MATH Section 10.6

Find the length of the shortest path between the given pairs of vertices in the following weighted graph: a and f The length of the shortest path between the vertices a and f is. Explanation The length of the shortest path is 11. Using Dijkstra's algorithm, the shortest path between a and f is as follows: a, c, d, f.

11

Find the length of the shortest path between the given pairs of vertices in the following weighted graph: b and z The length of the shortest path between the vertices b and z is. Explanation The length of the shortest path is 15. Using Dijkstra's algorithm, the shortest path between b and z is as follows: b, d, e, g, z.

15

Find the length of the shortest path between the given pairs of vertices in the following weighted graph: a and d The length of the shortest path between the vertices a and d is. Explanation The length of the shortest path is 6. Using Dijkstra's algorithm, the shortest path between a and d is as follows: a, c, d.

6

Find the length of the shortest path between the given pairs of vertices in the following weighted graph: c and f The length of the shortest path between the vertices c and f is. Explanation The length of the shortest path is 8. Using Dijkstra's algorithm, the shortest path between c and f is as follows: c, d, f.

8

Identify the route with the least total airfare that visits each of the cities in this graph, where the weight on an edge is the least price available for a flight between the two cities. (Check all that apply.) Explanation Let us use the first letters of the cities to represent them, except for New Orleans which we represent by O, as follows: S(Seattle), B(Boston), N(New York), O (New Orleans), and P(Phoenix). The following table shows the twelve different Hamilton circuit and their weights: Circuit Weight S-B-N-O-P-S 409 + 109 + 229 + 309 + 119 = 1175 S-B-N-P-O-S 409 + 109 + 319 + 309 + 429 = 1575 S-B-O-N-P-S 409 + 239 + 229 + 319 + 119 = 1315 S-B-O-P-N-S 409 + 239 + 309 + 319 + 389 = 1665 S-B-P-N-O-S 409 + 379 + 319 + 229 + 429 = 1765 S-B-P-O-N-S 409 + 379 + 309 + 229 + 389 = 1715 S-N-B-O-P-S 389 + 109 + 239 + 309 + 119 = 1165 S-N-B-P-O-S 389 + 109 + 379 + 309 + 429 = 1615 S-N-O-B-P-S 389 + 229 + 239 + 379 + 119 = 1355 S-N-P-B-O-S 389 + 319 + 379 + 239 + 429 = 1755 S-O-B-N-P-S 429 + 239 + 109 + 319 + 119 = 1215 S-O-N-B-P-S 429 + 229 + 109 + 379 + 119 = 1265 We see that the circuit S-N-B-O-P-S is the path with a minimum total weight of $1165. The same circuit starting at some other point but traversing the vertices in the same or exactly opposite order will also be a path with a minimum weight of $1165.

The correct routes are as follows: S-N-B-O-P-S N-B-O-P-S-N P-O-B-N-S-P S-P-O-B-N-S

Find a shortest path (in mileage) between each of the following pairs of cities in the airline system shown in the following figure. New York and Los Angeles Explanation Using Dijkstra's algorithm, the shortest path between New York and Los Angeles is the direct path.

The direct path is the shortest.

Find a shortest path (in mileage) between each of the following pairs of cities in the airline system shown in the following figure. Miami and Denver Explanation Using Dijkstra's algorithm, the shortest path between Miami and Denver is the path via Atlanta and Chicago.

The path via Atlanta and Chicago is the shortest.

Find a shortest path (in mileage) between each of the following pairs of cities in the airline system shown in the following figure. Miami and Los Angeles Explanation Using Dijkstra's algorithm, the shortest path between Miami and Los Angeles is the path via Atlanta, Chicago, and Denver.

The path via Atlanta, Chicago, and Denver is the shortest.

Find a shortest path (in mileage) between each of the following pairs of cities in the airline system shown in the following figure. Boston and San Francisco Explanation Using Dijkstra's algorithm, the shortest path between Boston and San Francisco is the path via Chicago.

The path via Chicago is the shortest.

Solve the traveling salesperson problem for this graph by finding the total weight of all Hamilton circuits and determining a circuit with minimum total weight. What is the minimum total weight of the Hamilton circuits for this graph? Select the circuit of minimum total weight. Explanation Circuit Weights a, b, c, d, e, a 3 + 10 + 6 + 1 + 7 = 27 a, b, c, e, d, a 3 + 10 + 5 + 1 + 4 = 23 a, b, d, c, e, a 3 + 9 + 6 + 5 + 7 = 30 a, b, d, e, c, a 3 + 9 + 1 + 5 + 8 = 26 a, b, e, c, d, a 3 + 2 + 5 + 6 + 4 = 20 a, b, e, d, c, a 3 + 2 + 1 + 6 + 8 = 20 a, c, b, d, e, a 8 + 10 + 9 + 1 + 7 = 35 a, c, b, e, d, a 8 + 10 + 2 + 1 + 4 = 25 a, c, d, b, e, a 8 + 6 + 9 + 2 + 7 = 32 a, c, e, b, d, a 8 + 5 + 2 + 9 + 4 = 28 a, d, b, c, e, a 4 + 9 + 10 + 5 + 7 = 35 a, d, c, b, e, a 4 + 6 + 10 + 2 + 7 = 29 The circuits a, b, e, c, d, a and a, b, e, d, c, a (or the same circuits starting at some other point but traversing the vertices in the same or exactly opposite order) are the ones with minimum total weight.

What is the minimum total weight of the Hamilton circuits for this graph? 20 Select the circuit of minimum total weight. a, b, e, d, c, a a, b, e, c, d, a