9 research outputs found
Reordering buffer management with advice
In the reordering buffer management problem, a sequence of colored items arrives at a service station to be processed. Each color change between two consecutively processed items generates some cost. A reordering buffer of capacity k items can be used to preprocess the input sequence in order to decrease the number of color changes. The goal is to find a scheduling strategy that, using the reordering buffer, minimizes the number of color changes in the given sequence of items. We consider the problem in the setting of online computation with advice. In this model, the color of an item becomes known only at the time when the item enters the reordering buffer. Additionally, together with each item entering the buffer, we get a fixed number of advice bits, which can be seen as information about the future or as information about an optimal solution (or an approximation thereof) for the whole input sequence. We show that for any ε>0 there is a (1+ε)-competitive algorithm for the problem which uses only a constant (depending on ε) number of advice bits per input item. This also immediately implies a (1+ε)-approximation algorithm which has 2O(nlog1/ε) running time (this should be compared to the trivial optimal algorithm which has a running time of kO(n)). We complement the above result by presenting a lower bound of Ω(logk) bits of advice per request for any 1-competitive algorithm
Optimal Online Edge Coloring of Planar Graphs with Advice
Using the framework of advice complexity, we study the amount of knowledge
about the future that an online algorithm needs to color the edges of a graph
optimally, i.e., using as few colors as possible. For graphs of maximum degree
, it follows from Vizing's Theorem that bits of
advice suffice to achieve optimality, where is the number of edges. We show
that for graphs of bounded degeneracy (a class of graphs including e.g. trees
and planar graphs), only bits of advice are needed to compute an optimal
solution online, independently of how large is. On the other hand, we
show that bits of advice are necessary just to achieve a
competitive ratio better than that of the best deterministic online algorithm
without advice. Furthermore, we consider algorithms which use a fixed number of
advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to
this class of algorithms). We show that for bipartite graphs, any such
algorithm must use at least bits of advice to achieve
optimality.Comment: CIAC 201
On the Power of Advice and Randomization for Online Bipartite Matching
While randomized online algorithms have access to a sequence of uniform
random bits, deterministic online algorithms with advice have access to a
sequence of advice bits, i.e., bits that are set by an all powerful oracle
prior to the processing of the request sequence. Advice bits are at least as
helpful as random bits, but how helpful are they? In this work, we investigate
the power of advice bits and random bits for online maximum bipartite matching
(MBM).
The well-known Karp-Vazirani-Vazirani algorithm is an optimal randomized
-competitive algorithm for \textsc{MBM} that requires access
to uniform random bits. We show that
advice bits are necessary and
sufficient in order to obtain a
-competitive deterministic advice algorithm. Furthermore, for a
large natural class of deterministic advice algorithms, we prove that
advice bits are required in order to improve on the
-competitiveness of the best deterministic online algorithm, while
it is known that bits are sufficient.
Last, we give a randomized online algorithm that uses random bits, for
integers , and a competitive ratio that approaches
very quickly as is increasing. For example if , then the difference
between and the achieved competitive ratio is less than
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
Randomization can be as helpful as a glimpse of the future in online computation
We provide simple but surprisingly useful direct product theorems for proving
lower bounds on online algorithms with a limited amount of advice about the
future. As a consequence, we are able to translate decades of research on
randomized online algorithms to the advice complexity model. Doing so improves
significantly on the previous best advice complexity lower bounds for many
online problems, or provides the first known lower bounds. For example, if
is the number of requests, we show that:
(1) A paging algorithm needs bits of advice to achieve a
competitive ratio better than , where is the cache
size. Previously, it was only known that bits of advice were
necessary to achieve a constant competitive ratio smaller than .
(2) Every -competitive vertex coloring algorithm must
use bits of advice. Previously, it was only known that
bits of advice were necessary to be optimal.
For certain online problems, including the MTS, -server, paging, list
update, and dynamic binary search tree problem, our results imply that
randomization and sublinear advice are equally powerful (if the underlying
metric space or node set is finite). This means that several long-standing open
questions regarding randomized online algorithms can be equivalently stated as
questions regarding online algorithms with sublinear advice. For example, we
show that there exists a deterministic -competitive -server
algorithm with advice complexity if and only if there exists a
randomized -competitive -server algorithm without advice.
Technically, our main direct product theorem is obtained by extending an
information theoretical lower bound technique due to Emek, Fraigniaud, Korman,
and Ros\'en [ICALP'09]
Reordering Buffer Management with Advice
In the reordering buffer management problem, a sequence of colored items arrives at a service station to be processed. Each color change between two consecutively processed items generates some cost. A reordering buffer of capacity k items can be used to preprocess the input sequence in order to decrease the number of color changes. The goal is to find a scheduling strategy that, using the reordering buffer, minimizes the number of color changes in the given sequence of items. We consider the problem in the setting of online computation with advice. In this model, the color of an item becomes known only at the time when the item enters the reordering buffer. Additionally, together with each item entering the buffer, we get a fixed number of advice bits, which can be seen as information about the future or as information about an optimal solution (or an approximation thereof) for the whole input sequence. We show that for any ε>0 there is a (1+ε)-competitive algorithm for the problem which uses only a constant (depending on ε) number of advice bits per input item. This also immediately implies a (1+ε)-approximation algorithm which has 2O(nlog1/ε) running time (this should be compared to the trivial optimal algorithm which has a running time of kO(n)). We complement the above result by presenting a lower bound of Ω(logk) bits of advice per request for any 1-competitive algorithm