The Traveling Salesman Problem Nearest-Neighbor...

The Traveling Salesman ProblemNearest-Neighbor Algorithm

Lecture 31Sections 6.4

Robb T. Koether

Hampden-Sydney College

Mon, Nov 6, 2017

Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor AlgorithmMon, Nov 6, 2017 1 / 15

1 Greedy and Approximate Algorithms

2 The Nearest-Neighbor Algorithm

3 The Repetitive Nearest-Neighbor Algorithm

4 Assignment

Outline

4 Assignment

Greedy Algorithms

Definition (Greedy Algorithms)A greedy algorithm is an algorithm that, like greedy people, grabs whatlooks 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.

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 agood 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)

The benefit of approximate algorithms is that they can produce agood 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)

The benefit of approximate algorithms is that they can produce agood 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)

The benefit of approximate algorithms is that they can produce agood 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)

We will look at three greedy, approximate algorithms to handle theTraveling Salesman Problem.

The Nearest-Neighbor AlgorithmThe Repetitive Nearest-Neighbor AlgorithmThe Cheapest-Link Algorithm

Outline

4 Assignment

The Nearest-Neighbor Algorithm

Definition (Nearest-Neighbor Algorithm)The Nearest-Neighbor Algorithm begins at any vertex and follows theedge of least weight from that vertex. At every subsequent vertex, itfollows the edge of least weight that leads to a city not yet visited, untilit returns to the starting point.

Example

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Example (Nearest-Neighbor Algorithm)We ended up with the circuit ABJCEDFGHIA.The length is 94 miles.Is it possible to do better?

Example (Nearest-Neighbor Algorithm)We ended up with the circuit ABJCEDFGHIA.The length is 94 miles.Is it possible to do better?Yes.

Example (Nearest-Neighbor Algorithm)Re-do the previous example, starting at city B.Re-do the previous example, starting at city C.Did we get a better solution?

Outline

4 Assignment

The Repetitive Nearest-Neighbor Algorithm

Definition (Repetitive Nearest-Neighbor Algorithm)The Repetitive Nearest-Neighbor Algorithm applies thenearest-neighbor algorithm repeatedly, using each of the vertices as astarting point. It selects the starting point that produced the shortestcircuit.

Outline

4 Assignment

Assignment

AssignmentChapter 6: Exercises 35, 36, 37, 41, 45.

The Traveling Salesman Problem Nearest-Neighbor...

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