We study the following online preemptive scheduling problem: given a set of jobs with release times, deadlines, processing times and weights, schedule them so as to maximize the total value obtained. Unlike traditional scheduling problems, partially completed jobs can get partial values proportional to their amounts processed. Recently Chrobak et al. gave improved lower and upper bounds [1.236, 1.8] on the competitive ratio for this problem, the upper bound being achieved by using timesharing to simulate two equal-speed processors. In this paper we (1) give a new algorithm MIXED-k with competitive ratio 1/(1 − (k/(k + 1)) k ) which approaches e/(e−1) ≈ 1.582 when k → ∞, by using timesharing to simulate k equal-speed processors;(2) give an equivalent but much more practical algorithm MIX, which is e/(e − 1)-competitive (independent of k), by timesharing the processor with different speeds (depending on the job weights), and use its interesting properties to devise an efficient implementation; (3) improve the lower bound to 1.25 by showing an identical lower bound for randomized algorithms; and (4) prove a lower bound of 1.618 on the competitive ratio when timesharing is not allowed, thus answering an open problem raised by Chang and Yap, showing that timesharing provably helps in giving better algorithms for this problem.
We study an online unit-job scheduling problem arising in buffer management. Each job is specified by its release time, deadline, and a nonnegative weight. Due to overloading conditions, some jobs have to be dropped. The goal is to maximize the total weight of scheduled jobs. We present several competitive online algorithms for various versions of unit-job scheduling, as well as some lower bounds on the competitive ratios.We first give a randomized algorithm RMIX with competitive ratio of e/(e − 1) ≈ 1.582. This is the first algorithm for this problem with competitive ratio smaller than 2.Then we consider s-bounded instances, where the span of each job (deadline minus release time) is at most s. We give a 1.25-competitive randomized algorithm for 2-bounded instances, matching the known lower bound. We also give a deterministic algorithm EDF α , whose competitive ratio on s-bounded instances is 2 − 2/s + o(1/s). For 3-bounded instances its ratio is φ ≈ 1.618, matching the known lower bound.In s-uniform instances, the span of each job is exactly s. We show that no randomized algorithm can be better than 1.25-competitive on s-uniform instances, if the span s is unbounded. For s = 2, our proof gives a lower bound of 4 − 2 √ 2 ≈ 1.172. Also, in the 2-uniform case, we prove a lower bound of √ 2 ≈ 1.414 for deterministic memoryless algorithms, matching a known upper bound. Finally, we investigate the multiprocessor case and give a 1/(1 − ( m m+1 ) m )-competitive algorithm for m processors. We also show improved lower bounds for the general and s-uniform cases.
Abstract. In this paper, a linear-time algorithm, which is optimal, is presented to solve the haplotype inference problem for pedigree data when there are no recombinations and the pedigree has no mating loops. The approach is based on the use of graphs to capture SNP, Mendelian and parity constraints of the given pedigree.
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