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Papers by alexander souza
We study a variant of Naor’s [23] online packet buffering model: We are given a (non-preemptive) ... more We study a variant of Naor’s [23] online packet buffering model: We are given a (non-preemptive) fifo buffer (e.g., in a network switch or a router) and packets that request transmission arrive over time. Any packet has an intrinsic value R and we have to decide whether to accept or reject it. In each time-step, the first packet in the buffer
Computing Research Repository, 2011
We analyze a natural greedy algorithm, GREEDY, which sends in each time step a packet with the gr... more We analyze a natural greedy algorithm, GREEDY, which sends in each time step a packet with the greatest value. For general packet values (v1<cdots<vm)(v_1 < \cdots < v_m)(v1<cdots<vm), we show that GREEDY is (1+r)(1+r)(1+r)-competitive, where r=max1leilem−1vi/vi+1r = \max_{1\le i \le m-1} \{v_i/v_{i+1}\}r=max1leilem−1vi/vi+1. Furthermore, we show a lower bound of 2−vm/sumi=1mvi2 - v_m / \sum_{i=1}^m v_i2−vm/sumi=1mvi on the competitiveness of any deterministic online algorithm. In the special case of two packet values (1 and alpha>1\alpha > 1alpha>1), GREEDY is shown to be optimal with a competitive ratio of (alpha+2)/(alpha+1)(\alpha + 2)/(\alpha + 1)(alpha+2)/(alpha+1).
Combinatorics, Probability & Computing, 2007
A set of n independent jobs is to be scheduled without preemption on m identical parallel machine... more A set of n independent jobs is to be scheduled without preemption on m identical parallel machines. For each job j, a so called diffuse adversary chooses the distribution F j of the random processing time P j from a certain class of distributions \(\mathcal{F}_{j}\) . The scheduler is given the expectation \(\mu_{j}=\mathbb{E}[P_{j}]\) , but the actual duration is not known in advance. A positive weight w j is associated with each job j and all jobs are ready for execution at time zero. The objective is to minimise the expected competitive ratio max \(_{F\in f} \mathbb{E}[\frac{\Sigma_{j}\omega_{j}C_{j}}{OPT}]\) , where C j denotes the completion time of job j and OPT the offline optimum value. The scheduler determines a list of jobs, which is then scheduled in non-preemptive static list policy. We show a general bound on the expected competitive ratio for list scheduling algorithms, which holds for a class of so called new-better-than-used processing time distributions. This class includes, among others the exponential distribution. Our bound depends on the probability of any pair of jobs being in the wrong order in the list of an arbitrary list scheduling algorithm, compared to an optimum list. As a special case, we show that the so called WSEPT algorithm achieves \(\mathbb{E}[\frac{WSEPT}{OPT}]\leq 3-{\frac{1}{m}}\) for exponential distributed processing times.
We consider the classical problem of scheduling preemptible jobs, that arrive over time, on ident... more We consider the classical problem of scheduling preemptible jobs, that arrive over time, on identical parallel machines. The goal is to minimize the total completion time of the jobs. In standard scheduling notation of Graham et al. [5], this problem is denoted P | r j , pmtn | j c j . A popular algorithm called SRPT, which always schedules the unfinished jobs with shortest remaining processing time, is known to be 2-competitive, see Phillips et al. [12,. This is also the best known competitive ratio for any online algorithm. However, it is conjectured that the competitive ratio of SRPT is significantly less than 2. Even breaking the barrier of 2 is considered a significant step towards the final answer of this classical online problem. We improve on this open problem by showing that SRPT is 1.86competitive. This result is obtained using the following method, which might be of general interest: We define two dependent random variables that sum up to the difference between the cost of an SRPT schedule and the cost of an optimal schedule. Then we bound the sum of the expected values of these random variables with respect to the cost of the optimal schedule, yielding the claimed competitiveness. Furthermore, we show a lower bound of 21/19 for SRPT, improving on the previously best known 12/11 due to Lu et In this paper, we study the classical problem of online scheduling preemptible jobs, arriving over time, on identical machines. The goal is to minimize the total completion time of the jobs. Our performance measure is the competitive ratio, i.e., the worst-case ratio of the objective value achieved by an online algorithm and the offline optimum. Specifically, we are given m identical machines and jobs J = {1, . . . , n}, which arrive over time, where each job j becomes known at its release time r j ≥ 0. At time r j we also learn the processing time p j > 0 of job j. Preemption is allowed, i.e., at any time we may interrupt any job that is currently running and resume it later, possibly on a different machine. A schedule σ assigns (pieces of) jobs to time-intervals on machines, and the time when job j completes is denoted c j . We seek to minimize the total completion time j c j . In the standard scheduling notation due to Graham et al. [5], this problem is denoted P | r j , pmtn | j c j .
Discrete Mathematics, Algorithms and Applications, 2009
This paper investigates the problem of scheduling jobs on multiple speedscaled processors without... more This paper investigates the problem of scheduling jobs on multiple speedscaled processors without migration, i.e., we have constant α > 1 such that running a processor at speed s results in energy consumption s α per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines and flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized βB α -approximation algorithm for multiple processors without migration, where B α is the αth Bell number, that is, the number of partitions of a set of size α. Analogously, we show that any βcompetitive online algorithm for a single processor yields a βB α -competitive online algorithm for multiple processors without migration. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic βB α -approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes.
