Chapter 3: Optimal Trees and Branchings - UKP

More documents

Recommendations

Info

Optimal Trees and Branchings Correctness of Reverse Kruskal Regard an arbitrary edge e = (v, w) that is removed during the algorithm At the time of removal, e is part of a cycle C. Amongst all edges within C, e is the first edge to consider, that is e is the most expensive edge on C Due to the cycle property, the removal has been done deservedly. The output T of the algorithm T is connected due to the fact that at no time an edge is removed from T that would destroy connectivity (only edges from cycles are removed). In the end, T does not contain a cycle anymore because the most expensive edge on this cycle would have been removed. Efficient Graph Algorithms | Wolfgang Stille | WS 2011/2012 | Chapter III - Optimal Trees and Branchings | 25
Optimal Trees and Branchings Implementation of Kruskal’s Algorithm (1) Kruskal’s Algorithm has an obvious invariant: the edge set T reflects a forest in G at any time of the algorithm. Obviously, the time critical operation is the test, if a cycle emerges in T ∪ {e i } by the insertion of an edge e i into T . A cycle emerges if and only if for en edge e i = (u, v), the two nodes u and v are part of the same connected component of T . We already know techniques to efficiently compute the connected components of a graph: BFS and DFS. These have linear complexity O(m + n). If we had to compute the CCs in every iteration of Kruskal’s algorithm from scratch, this would amount to a total complexity of O(m 2 ). In order to avoid this, we would like to make use of a data structure that supports the operations of the algorithm efficiently. Efficient Graph Algorithms | Wolfgang Stille | WS 2011/2012 | Chapter III - Optimal Trees and Branchings | 26
Page 1 and 2: Efficient Graph Algorithms III Opti
Page 3 and 4: Optimal Trees and Branchings MAX-FO
Page 5 and 6: Optimal Trees and Branchings MST
Page 7 and 8: Optimal Trees and Branchings MST(G,
Page 9 and 10: Optimal Trees and Branchings Correc
Page 11 and 12: Optimal Trees and Branchings Matroi
Page 13 and 14: Optimal Trees and Branchings Pairwi
Page 15 and 16: Optimal Trees and Branchings Dualit
Page 17 and 18: Optimal Trees and Branchings Borův
Page 19 and 20: Optimal Trees and Branchings Correc
Page 21 and 22: Optimal Trees and Branchings Kruska
Page 23: Optimal Trees and Branchings Revers
Page 27 and 28: Optimal Trees and Branchings Comple
Page 29 and 30: Optimal Trees and Branchings Prim
Page 31 and 32: Optimal Trees and Branchings Implem
Page 33 and 34: Optimal Trees and Branchings Implem
Page 35 and 36: Optimal Trees and Branchings Round
Page 37 and 38: Optimal Trees and Branchings MST Al
Page 39 and 40: Optimal Trees and Branchings - Part
Page 41 and 42: Optimal Trees and Branchings Delaye
Page 43 and 44: Optimal Trees and Branchings Critic
Page 51 and 52: Optimal Trees and Branchings Shrink
Page 53 and 54: Optimal Trees and Branchings Branch
Page 55 and 56: Optimal Trees and Branchings Edmond
Page 57 and 58: Optimal Trees and Branchings Exampl
Page 59: Optimal Trees and Branchings Select

Chapter 3: Optimal Trees and Branchings - UKP

Create successful ePaper yourself

Delete template?

Save as template?