1,415 research outputs found
A Bicriteria Approximation for the Reordering Buffer Problem
In the reordering buffer problem (RBP), a server is asked to process a
sequence of requests lying in a metric space. To process a request the server
must move to the corresponding point in the metric. The requests can be
processed slightly out of order; in particular, the server has a buffer of
capacity k which can store up to k requests as it reads in the sequence. The
goal is to reorder the requests in such a manner that the buffer constraint is
satisfied and the total travel cost of the server is minimized. The RBP arises
in many applications that require scheduling with a limited buffer capacity,
such as scheduling a disk arm in storage systems, switching colors in paint
shops of a car manufacturing plant, and rendering 3D images in computer
graphics.
We study the offline version of RBP and develop bicriteria approximations.
When the underlying metric is a tree, we obtain a solution of cost no more than
9OPT using a buffer of capacity 4k + 1 where OPT is the cost of an optimal
solution with buffer capacity k. Constant factor approximations were known
previously only for the uniform metric (Avigdor-Elgrabli et al., 2012). Via
randomized tree embeddings, this implies an O(log n) approximation to cost and
O(1) approximation to buffer size for general metrics. Previously the best
known algorithm for arbitrary metrics by Englert et al. (2007) provided an
O(log^2 k log n) approximation without violating the buffer constraint.Comment: 13 page
Run Generation Revisited: What Goes Up May or May Not Come Down
In this paper, we revisit the classic problem of run generation. Run
generation is the first phase of external-memory sorting, where the objective
is to scan through the data, reorder elements using a small buffer of size M ,
and output runs (contiguously sorted chunks of elements) that are as long as
possible.
We develop algorithms for minimizing the total number of runs (or
equivalently, maximizing the average run length) when the runs are allowed to
be sorted or reverse sorted. We study the problem in the online setting, both
with and without resource augmentation, and in the offline setting.
(1) We analyze alternating-up-down replacement selection (runs alternate
between sorted and reverse sorted), which was studied by Knuth as far back as
1963. We show that this simple policy is asymptotically optimal. Specifically,
we show that alternating-up-down replacement selection is 2-competitive and no
deterministic online algorithm can perform better.
(2) We give online algorithms having smaller competitive ratios with resource
augmentation. Specifically, we exhibit a deterministic algorithm that, when
given a buffer of size 4M , is able to match or beat any optimal algorithm
having a buffer of size M . Furthermore, we present a randomized online
algorithm which is 7/4-competitive when given a buffer twice that of the
optimal.
(3) We demonstrate that performance can also be improved with a small amount
of foresight. We give an algorithm, which is 3/2-competitive, with
foreknowledge of the next 3M elements of the input stream. For the extreme case
where all future elements are known, we design a PTAS for computing the optimal
strategy a run generation algorithm must follow.
(4) Finally, we present algorithms tailored for nearly sorted inputs which
are guaranteed to have optimal solutions with sufficiently long runs
Online Service with Delay
In this paper, we introduce the online service with delay problem. In this
problem, there are points in a metric space that issue service requests
over time, and a server that serves these requests. The goal is to minimize the
sum of distance traveled by the server and the total delay in serving the
requests. This problem models the fundamental tradeoff between batching
requests to improve locality and reducing delay to improve response time, that
has many applications in operations management, operating systems, logistics,
supply chain management, and scheduling.
Our main result is to show a poly-logarithmic competitive ratio for the
online service with delay problem. This result is obtained by an algorithm that
we call the preemptive service algorithm. The salient feature of this algorithm
is a process called preemptive service, which uses a novel combination of
(recursive) time forwarding and spatial exploration on a metric space. We hope
this technique will be useful for related problems such as reordering buffer
management, online TSP, vehicle routing, etc. We also generalize our results to
servers.Comment: 30 pages, 11 figures, Appeared in 49th ACM Symposium on Theory of
Computing (STOC), 201
Reordering Buffer Management with a Logarithmic Guarantee in General Metric Spaces
In the reordering buffer management problem a sequence of requests arrive online in a finite metric space, and have to be processed by a single server. This server is equipped with a request buffer of size k and can decide at each point in time, which request from its buffer to serve next. Servicing of a request is simply done by moving the server to the location of the request. The goal is to process all requests while minimizing the total distance that the server is traveling inside the metric space.
In this paper we present a deterministic algorithm for the reordering buffer management problem that achieves a competitive ratio of O(log Delta + min {log n,log k}) in a finite metric space of n points and aspect ratio Delta. This is the first algorithm that works for general metric spaces and has just a logarithmic dependency on the relevant parameters. The guarantee is memory-robust, i.e., the competitive ratio decreases only slightly when the buffer-size of the optimum is increased to h=(1+epsilon)k. For memory robust guarantees our bounds are close to optimal
Polylogarithmic guarantees for generalized reordering buffer management
In the Generalized Reordering Buffer Management Problem (GRBM) a sequence of items located in a metric space arrives online, and has to be processed by a set of k servers moving within the space. In a single step the first b still unprocessed items from the sequence are accessible, and a scheduling strategy has to select an item and a server. Then the chosen item is processed by moving the chosen server to its location. The goal is to process all items while minimizing the total distance travelled by the servers. This problem was introduced in [Chan, Megow, Sitters, van Stee TCS 12] and has been subsequently studied in an online setting by [Azar, Englert, Gamzu, Kidron STACS 14]. The problem is a natural generalization of two very well-studied problems: the k-server problem for b=1 and the Reordering Buffer Management Problem (RBM) for k=1. In this paper we consider the GRBM problem on a uniform metric in the online version. We show how to obtain a competitive ratio of O(log k(log k+loglog b)) for this problem. Our result is a drastic improvement in the dependency on b compared to the previous best bound of O(√b log k), and is asymptotically optimal for constant k, because Ω(log k + loglog b) is a lower bound for GRBM on uniform metrics
- …