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Hence, the optimal value of (15), which is the result from the optimized pick-l, can be smaller than or equal to that of (11), which is the basic pick-l.
(19) is just the optimized pick-p algorithm.
The optimized pick-l algorithm provides a tighter or at least the same upper bound than the optimized pick-p algorithm when l>p.
Thus, when l>p, the optimized pick-l algorithm provides a tighter or at least the same upper bound than the optimized pick-p algorithm.
The optimized pick-l algorithm provides an upper bound of α k.
This optimized pick-l algorithm is mostly useful from a theoretical perspective.
We are able to show that when l increases, the optimized pick-l algorithm provides tighter upper bound on α k.
In this section, we call the pick-l algorithms introduced in Section 2 and 3 as simply the (basic) pick-l and the optimized pick-l algorithms respectively.
The optimized pick-l algorithm provides a tighter, or at least the same, upper bound of α k than the basic pick-l algorithm introduced in (11).
In spite of the increased complexity of the optimized pick-l algorithm, it has an important theoretical merit, which is Lemma 6.
Therefore, the optimized pick-l algorithm, which is (15), provides a tighter or at least the same upper bound than the basic pick-l algorithm.
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