Semi-online Algorithm Research Articles

Algorithms for the semi-online MapReduce scheduling problem on two identical machines

We investigate the semi-online problem of MapReduce scheduling on two parallel machines. We aim to minimize the makespan. Jobs are released over-list, and each job includes a map task and a reduce task. The job’s map task can be preemptive and scheduled simultaneously onto different machines, however, the reduce task is non-preemptive. The job’s reduce task needs to wait for its map task to complete before starting. We consider the following two versions: Firstly, we know the processing time of the largest reduce task beforehand, and then design a 4/3-competitive optimal semi-online algorithm. Secondly, we know in advance the value of the reduce task with the largest processing time and the the total sum of the processing times. Then we present a 4/3-competitive semi-online algorithm. We conclude that the algorithm is the best possible when the largest reduce task meets certain conditions.

Colouring bottomless rectangles and arborescences

We study problems related to colouring families of bottomless rectangles in the plane, in an attempt to improve the polychromatic k-colouring numbermk⁎. This number is the smallest m such that any collection of bottomless rectangles can be k-coloured so that any m-fold covered point is covered by all k colours. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, or bottomless rectangles whose left corners lie on a line, mk⁎ is linear in k. We present the lower bound mk⁎≥2k−1 for general families.We also investigate semi-online colouring algorithms, which need not colour each vertex immediately, but must maintain a proper colouring. We prove that for many sweeping orders, for any positive integers m,k, there is no semi-online algorithm that can k-colour bottomless rectangles presented in that order, so that any m-fold covered point is covered by at least two colours. This holds even for translates of quadrants, and is a corollary of a stronger result for arborescence colourings: Any semi-online colouring algorithm that colours an arborescence presented in post-order may produce arbitrarily long monochromatic paths.

Minimising total weighted completion time for semi-online single machine scheduling with known arrivals and bounded processing times

This paper addresses the semi-online scheduling problem of minimising the total weighted completion time on a single machine, where a combination of information on jobs release dates and processing times is considered. In this study, jobs can only arrive at known future times and a lower bound on jobs processing times is known in advance. A new semi-online algorithm is presented and is shown to be the best possible for the considered problem. In order to make this statement, a new lower bound on the competitive ratio of any semi-online algorithm for the problem is developed and, using competitive analysis, the proposed semi-online algorithm is shown to have a competitive ratio that matches the lower bound.

Minimizing Total Weighted Completion Times for Semi-online Single Machine Scheduling

This paper addresses the semi-online scheduling problem for total weighted completion time minimization on a single machine, where a combination of information on jobs’ release dates and processing times is considered. In this study, jobs can only arrive at known future times and a lower bound on jobs’ processing times is known in advance. A new semi-online algorithm, called MCSWPT (Modified Critical Shortest Weighted Processing Time), that makes use of the available pieces of information in order to produce better schedules compared to its online peers is presented. A numerical analysis is presented, showing the impact of having the considered partial information.

Virtual network function placement with bounded migrations

With the penetration of Network Function Virtualization (NFV), network functions, traditionally deployed as proprietary physical equipment like firewalls, Network Address Translations (NATs), are gradually being implemented as software and deployed on standardized hardware. One of the crucial challenges in this paradigm is how to place the software implemented network functions to minimize the number of used physical servers. In this paper, we study the problem of how to optimally place Virtual Network Functions (VNFs) in networks where it is allowed to migrate already placed VNFs to decrease used servers. We first formulate the offline problem as an Integer Linear Programming (ILP) problem, and then propose a semi-online algorithm to solve the online variant of the problem. We name the proposed algorithm Semi-onlIne Vnf plAcement (SIVA). In particular, SIVA is based on a bin packing algorithm that solves online bin packing problem while taking care of migrations. According to our theoretical analysis, SIVA migrates at most $$\lambda$$ VNFs each step, and it has Asymptotic Competitive Ratio (ACR) of 3/2 if $$k \rightarrow \infty$$ , where $$\lambda = k \cdot |N|$$ , k is a tunable parameter, and $$|N|$$ is the the number of supported VNF types. We conduct extensive numerical simulations to evaluate the performances of SIVA. The experiment results validate the theoretical analysis and show that SIVA outperforms the state-of-the-art algorithms by achieving near-optimal performance with minor VNF migrations.

