Sublinear Approximation Schemes for Scheduling Precedence Graphs of Bounded Depth
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We study the classical scheduling problem on parallel machines precedence constraints where the precedence graph has the bounded depth h. Our goal is to minimize the maximum completion time. We focus on developing approximation algorithms that use only sublinear space or sublinear time. We develop the first one-pass streaming approximation schemes using sublinear space when all jobs' processing times differ no more than a constant factor c and the number of machines m is at most 2n ϵ3 h c. This is so far the best approximation we can have in terms of m, since no polynomial time approximation better than 43 exists when m = n3 unless P=NP. of 43 when m = n3 even if all jobs have equal processing time. The algorithms are then extended to the more general problem where the largest α n jobs have no more than c factor difference. for some constant 0 < α≤ 1. We also develop the first sublinear time algorithms for both problems. For the more general problem, when m ≤α n ϵ20 c^2 · h, our algorithm is a randomized (1+ϵ)-approximation scheme that runs in sublinear time. This work not only provides an algorithmic solution to the studied problem under big data and cloud computing environment, but also gives a methodological framework for designing sublinear approximation algorithms for other scheduling problems.
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