Design and Analysis of Algorithms

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In the bin packing problem, the input consists of a sequence of items I = {1, . . . , n} where each item i has a size, which is a real number 0 ≤ ai ≤ 1. The goal is to “pack” the items in the smallest possible number of bins of unit size. Formally, the items should be partitioned in disjoint subsets (bins), such that the total size in each bin is at most.

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Design and Analysis of Algorithms510.6401 Design and Analysis of AlgorithmsJanuary 21, 2008Problem Set 1Due: February 4, 2008. 1. In the bin packing problem, the input consists of a sequence of items I = {1, . . . , n} where each item i has a size, which is a real number 0 ai 1. The goal is to pack the items in the smallest possible number of bins of unit size. Formally, the items should be partitioned in disjoint subsets (bins), such that the total size in each bin is at most 1. The rst t heuristic scans the items one by one, and each item is assigned to the rst bin that it can t in. Prove that rst-t is a 2-approximation algorithm for bin packing. Hint. Bound from below the number of bins used by an optimal solution; and bound from above the number of bins used by rst t, using the observation that nearly all bins are at least half-full. 2.

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