Maximum Flows of Minimum Cost
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1 Maximum Flows of Minimum Cost Figure 8-24 Two possible maximum flows for the same network Data Structures and Algorithms in Java 1
2 Maximum Flows of Minimum Cost (continued) Figure 8-25 Finding a maximum flow of minimum cost Data Structures and Algorithms in Java 2
3 Maximum Flows of Minimum Cost (continued) Figure 8-25 Finding a maximum flow of minimum cost (continued) Data Structures and Algorithms in Java 3
4 Maximum Flows of Minimum Cost (continued) Figure 8-25 Finding a maximum flow of minimum cost (continued) Data Structures and Algorithms in Java 4
5 Matching Maximum matching is a matching that contains a maximum number of edges so that the number of unmatched vertices (that is, vertices not incident with edges in M) is minimal A matching problem consists in finding a maximum matching for a certain graph G The problem of finding a perfect matching is also called the marriage problem Data Structures and Algorithms in Java 5
6 Matching (continued) Figure 8-26 Matching five applicants with five jobs Data Structures and Algorithms in Java 6
7 Matching (continued) Figure 8-27 A graph with matchings M 1 = {edge(ab), edge(ef)} and M 2 = {edge(ab), edge(de), edge(fh)} Data Structures and Algorithms in Java 7
8 Matching (continued) Figure 8-28 (a) Two matchings M and N in a graph G = (V,E) and (b) the graph G = (V, M N) Data Structures and Algorithms in Java 8
9 Matching (continued) Figure 8-29 (a) Augmenting path P and a matching M and (b) the matching M P Data Structures and Algorithms in Java 9
10 Matching (continued) Figure 8-30 Application of the findmaximummatching() algorithm. Matched vertices are connected with solid lines. Data Structures and Algorithms in Java 10
11 Matching (continued) Figure 8-30 Application of the findmaximummatching() algorithm. Matched vertices are connected with solid lines (continued). Data Structures and Algorithms in Java 11
12 Assignment Problem Figure 8-31 An example of application of the optimalassignment() algorithm Data Structures and Algorithms in Java 12
13 Matching in Nonbipartite Graphs Figure 8-32 Application of the findmaximummatching() algorithm to a nonbipartite graph Data Structures and Algorithms in Java 13
14 Matching in Nonbipartite Graphs (continued) Figure 8-33 Processing a graph with a blossom Data Structures and Algorithms in Java 14
15 Matching in Nonbipartite Graphs (continued) Figure 8-33 Processing a graph with a blossom (continued) Data Structures and Algorithms in Java 15
16 Eulerian Graphs An Eulerian trail in a graph is a path that includes all edges of the graph only once An Eulerian cycle is a cycle that is also an Eulerian trail A graph that has an Eulerian cycle is called an Eulerian graph Data Structures and Algorithms in Java 16
17 Eulerian Graphs (continued) Figure 8-34 Finding an Eulerian cycle Data Structures and Algorithms in Java 17
18 The Chinese Postman Problem Figure 8-35 Solving the Chinese postman problem Data Structures and Algorithms in Java 18
19 The Chinese Postman Problem Figure 8-35 Solving the Chinese postman problem (continued) Data Structures and Algorithms in Java 19
20 The Chinese Postman Problem Figure 8-35 Solving the Chinese postman problem (continued) Data Structures and Algorithms in Java 20
21 The Chinese Postman Problem Figure 8-35 Solving the Chinese postman problem (continued) Data Structures and Algorithms in Java 21
22 Hamiltonian Graphs A Hamiltonian cycle in a graph is a cycle that passes through all the vertices of the graph A graph is called a Hamiltonian graph if it includes at least one Hamiltonian cycle Data Structures and Algorithms in Java 22
23 Hamiltonian Graphs (continued) Figure 8-36 Crossover edges Data Structures and Algorithms in Java 23
24 Hamiltonian Graphs (continued) Figure 8-37 Finding a Hamiltonian cycle Data Structures and Algorithms in Java 24
25 Hamiltonian Graphs (continued) Figure 8-37 Finding a Hamiltonian cycle (continued) Data Structures and Algorithms in Java 25
26 The Traveling Salesman Problem Figure 8-38 Using a minimum spanning tree to find a minimum salesman tour Data Structures and Algorithms in Java 26
27 The Traveling Salesman Problem (continued) Figure 8-38 Using a minimum spanning tree to find a minimum salesman tour (continued) Data Structures and Algorithms in Java 27
28 The Traveling Salesman Problem (continued) Figure 8-38 Using a minimum spanning tree to find a minimum salesman tour (continued) Data Structures and Algorithms in Java 28
