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Improved online algorithms for the sorting buffer problem
 In Proceedings of the 24th Symposium on Theoretical Aspects of Computer Science (STACS
, 2007
"... Abstract. An instance of the sorting buffer problem consists of a metric space and a server, equipped with a finitecapacity buffer capable of holding a limited number of requests. An additional ingredient of the input is an online sequence of requests, each of which is characterized by a destinatio ..."
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Abstract. An instance of the sorting buffer problem consists of a metric space and a server, equipped with a finitecapacity buffer capable of holding a limited number of requests. An additional ingredient of the input is an online sequence of requests, each of which is characterized by a destination in the given metric; whenever a request arrives, it must be stored in the sorting buffer. At any point in time, a currently pending request can be served by drawing it out of the buffer and moving the server to its corresponding destination. The objective is to serve all input requests in a way that minimizes the total distance traveled by the server. In this paper, we focus our attention on instances of the problem in which the underlying metric is either an evenlyspaced or a continuous line metric. Our main findings can be briefly summarized as follows: 1. We present a deterministic O(log n) competitive algorithm for npoint evenlyspaced line metrics. This result improves on a randomized O(log 2 n) competitive algorithm due to Khandekar and Pandit. 2. We devise a deterministic O(log N log log N) competitive algorithm for continuous line metrics, where N is the input sequence length. 3. We establish the first nontrivial lower bound for the evenlyspaced case, by proving that the competitive ratio of any deterministic algorithm is at least 2+ √ √
The power of reordering for online minimum makespan scheduling
 In Proc. 49th FOCS
"... In the classic minimum makespan scheduling problem, we are given an input sequence of jobs with processing times. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, w ..."
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Cited by 7 (1 self)
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In the classic minimum makespan scheduling problem, we are given an input sequence of jobs with processing times. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we do not require that each arriving job has to be assigned immediately to one of the machines. A reordering buffer with limited storage capacity can be used to reorder the input sequence in a restricted fashion so as to schedule the jobs with a smaller makespan. This is a natural extension of lookahead. We present an extensive study of the power and limits of online reordering for minimum makespan scheduling. As main result, we give, for m identical machines, tight and, in comparison to the problem without reordering, much improved bounds on the competitive ratio for minimum makespan scheduling with reordering buffers. Depending on m, the achieved competitive ratio lies between 4/3 and 1.4659. This optimal ratio is achieved with a buffer of size Θ(m). We show that larger buffer sizes do not result in an additional advantage and that a buffer of size Ω(m) is necessary to achieve this competitive ratio. Further, we present several algorithms for different buffer sizes. Among others, we introduce, for every buffer size k ∈ [1,(m + 1)/2], a (2 − 1/(m − k + 1))competitive algorithm, which nicely generalizes the wellknown result of Graham. For m uniformly related machines, we give a scheduling algorithm that achieves a competitive ratio of 2 with a reordering buffer of size m. Considering that the best known ∗ Supported by DFG grant WE 2842/1. competitive ratio for uniformly related machines without reordering is 5.828, this result emphasizes the power of online reordering further more. 1.
Online and offline algorithms for the Sorting Buffers Problem on the line metric
"... We consider the sorting buffers problem. Input to this problem is a sequence of requests, each specified by a point in a metric space. There is a “server ” that moves from point to point to serve these requests. To serve a request, the server needs to visit the point corresponding to that request. T ..."
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We consider the sorting buffers problem. Input to this problem is a sequence of requests, each specified by a point in a metric space. There is a “server ” that moves from point to point to serve these requests. To serve a request, the server needs to visit the point corresponding to that request. The objective is to minimize the total distance traveled by the server in the metric space. In order to achieve this, the server is allowed to serve the requests in any order that requires to “buffer ” at most k requests at any time. Thus a valid reordering can serve a request only after serving all but k previous requests. In this paper, we consider this problem on the line metric which is motivated by its application to the disc scheduling problem. We present first approximation algorithms with nontrivial approximation ratios in both online and offline settings. On a line metric with n uniformly spaced points, we give a randomized online algorithm with a competitive ratio of O(log 2 n) in expectation against an oblivious adversary. In the offline setting, our algorithm yields the first constantfactor approximation and runs in quasipolynomial time N · n · k O(log n) where N is the total number of requests. Our approach is based on a dynamic program that keeps track of the number of pending requests in each of O(log n) line segments that are geometrically increasing in length.
