26,536 research outputs found
Deterministic Graph Exploration with Advice
We consider the task of graph exploration. An -node graph has unlabeled
nodes, and all ports at any node of degree are arbitrarily numbered
. A mobile agent has to visit all nodes and stop. The exploration
time is the number of edge traversals. We consider the problem of how much
knowledge the agent has to have a priori, in order to explore the graph in a
given time, using a deterministic algorithm. This a priori information (advice)
is provided to the agent by an oracle, in the form of a binary string, whose
length is called the size of advice. We consider two types of oracles. The
instance oracle knows the entire instance of the exploration problem, i.e., the
port-numbered map of the graph and the starting node of the agent in this map.
The map oracle knows the port-numbered map of the graph but does not know the
starting node of the agent.
We first consider exploration in polynomial time, and determine the exact
minimum size of advice to achieve it. This size is ,
for both types of oracles.
When advice is large, there are two natural time thresholds:
for a map oracle, and for an instance oracle, that can be achieved
with sufficiently large advice. We show that, with a map oracle, time
cannot be improved in general, regardless of the size of advice.
We also show that the smallest size of advice to achieve this time is larger
than , for any .
For an instance oracle, advice of size is enough to achieve time
. We show that, with any advice of size , the time of
exploration must be at least , for any , and with any
advice of size , the time must be .
We also investigate minimum advice sufficient for fast exploration of
hamiltonian graphs
Relaxing the Irrevocability Requirement for Online Graph Algorithms
Online graph problems are considered in models where the irrevocability
requirement is relaxed. Motivated by practical examples where, for example,
there is a cost associated with building a facility and no extra cost
associated with doing it later, we consider the Late Accept model, where a
request can be accepted at a later point, but any acceptance is irrevocable.
Similarly, we also consider a Late Reject model, where an accepted request can
later be rejected, but any rejection is irrevocable (this is sometimes called
preemption). Finally, we consider the Late Accept/Reject model, where late
accepts and rejects are both allowed, but any late reject is irrevocable. For
Independent Set, the Late Accept/Reject model is necessary to obtain a constant
competitive ratio, but for Vertex Cover the Late Accept model is sufficient and
for Minimum Spanning Forest the Late Reject model is sufficient. The Matching
problem has a competitive ratio of 2, but in the Late Accept/Reject model, its
competitive ratio is 3/2
Topology recognition with advice
In topology recognition, each node of an anonymous network has to
deterministically produce an isomorphic copy of the underlying graph, with all
ports correctly marked. This task is usually unfeasible without any a priori
information. Such information can be provided to nodes as advice. An oracle
knowing the network can give a (possibly different) string of bits to each
node, and all nodes must reconstruct the network using this advice, after a
given number of rounds of communication. During each round each node can
exchange arbitrary messages with all its neighbors and perform arbitrary local
computations. The time of completing topology recognition is the number of
rounds it takes, and the size of advice is the maximum length of a string given
to nodes.
We investigate tradeoffs between the time in which topology recognition is
accomplished and the minimum size of advice that has to be given to nodes. We
provide upper and lower bounds on the minimum size of advice that is sufficient
to perform topology recognition in a given time, in the class of all graphs of
size and diameter , for any constant . In most
cases, our bounds are asymptotically tight
Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach
In this paper, we study the -forest problem in the model of resource
augmentation. In the -forest problem, given an edge-weighted graph ,
a parameter , and a set of demand pairs , the
objective is to construct a minimum-cost subgraph that connects at least
demands. The problem is hard to approximate---the best-known approximation
ratio is . Furthermore, -forest is as hard to
approximate as the notoriously-hard densest -subgraph problem.
While the -forest problem is hard to approximate in the worst-case, we
show that with the use of resource augmentation, we can efficiently approximate
it up to a constant factor.
First, we restate the problem in terms of the number of demands that are {\em
not} connected. In particular, the objective of the -forest problem can be
viewed as to remove at most demands and find a minimum-cost subgraph that
connects the remaining demands. We use this perspective of the problem to
explain the performance of our algorithm (in terms of the augmentation) in a
more intuitive way.
Specifically, we present a polynomial-time algorithm for the -forest
problem that, for every , removes at most demands and has
cost no more than times the cost of an optimal algorithm
that removes at most demands
Compressing DNA sequence databases with coil
Background: Publicly available DNA sequence databases such as GenBank are large, and are
growing at an exponential rate. The sheer volume of data being dealt with presents serious storage
and data communications problems. Currently, sequence data is usually kept in large "flat files,"
which are then compressed using standard Lempel-Ziv (gzip) compression – an approach which
rarely achieves good compression ratios. While much research has been done on compressing
individual DNA sequences, surprisingly little has focused on the compression of entire databases
of such sequences. In this study we introduce the sequence database compression software coil.
Results: We have designed and implemented a portable software package, coil, for compressing
and decompressing DNA sequence databases based on the idea of edit-tree coding. coil is geared
towards achieving high compression ratios at the expense of execution time and memory usage
during compression – the compression time represents a "one-off investment" whose cost is
quickly amortised if the resulting compressed file is transmitted many times. Decompression
requires little memory and is extremely fast. We demonstrate a 5% improvement in compression
ratio over state-of-the-art general-purpose compression tools for a large GenBank database file
containing Expressed Sequence Tag (EST) data. Finally, coil can efficiently encode incremental
additions to a sequence database.
Conclusion: coil presents a compelling alternative to conventional compression of flat files for the
storage and distribution of DNA sequence databases having a narrow distribution of sequence
lengths, such as EST data. Increasing compression levels for databases having a wide distribution of
sequence lengths is a direction for future work
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
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