29 The Traveling Salesman Problem (continued) Figure 8-39 Applying the nearest insertion algorithm to the cities in Figure 8.38a Data Structures and Algorithms in Java 29
30 The Traveling Salesman Problem (continued) Figure 8-39 Applying the nearest insertion algorithm to the cities in Figure 8.38a (continued) Data Structures and Algorithms in Java 30
31 Graph Coloring Sequential coloring establishes the sequence of vertices and a sequence of colors before coloring them, and then color the next vertex with the lowest number possible sequentialcoloringalgorithm(graph = (V, E)) put vertices in a certain order v P1, v P2,..., v Pv ; put colors in a certain order c P1, c 2,..., c k ; for i = 1 to V j = the smallest index of color that does not appear in any neighbor of v Pi ; color(v Pi ) = c j ; Data Structures and Algorithms in Java 31
32 Graph Coloring (continued) Figure 8-40 (a) A graph used for coloring; (b) colors assigned to vertices with the sequential coloring algorithm that orders vertices by index number; (c) vertices are put in the largest first sequence; (d) graph coloring obtained with the Brélaz algorithm Data Structures and Algorithms in Java 32
33 The Clique Problem Figure 8-41 A graph corresponding to Boolean expression Data Structures and Algorithms in Java 33
34 The 3-Colorability Problem Figure 8-42 (a) A 9-subgraph; (b) a graph corresponding to Boolean expression Data Structures and Algorithms in Java 34
35 The Vertex Cover Problem Figure 8-43 (a) A graph with a clique; (b) a complement graph Data Structures and Algorithms in Java 35
36 The Hamiltonian Cycle Problem Figure 8-44 (a) A 12-subgraph; (b) a graph G and (c) its transformation, graph G H Data Structures and Algorithms in Java 36
37 The Hamiltonian Cycle Problem Figure 8-44 (a) A 12-subgraph; (b) a graph G and (c) its transformation, graph G H (continued) Data Structures and Algorithms in Java 37
38 Case Study: Distinct Representatives Figure 8-45 (a) A network representing membership of three committees, C 1, C 2, and C 3, and (b) the first augmenting path found in this network Data Structures and Algorithms in Java 38
39 Case Study: Distinct Representatives (continued) Figure 8-46 The network representation created by FordFulkersonMaxFlow() Data Structures and Algorithms in Java 39
40 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem Data Structures and Algorithms in Java 40
41 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 41
42 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 42
43 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 43
44 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 44
45 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 45
46 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 46
47 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 47
48 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 48
49 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 49
50 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 50
51 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 51
52 Case Study: Distinct Representatives (continued) Figure 8-47 An implementation of the distinct representatives problem (continued) Data Structures and Algorithms in Java 52
53 Summary A graph is a collection of vertices (or nodes) and the connections between them A multigraph is a graph in which two vertices can be joined by multiple edges A pseudograph is a multigraph with the condition v i v j removed, which allows for loops to occur The sets used to solve the union-find problem are implemented with circular linked lists Data Structures and Algorithms in Java 53
54 Summary (continued) A spanning tree is an algorithm that guarantees generating a tree (or a forest, a set of trees) that includes or spans over all vertices of the original graph An undirected graph is called connected when there is a path between any two vertices of the graph A network is a digraph with one vertex s, called the source, with no incoming edges, and one vertex t, called the sink, with no outgoing edges Data Structures and Algorithms in Java 54
55 Summary (continued) Maximum matching is a matching that contains a maximum number of edges so that the number of unmatched vertices (that is, vertices not incident with edges in M) is minimal A Hamiltonian cycle in a graph is a cycle that passes through all the vertices of the graph Sequential coloring establishes the sequence of vertices and a sequence of colors before coloring them, and then color the next vertex with the lowest number possible Data Structures and Algorithms in Java 55
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