An Improved Competitive Algorithm for Reordering Buffer Management §
"... log k log log k We design and analyze an online reordering buffer) management algorithm with improved O competitive ratio for nonuniform costs, where k is the buffer size. This improves on the best previous result (even for uniform costs) of Englert and Westermann (ICALP 2005) giving O(log k) comp ..."
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log k log log k We design and analyze an online reordering buffer) management algorithm with improved O competitive ratio for nonuniform costs, where k is the buffer size. This improves on the best previous result (even for uniform costs) of Englert and Westermann (ICALP 2005) giving O(log k) competitive ratio, which was also the best (offline) polynomial time approximation guarantee for this problem. Our analysis is based on an intricate dual fitting argument using a linear programming relaxation for the problem that we introduce in this paper. 1
Buffer Management for Colored Packets with Deadlines
"... We consider buffer management of unit packets with deadlines for a multiport device with reconfiguration overhead. The goal is to maximize the throughput of the device, i.e., the number of packets delivered by their deadline. For a single port or with free reconfiguration, the problem reduces to th ..."
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We consider buffer management of unit packets with deadlines for a multiport device with reconfiguration overhead. The goal is to maximize the throughput of the device, i.e., the number of packets delivered by their deadline. For a single port or with free reconfiguration, the problem reduces to the wellknown packets scheduling problem, where the celebrated earliestdeadlinefirst (EDF) strategy is optimal 1competitive. However, EDF is not 1competitive when there is a reconfiguration overhead. We design an online algorithm that achieves a competitive ratio of 1 − o(1) when the ratio between the minimum laxity of the packets and the number of ports tends to infinity. This is one of the rare cases where one can design an almost 1competitive algorithm. One ingredient of our analysis, which may be interesting on its own right, is a perturbation theorem on EDF for the classical packets scheduling problem. Specifically, we show that a small perturbation in the release and deadline times cannot significantly degrade the optimal throughput. This implies
Optimal Online Buffer Scheduling for Block Devices * ABSTRACT
"... We introduce a buffer scheduling problem for block operation devices in an online setting. We consider a stream of items of different types to be processed by a block device. The block device can process all items of the same type in a single step. To improve the performance of the system a buffer o ..."
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We introduce a buffer scheduling problem for block operation devices in an online setting. We consider a stream of items of different types to be processed by a block device. The block device can process all items of the same type in a single step. To improve the performance of the system a buffer of size k is used to store items in order to reduce the number of operations required. Whenever the buffer becomes full a buffer scheduling strategy has to select one type and then a block operation on all elements with this type that are currently in the buffer is performed. The goal is to design a scheduling strategy that minimizes the number of block operations required. In this paper we consider the online version of this problem, where the buffer scheduling strategy must make decisions without knowing the future items that appear in the input stream. Our main result is the design of an O(log log k)competitive online randomized buffer scheduling strategy. The bound is asymptotically tight. As a byproduct of our LPbased techniques, we obtain a randomized offline algorithm that approximates the optimal number of block operations to within a constant factor.
A note on sorting buffers offline
"... We consider the offline sorting buffer problem. The input is a sequence of items of different types. All items must be processed one by one by a server. The server is equipped with a randomaccess buffer of limited capacity which can be used to rearrange items. The problem is to design a scheduling ..."
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We consider the offline sorting buffer problem. The input is a sequence of items of different types. All items must be processed one by one by a server. The server is equipped with a randomaccess buffer of limited capacity which can be used to rearrange items. The problem is to design a scheduling strategy that decides upon the order in which items from the buffer are sent to the server. Each type change incurs unit cost, and thus, the objective is to minimize the total number of type changes for serving the entire sequence. This problem is motivated by various applications in manufacturing processes and computer science, and it has attracted significant attention in the last few years. The main focus has been on online competitive algorithms. Surprisingly little is known on the basic offline problem. In this paper, we show that the sorting buffer problem with uniform cost is NPhard and, thus, close one of the most fundamental questions for the offline problem. On the positive side, we give an O(1)approximation algorithm when the scheduler is given a buffer only slightly larger than double the original size. We also sketch a fast dynamic programming algorithm for the special case of buffer size 2. Keywords: NPhard, approximation algorithm, resource augmentation, buffer sorting