A Semi-Online Algorithm for Single Machine Scheduling with Rejection

In this paper, a single-machine semi-online scheduling problem with rejection is addressed. In this model, “semi-online” implies that [Formula: see text], where [Formula: see text] and [Formula: see text] are the maximum processing time and the minimum one among all jobs, respectively, [Formula: see text]. In this setting, each job arrives online over time, and rejection is allowable. Our goal is minimizing the total penalty cost of rejected jobs plus the total completion time of processed jobs. The seminal result of this study is that we offer an algorithm with competitive ratio [Formula: see text], which matches the result of the problem without rejection.

Semi-online Scheduling for Minimizing the Total Completion Time with Known Release Dates

The single machine scheduling problem with the objective to minimize the total completion time is considered in the semi-online setting, where release dates of jobs are known at the beginning of the decision process. A semi-online algorithm which exploits this information is presented and a numerical study is conducted in order to evaluate its performance for a variety of jobs sequences, a comparison with existing online and semi-online algorithms is presented. This study can help practitioners decide which algorithm to choose depending on the instance characteristics.

A semi-online algorithm and its competitive analysis for parallel-machine scheduling problem with rejection

In this paper, we focus on a semi-online scheduling problem with rejection on identical parallel machines, where “semi-online” means that the ratio of the longest processing time among all jobs to the shortest one is no more than γ with γ ≥ 1. In particular, in this setting, there are a coupling of independent jobs arriving online over time with the flexibility of rejection, which implies that each job will be either accepted and scheduled on one of identical machines or rejected at the cost of penalty cost. Our objective is minimizing the total completion time of the accepted jobs plus the total penalty cost of the rejected jobs. For this problem, we design a deterministic polynomial time semi-online algorithm entitled as α Delayed Shortest Processing Time with Rejection (ADSPTR). In competitive analysis, by adopting the approach “Improved Instance Reduction”, we obtain the competitive ratio of ADSPTR is at most 1+1+γ(γ−1)−1γ.

An improved semi-online algorithm for scheduling on a single machine with unexpected breakdown

We consider a semi-online scheduling problem on a single machine with an unexpected breakdown period. In the problem, each job has a processing time and a subsequent delivery time. All these data are known in the beginning. The scheduler has to determine a sequence S of these jobs for processing. After S is given, a machine unavailability period may occur where its starting time and length are not known in advance. The objective is to minimize the time by which all jobs are delivered. We present an online algorithm with a competitive ratio of $$\frac{\sqrt{5}+1}{2}\approx 1.618$$, which improves the previously known algorithm with $$\frac{2+\sqrt{2}}{2}\approx 1.707$$-competitive.

Efficient Semi-Online Algorithms for Multiple Objects in Computational Offloading

Task scheduling between edge devices and remote servers is a common application scenario in edge computing or cloud computing, also known as computational offloading. A reasonable scheduling strategy can effectively shorten task completion time, reduce energy consumption, and improve user experience. However, the traditional offline task scheduling algorithm is NP-hard, and the decision requires obtaining all the information of the task and the device (such as task computing amount, data amount, device computing resources, etc.), which is challenging to meet in practical applications. The semi-online algorithm describes the task scheduling method when the system cannot obtain all the information. In this paper, we propose an Efficient Semi-online algorithm for Multi-users task offloading (ESaM), which includes two specific implementations: ESaM-I as known server-side idle time, and ESaM-O for known task computing amount. Because ESaM-I has obtained server information, it is better than ESaM-O in performance for most of the scenarios. The experimental results show that ESaM-I and ESaM-O are superior to the well-known semi-online scheduling algorithm SPaC in task completion time. As the remote processor computing ability increases, the average makespan converges to 0.875, 0.742, 0.782 for SPaC-M, ESaM-O, and ESaM-I in the simulation.

New Heuristic For Single Machine Semi-online Total Completion Time Minimization

Abstract This paper addresses a semi-online setting of the minimization of the total completion time scheduling problem on a single machine, where jobs arrive over-time, i.e., each job has a corresponding release date at which it becomes available for processing. In this study, the case where the release dates of the jobs are known at the beginning of the decision process is considered while processing times remain unknown. A semi-online algorithm that makes use of the available information in order to produce better schedules compared to its online peers is presented. A numerical analysis is established, showing the impact of having this information about release dates.

