CHAPTER 25 Weighted Graphs and Applications Objectives â¢ To ...

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To 1: 3 1 (cost: 3) To 2: 3 2 (cost: 4) To 3: 3 (cost: 0) To 4: 3 4 (cost: 6) The program creates a weighted graph for Figure 25.1 in line 67. It then invokes the getShortestPath(5) function to return a ShortestPathTree object that contains all shortest paths from vertex 5 (i.e., Chicago) (line 68). The printAllPaths function displays all the paths (line 69). The graphical illustration of all shortest paths from Chicago is shown in Figure 25.20. The shortest paths from Chicago to the cities are found in this order: Kansas City, New York, Boston, Denver, Dallas, Houston, Atlanta, Los Angeles, Miami, Seattle, and San Francisco. Seattle 10 807 1331 2097 1003 Chicago 787 983 3 2 Boston 214 New York 11 San Francisco 381 8 1267 1015 1663 Denver 4 599 Kansas City 533 1 1260 864 888 Los Angeles 1435 496 5 Dallas 239 6 781 810 7 Atlanta 661 Houston 1187 9 Miami Figure 25.20 The shortest paths from Chicago to all other cities are highlighted. Check point 25.9 Trace Dijkstra’s algorithm for finding shortest paths from Boston to all other cities in Figure 25.1. 25.10 Is the shortest path between two vertices unique if all edges have different weights? 25.11 If you use an adjacency matrix to represent weighted edges, what would be the time complexity for Dijkstra’s algorithm? 38
25.12 What happens to the getShortestPath() function in WeightedGraph if the source vertex cannot reach all vertieces in the graph? Verify your answer by writing a test program that creates an unconnected graph and invoke the getShortestPath() function. How do you fix the problem by obtaining a partial shortest path tree? 25.13 If there is no path from vertex v to the source vertex, what will be cost[v]? 25.14 Assume that the graph is connected; will the getShortestPath function find the shortest paths correctly if lines 228-231 in WeightedGraph are deleted? 25.6 Case Study: The Weighted Nine Tail Problem Key Point: The weighted nine tails problem can be reduced to the weighted shortest path problem. Section 24.8 presented the nine tail problem and solved it using the BFS algorithm. This section presents a variation of the problem and solves it using the shortest-path algorithm. The nine tail problem is to find the minimum number of the moves that lead to all coins being face down. Each move flips a head coin and its neighbors. The weighted nine tail problem assigns the number of flips as a weight on each move. For example, you can move from the coins in Figure 25.21a to Figure 25.21b by flipping the three coins. So the weight for this move is 3. H H H T T T H H H T T H H T T H H H (a) (b) Figure 25.21 The weight for each move is the number of flips for the move. The weighted nine tail problem is to find the minimum number of flips that lead to all coins face down. The problem can be reduced to finding the shortest path from a starting node to the target node in an edgeweighted graph. The graph has 512 nodes. Create an edge from node v to u if there is a move from node u to node v. Assign the number of flips to be the weight of the edge. 39
Page 1 and 2: CHAPTER 25 Weighted Graphs and Appl
Page 3 and 4: acquainted with weighted graphs usi
Page 5 and 6: 7 { 8 public: 9 double weight; // T
Page 7 and 8: Graph Defined in Figure 24.7. Weigh
Page 9 and 10: 72 for (unsigned i = 0; i < edges.s
Page 11 and 12: 189 u = i; 190 } 191 } 192 193 if (
Page 13 and 14: 72 edges2[i][1], edges2[i][2])); 73
Page 15 and 16: vector neighbors; neighbors[0].push
Page 17 and 18: 10 Add u to T; 11 for (each v not i
Page 19 and 20: 1 10 2 cost 0 9 5 9 7 5 8 T 0 1 2 3
Page 21 and 22: (g) NOTE: A minimum spanning tree i
Page 23 and 24: The MST class extends the Tree clas
Page 25 and 26: 81 vector edgeVector2; 82 for (int
Page 27 and 28: 25.5 Finding Shortest Paths The sh
Page 29 and 30: (a) Before moving u to T (b) After
Page 31 and 32: Now T contains {1, 2, 0}. Vertex 6
Page 33 and 34: As you see, the algorithm essential
Page 35 and 36: priority queues, we can implement t
Page 37: 57 const int NUMBER_OF_EDGES = 46;
Page 41 and 42: and flipACell(&node, row, column) d
Page 43 and 44: The getNumberOfFlips(int u) functio
Page 45 and 46: 4. Dijkstra’s algorithm starts se
Page 47 and 48: The vertices and edges of a weighte
Page 49 and 50: Input: a weighted graph G = (V, E)

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CHAPTER 25 Weighted Graphs and Applications Objectives â¢ To ...