By Sorensen D.C., Wets R.J.-B. (eds.)
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Extra resources for Algorithms and Theory in Filtering and Control, part 1
Aggarwal, A. Deshpande, and R. Kannan k−1 ≤ i(n − k)δ 2 2n(k − i)Δ2 1− i=1 ≤ 1− = 1− ≤ 1− ≤ 1− = 1− 1 2 k−1 i=1 i(n − k)δ 2 2n(k − i)Δ2 n − k δ2 2n Δ2 2 δ 4Δ2 k−1 i=1 δ2 k 8Δ2 k−1 i=1 i=1 kδ i k−i k−i i k−1 for Δ for n k 1 −1 i 2 δ k log k. 8Δ2 Thus Pr (adaptive sampling covers all S1 , S2 , . . , Sk ) = 1 − Θ δ2 k log k . Δ2 If our adaptive sampling covers all S1 , S2 , . . , do not cover) one of the Si ’s the error is at least Errsome miss ≥ n 2 Δ . k So the expected error for adaptive sampling is given by δ2 δ2 k log k Err + Θ k log k Errsome miss no miss Δ2 Δ2 δ2 δ2 n 2 2 Δ ≥ 1−Θ k log k (n − k)δ + Θ k log k 2 2 Δ Δ k 1 ≥ (n − k)δ 2 + 2 · some term + Θ(log k)nδ 2 Δ n−k 2 δ = Ω(log k) using n k and Δ → ∞ 2 = Ω(log k)OPT.
Notice that if we cannot color the weight2, it must be that each weight-1 is blocking colors 2/3, 4/5, 6/7, 8/9. Thus, WLOG, we assume that these weight-1s are in colors 2, 4, 6, and 8. Then one of the following cases must hold: 1. All weight-2 intervals seen so far are colored 0/1. 2. Let t be the ending time of the latest weight-2 that isn’t colored 0/1 (without loss of generality, we’ll assume its colored 2/3). At least one of the colors 4 through 9 either have a weight-1 interval starting at t OR are unoccupied between t and t + 1.
26(6), 192–203 (1991) 7. : Register allocation and spilling via graph coloring. SIGPLAN Notices 17, 98–105 (1982) 8. : Register allocation via coloring. Computer Languages 6, 47–57 (1981) 9. : Register allocation by priority-based coloring. SIGPLAN Not. 19(6), 222–232 (1984) 10. : Engineering a Compiler. Morgan Kaufmann, San Francisco (2003) 11. : A threshold of ln n for approximating set cover. Journal of the ACM 45(4), 634–652 (1998) 12. : Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph.