Download Algorithm Theory – SWAT 2006: 10th Scandinavian Workshop on by Raimund Seidel (auth.), Lars Arge, Rusins Freivalds (eds.) PDF

By Raimund Seidel (auth.), Lars Arge, Rusins Freivalds (eds.)

This booklet constitutes the refereed court cases of the tenth Scandinavian Workshop on set of rules concept, SWAT 2006, held in Riga, Latvia, in July 2006.

The court cases contains 36 revised complete papers provided including three invited papers, addressing problems with theoretical algorithmics and purposes in a number of fields together with graph algorithms, computational geometry, scheduling, approximation algorithms, community algorithms, info garage and manipulation, combinatorics, sorting, looking, on-line algorithms, optimization, amd more.

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Theorem 3. For the Grid Scheduling Problem, CRFFD ≥ 2 − Θ log M M . Proof. Let k be the largest integer such that 2k+1 ≤ M , let M = 2k , and let n be a large integer. Consider the following instance of the problem: Items: For 0 ≤ i ≤ k, 2i n items of size xi = ( 12 + 2−i )M . Bins: For 0 ≤ i ≤ k − 1, 2i n bins of size bi = (1 + 2−i )M , then n bins of size 3 2 M , and finally (M − 2)n bins of size M + 1. FFD packs the items of size xi in the bins of size bi , 0 ≤ i ≤ k − 1. The last 2k n = M n items are packed, two by two, in the n bins of size 32 M and, one by one, in the (M − 2)n bins of size M + 1.

Our algorithm for the large requests computes k + 1 solutions and picks the cheapest solution among these. The first solution is to pack all the large requests with a minimum number of unit capacity colors (using First-Fit on the sorted list of large requests). For each j = 1, 2, . . , k we define aj = 12 + jε and our (j + 1)-th solution is constructed as follows. We partition the large requests into two classes: the first class consists of all large requests with bandwidth at most aj , and the second class consists of all the remaining large requests.

Since the largest possible capacity of a color is 1, no two overlapping intervals can receive the same color, and therefore the algorithm is forced to use 3k −2 colors, whereas an optimal offline algorithm can use at most k colors, each of capacity 12 + ε. The second construction will use intervals of bandwidth 12 + jε for some 2 ≤ j ≤ P . In this construction as well the algorithm is forced to use 3k − 2 colors of capacity at least 12 + jε, whereas the construction is k-colorable. An optimal offline algorithm uses k colors of capacity 12 + jε each, and these colors are used 36 L.

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