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Register a new userResults on Competitiveness of Online Shortest Remaining Processing Time(SRPT) Scheduling with Special Classes of Inputs
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Debasis Dwibedy
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Shortest Remaining Processing Time (SRPT) is a well known preemptive scheduling algorithm for uniprocessor and multiprocessor systems. SRPT finds applications in the emerging areas such as scheduling of clients requests that are submitted to a web server for accessing static web pages, managing the access requests to files in multiuser database systems and routing of packets across several links as per bandwidth availability in data communications. SRPT has been proved to be optimal for the settings, where the objective is to minimize the mean response time of a list of jobs. According to our knowledge, there is less attention on the study of online SRPT with respect to the minimization of makespan as a performance criterion. In this paper, we study the SRPT algorithm for online scheduling in multiprocessor systems with makespan minimization as an objective. We obtain improved constant competitiveness results for algorithm SRPT for special classes of online job sequences based on practical real life applications.
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We consider the problem of online scheduling on a single machine in order to minimize weighted flow time. The existing algorithms for this problem (STOC 01, SODA 03, FOCS 18) all require exact knowledge of the processing time of each job. This assumption is crucial, as even a slight perturbation of the processing time would lead to polynomial competitive ratio. However, this assumption very rarely holds in real-life scenarios. In this paper, we present the first algorithm for weighted flow time which do not require exact knowledge of the processing times of jobs. Specifically, we introduce the Scheduling with Predicted Processing Time (SPPT) problem, where the algorithm is given a prediction for the processing time of each job, instead of its real processing time. For the case of a constant factor distortion between the predictions and the real processing time, our algorithms match all the best known competitiveness bounds for weighted flow time -- namely $O(log P), O(log D)$ and $O(log W)$, where $P,D,W$ are the maximum ratios of processing times, densities, and weights, respectively. For larger errors, the competitiveness of our algorithms degrades gracefully.
We consider the problem of efficiently scheduling jobs with precedence constraints on a set of identical machines in the presence of a uniform communication delay. In this setting, if two precedence-constrained jobs $u$ and $v$, with ($u prec v$), are scheduled on different machines, then $v$ must start at least $rho$ time units after $u$ completes. The scheduling objective is to minimize makespan, i.e. the total time between when the first job starts and the last job completes. The focus of this paper is to provide an efficient approximation algorithm with near-linear running time. We build on the algorithm of Lepere and Rapine [STACS 2002] for this problem to give an $Oleft(frac{ln rho}{ln ln rho} right)$-approximation algorithm that runs in $tilde{O}(|V| + |E|)$ time.
Consider an online facility assignment problem where a set of facilities $F = { f_1, f_2, f_3, cdots, f_{|F|} }$ of equal capacity $l$ is situated on a metric space and customers arrive one by one in an online manner on that space. We assign a customer $c_i$ to a facility $f_j$ before a new customer $c_{i+1}$ arrives. The cost of this assignment is the distance between $c_i$ and $f_j$. The objective of this problem is to minimize the sum of all assignment costs. Recently Ahmed et al. (TCS, 806, pp. 455-467, 2020) studied the problem where the facilities are situated on a line and computed competitive ratio of Algorithm Greedy which assigns the customer to the nearest available facility. They computed competitive ratio of algorithm named Algorithm Optimal-Fill which assigns the new customer considering optimal assignment of all previous customers. They also studied the problem where the facilities are situated on a connected unweighted graph. In this paper we first consider that $F$ is situated on the vertices of a connected unweighted grid graph $G$ of size $r times c$ and customers arrive one by one having positions on the vertices of $G$. We show that Algorithm Greedy has competitive ratio $r times c + r + c$ and Algorithm Optimal-Fill has competitive ratio $O(r times c)$. We later show that the competitive ratio of Algorithm Optimal-Fill is $2|F|$ for any arbitrary graph. Our bound is tight and better than the previous result. We also consider the facilities are distributed arbitrarily on a plane and provide an algorithm for the scenario. We also provide an algorithm that has competitive ratio $(2n-1)$. Finally, we consider a straight line metric space and show that no algorithm for the online facility assignment problem has competitive ratio less than $9.001$.
The determination of time-dependent collision-free shortest paths has received a fair amount of attention. Here, we study the problem of computing a time-dependent shortest path among growing discs which has been previously studied for the instance where the departure times are fixed. We address a more general setting: For two given points $s$ and $d$, we wish to determine the function $mathcal{A}(t)$ which is the minimum arrival time at $d$ for any departure time $t$ at $s$. We present a $(1+epsilon)$-approximation algorithm for computing $mathcal{A}(t)$. As part of preprocessing, we execute $O({1 over epsilon} log({mathcal{V}_{r} over mathcal{V}_{c}}))$ shortest path computations for fixed departure times, where $mathcal{V}_{r}$ is the maximum speed of the robot and $mathcal{V}_{c}$ is the minimum growth rate of the discs. For any query departure time $t geq 0$ from $s$, we can approximate the minimum arrival time at the destination in $O(log ({1 over epsilon}) + loglog({mathcal{V}_{r} over mathcal{V}_{c}}))$ time, within a factor of $1+epsilon$ of optimal. Since we treat the shortest path computations as black-box functions, for different settings of growing discs, we can plug-in different shortest path algorithms. Thus, the exact time complexity of our algorithm is determined by the running time of the shortest path computations.
Frei et al. [6] showed that the problem to decide whether a graph is stable with respect to some graph parameter under adding or removing either edges or vertices is $Theta_2^{text{P}}$-complete. They studied the common graph parameters $alpha$ (independence number), $beta$ (vertex cover number), $omega$ (clique number), and $chi$ (chromatic number) for certain variants of the stability problem. We follow their approach and provide a large number of polynomial-time algorithms solving these problems for special graph classes, namely for graphs without edges, complete graphs, paths, trees, forests, bipartite graphs, and co-graphs.
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