge s.t. u in TV and v not in TV; if (there is no such edge) break; add v to TV; add (u,v) to T; } If (T contains fewer than n-1 edges) printf(“No spanning tree\n”);

  6. Activity networks (6.5) • Definition: • Vertex i is a predecessor of j • Iff there is a directed path from i to j • Vertex i is an immediate predecessor of j • Iff <i,j> is an edge in G • Successor, immediate successor • Precedence relation, is a partial order relation • Transitive • Irreflexive • Can be represented by a Directed Acyclic Graph (DAG)? • Topological order • is a linear ordering of the vertices of a graph s.t. if i is a predecessor of j, then i precedes j in the linear ordering

  7. An example of activity network Activity on Edge networks Activity on Vertex networks 7 2 資結 1 9 finish 2 start 5 6 9 計概 邏設 OS 1 4 7 1 3 8 4 組語 計結 5 4 4 6 A topological order: 計概,邏設,組語,計結,資結,OS 2 Earliest time Latest time Critical activites

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