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1. kleinberg, jon. algorithm design (p. 189, q. 3). you are consulting for a trucking company that do es a large amount of business shipping packages b etween new york and boston. the volume is high enough that they have to send a numb er of trucks each day b etween the two lo cations. trucks have a xed limit w on the maximum amount of weight they are allowed to carry. boxes arrive at the new york station one by one, and each package i has a weight wi. the trucking station is quite small, so at most one truck can b e at the station at any time. company p olicy requires that b oxes are shipp ed in the order they arrive; otherwise, a customer might get upset up on seeing a b ox that arrived after his make it to boston faster. at the moment, the company is using a simple greedy algorithm for packing: they pack b oxes in the order they arrive, and whenever the next b ox do es not t, they send the truck on its way. prove that, for a given set of b oxes with sp ecied weights, the greedy algorithm currently in use actually minimizes the numb er of trucks that are needed. hint: use the stay ahead metho d.

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