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Unformatted text preview: Problem 3 0) Construction of the Algorithm We began with the observation that at the last step of the algorithm, there will be one stack of boxes on the truck and then one more stack somewhere else that needs to be moved en mass onto the truck in the optimal solution. Lemma: At any point in the algorithm to find the optimal solution, there must exists at least two locations x and y such that both have stacks of boxes and the boxes in each stack are completely disjoint with those of the other stack throughout every other step of the algorithm. Because if any element of x was already in a stack with y, and they were transfered to x, then they would have to be remerged at the end of the algorithm, which by the triangle inequality means that were taking more time than just staying with the x pile the entire way through, which means that that instance could not have been an optimum. So given the above lemma, we know that at that last step of the algorithm, there exists two contiguous stacks of boxes at y and x, where the sum of the height of the two stacks is n. Now because the two stacks are completely disjoint up until this step, we know that we can partition the problem into these two distinct cases. For each stack, we know that there exists a previous step where some subset of that stack was from another location...
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 Spring '08
 KLEINBERG
 Algorithms

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