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HW8_Solution

HW8_Solution - HW8 SOLUTION 1 Chapter 11 Problem 1 This...

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HW8 SOLUTION 1. Chapter 11 Problem 1: This problem is known as Bin Packing, and has a long history of study. We’ll only scratch the surface here, of course. (a) An example would be to have K = 10, and have items of weights w 1 = 6 , w 2 = 6 , w 3 = 4 , w 4 = 4. Then, we could pack items 1 and 1 on the truck, and 2 and 4 on another, for a total of two trucks. Instead, the greedy algorithm packs item 1 on a truck by itself, then puts 2 and 3 on another truck, and finally puts item 4 on a truck by itself. Thus, it uses 3 trucks. (b) Perhaps the easiest way to prove this is as follows. We let l j denote the load of the j - th truck, i.e., the total weight of all items that were put on truck j . Then, we always have that l j + l j +1 > K . For otherwise, the first item that ended up on truck j + 1 would have been put on truck j by the algorithm, as it would still have fit there. Let m be the number of trucks used by the algorithm and L = j l j be the total load. We can now rewrite L = j l j m/ 2 j =1 (1 2 j - 1+ l 2 j ) > m/ 2 j =1 K = m/ 2 K .

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