The Traveling Salesman Problem Nearest-Neighbor Algorithm
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1 The Traveling Salesman Problem Nearest-Neighbor Algorithm Lecture 31 Sections 6.4 Robb T. Koether Hampden-Sydney College Fri, Apr 6, 2018 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
2 1 Greedy and Approximate Algorithms 2 The Nearest-Neighbor Algorithm 3 The Repetitive Nearest-Neighbor Algorithm 4 Assignment Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
3 Outline 1 Greedy and Approximate Algorithms 2 The Nearest-Neighbor Algorithm 3 The Repetitive Nearest-Neighbor Algorithm 4 Assignment Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
4 Greedy Algorithms Definition (Greedy Algorithms) A greedy algorithm is an algorithm that, like greedy people, grabs what looks best in the short run, whether or not it is best in the long run. Greedy algorithms optimize locally, but not necessarily globally. The benefit of greedy algorithms is that they are simple and fast. They may or may not produce the optimal solution. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
5 Approximate Algorithms Definition (Approximate Algorithm) An approximate algorithm is an algorithm that gives a good solution, but not necessarily the best solution. The benefit of approximate algorithms is that they can produce a good solution very quickly. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
6 Approximate Algorithms Definition (Approximate Algorithm) An approximate algorithm is an algorithm that gives a good solution, but not necessarily the best solution. The benefit of approximate algorithms is that they can produce a good solution very quickly. They operate under the principle Good is good enough. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
7 Approximate Algorithms Definition (Approximate Algorithm) An approximate algorithm is an algorithm that gives a good solution, but not necessarily the best solution. The benefit of approximate algorithms is that they can produce a good solution very quickly. They operate under the principle Good is good enough. Also known as The perfect is the enemy of the good. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
8 Approximate Algorithms Definition (Approximate Algorithm) An approximate algorithm is an algorithm that gives a good solution, but not necessarily the best solution. The benefit of approximate algorithms is that they can produce a good solution very quickly. They operate under the principle Good is good enough. Also known as The perfect is the enemy of the good. Striving to be better, oft we mar what s well. (Shakespeare) Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
9 Approximate Algorithms We will look at three greedy, approximate algorithms to handle the Traveling Salesman Problem. The Nearest-Neighbor Algorithm The Repetitive Nearest-Neighbor Algorithm The Cheapest-Link Algorithm Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
10 Outline 1 Greedy and Approximate Algorithms 2 The Nearest-Neighbor Algorithm 3 The Repetitive Nearest-Neighbor Algorithm 4 Assignment Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
11 The Nearest-Neighbor Algorithm Definition (Nearest-Neighbor Algorithm) The Nearest-Neighbor Algorithm begins at any vertex and follows the edge of least weight from that vertex. At every subsequent vertex, it follows the edge of least weight that leads to a city not yet visited, until it returns to the starting point. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
12 The Nearest-Neighbor Algorithm Definition (Nearest-Neighbor Algorithm) The Nearest-Neighbor Algorithm begins at any vertex and follows the edge of least weight from that vertex. At every subsequent vertex, it follows the edge of least weight that leads to a city not yet visited, until it returns to the starting point. The result typically depends on the chosen starting point. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
13 Example B A K 1 J H I 4 G C F D 12 E Start at A Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
14 Example B A K 1 J H I 4 G C F D 12 E Distance = Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
15 Example B A K 1 J H I 4 G C F D 12 E Distance = 2 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
16 Example B A K 1 J H I 4 G C F D 12 E Distance = 31 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
17 Example B A K 1 J H I 4 G C F D 12 E Distance = 36 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
18 Example B A K 1 J H I 4 G C F D 12 E Distance = 46 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
19 Example B A K 1 J H I 4 G C F D 12 E Distance = 4 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
20 Example B A K 1 J H I 4 G C F D 12 E Distance = 6 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
21 Example B A K 1 J H I 4 G C F D 12 E Distance = Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
22 Example B A K 1 J H I 4 G C F D 12 E Distance = 111 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
23 Example B A K 1 J H I 4 G C F D 12 E Distance = 116 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
24 Example B A K 1 J H I 4 G C F D 12 E Distance = 0 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
25 The Nearest-Neighbor Algorithm Example (Nearest-Neighbor Algorithm) We ended up with the circuit ABIGHFCDEJKA. The length is 0 miles. Is it possible to do better? Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
26 The Nearest-Neighbor Algorithm Example (Nearest-Neighbor Algorithm) We ended up with the circuit ABIGHFCDEJKA. The length is 0 miles. Is it possible to do better? Yes. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
27 The Nearest-Neighbor Algorithm B A K 1 J H I 4 G C F D 12 E Re-do the previous example, starting at city B. Re-do the previous example, starting at city C. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
28 Outline 1 Greedy and Approximate Algorithms 2 The Nearest-Neighbor Algorithm 3 The Repetitive Nearest-Neighbor Algorithm 4 Assignment Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
29 The Repetitive Nearest-Neighbor Algorithm Definition (Repetitive Nearest-Neighbor Algorithm) The Repetitive Nearest-Neighbor Algorithm applies the nearestneighbor algorithm repeatedly, using each of the vertices as a starting point. It selects the starting point that produced the shortest circuit. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
30 Outline 1 Greedy and Approximate Algorithms 2 The Nearest-Neighbor Algorithm 3 The Repetitive Nearest-Neighbor Algorithm 4 Assignment Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, 2018 / 1
31 Assignment Assignment Chapter 6: Exercises 3, 36, 3, 41, 4. Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor Algorithm Fri, Apr 6, / 1
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