We study a variant of Naor’s [23] online packet buffering model: We are given a (non-preemptive) ... more We study a variant of Naor’s [23] online packet buffering model: We are given a (non-preemptive) fifo buffer (e.g., in a network switch or a router) and packets that request transmission arrive over time. Any packet has an intrinsic value R and we have to decide whether to accept or reject it. In each time-step, the first packet in the buffer
Computing Research Repository, 2011
We analyze a natural greedy algorithm, GREEDY, which sends in each time step a packet with the gr... more We analyze a natural greedy algorithm, GREEDY, which sends in each time step a packet with the greatest value. For general packet values (v1<cdots<vm)(v_1 < \cdots < v_m)(v1<cdots<vm), we show that GREEDY is (1+r)(1+r)(1+r)-competitive, where r=max1leilem−1vi/vi+1r = \max_{1\le i \le m-1} \{v_i/v_{i+1}\}r=max1leilem−1vi/vi+1. Furthermore, we show a lower bound of 2−vm/sumi=1mvi2 - v_m / \sum_{i=1}^m v_i2−vm/sumi=1mvi on the competitiveness of any deterministic online algorithm. In the special case of two packet values (1 and alpha>1\alpha > 1alpha>1), GREEDY is shown to be optimal with a competitive ratio of (alpha+2)/(alpha+1)(\alpha + 2)/(\alpha + 1)(alpha+2)/(alpha+1).
Combinatorics, Probability & Computing, 2007
A set of n independent jobs is to be scheduled without preemption on m identical parallel machine... more A set of n independent jobs is to be scheduled without preemption on m identical parallel machines. For each job j, a so called diffuse adversary chooses the distribution F j of the random processing time P j from a certain class of distributions \(\mathcal{F}_{j}\) . The scheduler is given the expectation \(\mu_{j}=\mathbb{E}[P_{j}]\) , but the actual duration is not known in advance. A positive weight w j is associated with each job j and all jobs are ready for execution at time zero. The objective is to minimise the expected competitive ratio max \(_{F\in f} \mathbb{E}[\frac{\Sigma_{j}\omega_{j}C_{j}}{OPT}]\) , where C j denotes the completion time of job j and OPT the offline optimum value. The scheduler determines a list of jobs, which is then scheduled in non-preemptive static list policy. We show a general bound on the expected competitive ratio for list scheduling algorithms, which holds for a class of so called new-better-than-used processing time distributions. This class includes, among others the exponential distribution. Our bound depends on the probability of any pair of jobs being in the wrong order in the list of an arbitrary list scheduling algorithm, compared to an optimum list. As a special case, we show that the so called WSEPT algorithm achieves \(\mathbb{E}[\frac{WSEPT}{OPT}]\leq 3-{\frac{1}{m}}\) for exponential distributed processing times.
We consider the classical problem of scheduling preemptible jobs, that arrive over time, on ident... more We consider the classical problem of scheduling preemptible jobs, that arrive over time, on identical parallel machines. The goal is to minimize the total completion time of the jobs. In standard scheduling notation of Graham et al. [5], this problem is denoted P | r j , pmtn | j c j . A popular algorithm called SRPT, which always schedules the unfinished jobs with shortest remaining processing time, is known to be 2-competitive, see Phillips et al. [12,. This is also the best known competitive ratio for any online algorithm. However, it is conjectured that the competitive ratio of SRPT is significantly less than 2. Even breaking the barrier of 2 is considered a significant step towards the final answer of this classical online problem. We improve on this open problem by showing that SRPT is 1.86competitive. This result is obtained using the following method, which might be of general interest: We define two dependent random variables that sum up to the difference between the cost of an SRPT schedule and the cost of an optimal schedule. Then we bound the sum of the expected values of these random variables with respect to the cost of the optimal schedule, yielding the claimed competitiveness. Furthermore, we show a lower bound of 21/19 for SRPT, improving on the previously best known 12/11 due to Lu et In this paper, we study the classical problem of online scheduling preemptible jobs, arriving over time, on identical machines. The goal is to minimize the total completion time of the jobs. Our performance measure is the competitive ratio, i.e., the worst-case ratio of the objective value achieved by an online algorithm and the offline optimum. Specifically, we are given m identical machines and jobs J = {1, . . . , n}, which arrive over time, where each job j becomes known at its release time r j ≥ 0. At time r j we also learn the processing time p j > 0 of job j. Preemption is allowed, i.e., at any time we may interrupt any job that is currently running and resume it later, possibly on a different machine. A schedule σ assigns (pieces of) jobs to time-intervals on machines, and the time when job j completes is denoted c j . We seek to minimize the total completion time j c j . In the standard scheduling notation due to Graham et al. [5], this problem is denoted P | r j , pmtn | j c j .
Discrete Mathematics, Algorithms and Applications, 2009
This paper investigates the problem of scheduling jobs on multiple speedscaled processors without... more This paper investigates the problem of scheduling jobs on multiple speedscaled processors without migration, i.e., we have constant α > 1 such that running a processor at speed s results in energy consumption s α per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines and flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized βB α -approximation algorithm for multiple processors without migration, where B α is the αth Bell number, that is, the number of partitions of a set of size α. Analogously, we show that any βcompetitive online algorithm for a single processor yields a βB α -competitive online algorithm for multiple processors without migration. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic βB α -approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes.