How to compute optimal paths for traveling salesman bitonic tour?

Question

UPDATED

After more reading, the solution can be given with the following recurrence relation:

(a) When i = 1 and j = 2, l(i; j) = di         


        

Answer 1

Okay, the key notions in a dynamic programming solution are:

• you pre-compute smaller problems
• you have a rule to let you combine smaller problems to find solutions for bigger problems
• you have a known property of the problems that let's you prove the solution is really optimal under some measure of optimality. (In this case, shortest.)

The essential property of a bitonic tour is that a vertical line in the coordinate system crosses a side of the closed polygon at most twice. So, what is a bitonic tour of exactly two points? Clearly, any two points form a (degenerate) bitonic tour. Three points have two bitonic tours ("clockwise" and "counterclockwise").

Now, how can you pre-compute the various smaller bitonic tours and combine them until you have all points included and still have a bitonic tour?

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Okay, you're on the righ track with your update. But now, in a dynamic programming solution, what you do with work it bottom-up: pre-compute and memoize (not "memorize") the optimal subproblems.