Single Restart with Time Stamps for Parallel Task Processing with Known and Unknown Processors

We study the problem of scheduling $n$ n tasks on $m+m^{\prime }$ m + m ' parallel processors, where the processing times on $m$ m processors are known while those on the remaining $m^{\prime }$ m ' processors are not known a priori. This semi-online model is an abstraction of certain heterogeneous computing systems, e.g., with the $m$ m known processors representing local CPU cores and the unknown processors representing remote servers with uncertain availability of computing cycles. Our objective is to minimize the makespan of all tasks. We initially focus on the case $m^{\prime }=1$ m ' = 1 and propose a semi-online algorithm termed Single Restart with Time Stamps (SRTS), which has time complexity $O(n \log n)$ O ( n log n ) . We derive its competitive ratio in comparison with the optimal offline solution. If the unknown processing times are deterministic, the competitive ratio of SRTS is shown to be either always constant or asymptotically constant in practice, respectively in cases where the processing times are independent and dependent on $m$ m . A similar result is obtained when the unknown processing times are random. Furthermore, extending the ideas of SRTS, we propose a heuristic algorithm termed SRTS-Multiple (SRTS-M) for the case $m^{\prime }>1$ m ' > 1 . Finally, where tasks arrive dynamically with unknown arrival times, we extend SRTS to Dynamic SRTS (DSRTS) and find its competitive ratio. Besides the proven competitive ratios, simulation results further suggest that SRTS and SRTS-M give superior performance on average over randomly generated task processing times, substantially reducing the makespan over the best known alternatives. Interestingly, the performance gain is more significant for task processing times sampled from heavy-tailed distributions.

Semi-online scheduling of two job types on a set of multipurpose machines

In the majority of studies of online scheduling on m multipurpose machines, there is complete uncertainty about the scheduling instance. In contrast, we consider a semi-online environment where there is prior knowledge about some parameters of the problem and the objective is to minimise the makespan. In our problem, there are two job types, each of which can be processed on a unique subset of an arbitrary number of machines and the processing sets are of arbitrary structure. We analyse three distinct cases, corresponding to prior knowledge of the following three values: (1) the optimal (offline) solution value; (2) the value of the total processing time; (3) the (constant) value of the largest processing time or an upper bound on the largest processing time. We provide a semi-online algorithm with a competitive ratio of 21/m for the first two variations. For the last case, we show a competitive ratio as a function of the processing set parameters. In this case, we prove that the algorithm is asymptotically optimal for any structure of the multipurpose machines and that the competitive ratio in the worst case tends to 4/3.

Multi-Tenant Radio Access Network Slicing: Statistical Multiplexing of Spatial Loads

This paper addresses the slicing of Radio Access Network (RAN) resources by multiple tenants, e.g., virtual wireless operators and service providers. We consider a criterion for dynamic resource allocation amongst tenants, based on a weighted proportionally fair objective, which achieves desirable fairness/protection across the network slices of the different tenants and their associated users. Several key properties are established, including: the Pareto optimality of user association to base stations, the fair allocation of base stations resources, and the gains resulting from dynamic resource sharing across slices, both in terms of utility gains and capacity savings. We then address algorithmic and practical challenges in realizing the proposed criterion. We show that the objective is NP-hard, making an exact solution impractical, and design a distributed semi-online algorithm which meets performance guarantees in equilibrium and can be shown to quickly converge to a region around the equilibrium point. Building on this algorithm, we devise a practical approach with limited computational, information, and handoff overheads. We use detailed simulations to show that our approach is indeed near-optimal and provides substantial gains both to tenants (in terms of capacity savings) and end-users (in terms of improved performance).

Tight bounds for NF-based bounded-space online bin packing algorithms

In Zheng et al. (J Comb Optim 30(2):360–369, 2015) modelled a surgery problem by the one-dimensional bin packing, and developed a semi-online algorithm to give an efficient feasible solution. In their algorithm they used a buffer to temporarily store items, having a possibility to lookahead in the list. Because of the considered practical problem they investigated the 2-parametric case, when the size of the items is at most 1 / 2. Using an NF-based online algorithm the authors proved an ACR of $$13/9 = 1.44\ldots $$ for any given buffer size not less than 1. They also gave a lower bound of $$4/3=1.33\ldots $$ for the bounded-space algorithms that use NF-based rules. Later, in Zhang et al. (J Comb Optim 33(2):530–542, 2017) an algorithm was given with an ACR of 1.4243, and the authors improved the lower bound to 1.4230. In this paper we present a tight lower bound of $$h_\infty (r)$$ for the r-parametric problem when the buffer capacity is 3. Since $$h_\infty (2) = 1.42312\ldots $$ , our result—as a special case—gives a tight bound for the algorithm-class given in 2017. To prove that the lower bound is tight, we present an NF-based online algorithm that considers the r-parametric problem, and uses a buffer with capacity of 3. We prove that this algorithm has an ACR that is equal to the lower bounds for arbitrary r.

Online Algorithms with Advice

In online scenarios requests arrive over time, and each request must be serviced in an irrevocable manner before the next request arrives. Online algorithms with advice is an area of research where one attempts to measure how much knowledge of future requests is necessary to achieve a given performance level, as defined by the competitive ratio. When this knowledge, the advice, is obtainable, this leads to practical algorithms, called semi-online algorithms. On the other hand, each negative result gives robust results about the limitations of a broad range of semi-online algorithms. This survey explains the models for online algorithms with advice, motivates the study in general, presents some examples of the work that has been carried out, and includes an extensive set of references, organized by problem studied.

Semi-Online Algorithms for Computational Task Offloading with Communication Delay

We study the scheduling of computational tasks on one local processor and one remote processor with communication delay. This problem has important application in cloud computing. Although the communication time to transmit a task can be inferred from the known data size of the task and the transmission bandwidth, the processing time of the task is generally unknown until it is processed to completion. Given a set of independent tasks with unknown processing times, our objective is to minimize makespan. We study the problem under two scenarios: 1) the communication times of the tasks to the remote processor are smaller than their corresponding processing times on the remote processor, and 2) the communication times of the tasks to the remote processor are larger than their corresponding processing times on the remote processor. For the first scenario we propose the Semi-online Partitioning and Communication (SPaC) algorithm, and for the second scenario we propose the SPaC-Restart (SPaC-R) algorithm. Even though the offline version of this problem, with a priori known processing times, is NP-hard, we show that the proposed semi-online algorithms achieve $O(1)$ competitive ratios for their intended scenarios. We also provide competitive ratios for both algorithms for more general communication times. We use simulation to demonstrate that SPaC and SPaC-R outperform online list scheduling and performs comparably well with the best known offline heuristics.

Semi-online hierarchical scheduling for $$l_p$$ l p -norm load balancing with buffer or rearrangements

In this paper, we consider the semi-online hierarchical scheduling for load balancing on two identical machines. In the problem, the jobs are available online over list and the objective is to minimize the \(l_p\)-norm of the two machines’ loads. Two semi-online versions are investigated: the buffer version and the rearrangement version. We design a unified optimal semi-online algorithm for both models.

Online Algorithms with Advice: A Survey

Online algorithms with advice is an area of research where one attempts to measure how much knowledge of the future is necessary to achieve a given competitive ratio. The lower bound results give robust bounds on what is possible using semi-online algorithms. On the other hand, when the advice is of an obtainable form, algorithms using advice can lead to semi-online algorithms. There are strong relationships between advice complexity and randomization, and advice complexity has led to the introduction of the first complexity classes for online problems. This survey concerning online algorithms with advice explains the models, motivates the study in general, presents some examples of the work that has been carried out, and includes a fairly complete set of references, organized by problem studied.

Online and semi-online scheduling to minimize makespan on single machine with an availability constraint

Online and semi-online scheduling problems on a single machine with an availability constraint are considered in this paper. The machine has one unavailable interval in which jobs cannot be processed. Preemption is not allowed. Jobs arrive over time. The objective is to minimize makespan. First we discuss the online version of the problem. After giving its lower bound, we prove that Earliest Release Date (ERD) algorithm is an optimal algorithm. Then we study some semi-online problems in which the largest processing time, the total processing time, the largest release date, or the optimal makespan is known in advance. For these semi-online problems, we give their lower bounds, design semi-online algorithms and prove their competitive ratios, respectively.