1,240 research outputs found
Arithmetic and Boolean Operations on Recursively Run-Length Compressed Natural Numbers
We study arithmetic properties of a new tree-based canonical num- ber representation, recursively run-length compressed natural numbers, defined by applying recursively a run-length encoding of their binary digits. We design arithmetic and boolean operations with recursively run- length compressed natural numbers that work a block of digits at a time and are limited only by the representation complexity of their operands, rather than their bitsizes. As a result, operations on very large numbers exhibiting a regular structure become tractable. In addition, we ensure that the average complexity of our operations is still within constant factors of the usual arithmetic operations on binary numbers. Arithmetic operations on our recursively run-length compressed are specified as pattern-directed recursive equations made executable by using a purely declarative subset of the functional language Haskell
Schumacher's quantum data compression as a quantum computation
An explicit algorithm for performing Schumacher's noiseless compression of
quantum bits is given. This algorithm is based on a combinatorial expression
for a particular bijection among binary strings. The algorithm, which adheres
to the rules of reversible programming, is expressed in a high-level pseudocode
language. It is implemented using two- and three-bit primitive
reversible operations, where is the length of the qubit strings to be
compressed. Also, the algorithm makes use of auxiliary qubits; however,
space-saving techniques based on those proposed by Bennett are developed which
reduce this workspace to while increasing the running time by
less than a factor of two.Comment: 37 pages, no figure
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Complexity of Computations with Matrices and Polynomials
We review the complexity of polynomial and matrix computations, as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms
Faster subsequence recognition in compressed strings
Computation on compressed strings is one of the key approaches to processing
massive data sets. We consider local subsequence recognition problems on
strings compressed by straight-line programs (SLP), which is closely related to
Lempel--Ziv compression. For an SLP-compressed text of length , and an
uncompressed pattern of length , C{\'e}gielski et al. gave an algorithm for
local subsequence recognition running in time . We improve
the running time to . Our algorithm can also be used to
compute the longest common subsequence between a compressed text and an
uncompressed pattern in time ; the same problem with a
compressed pattern is known to be NP-hard
Unary Pushdown Automata and Straight-Line Programs
We consider decision problems for deterministic pushdown automata over a
unary alphabet (udpda, for short). Udpda are a simple computation model that
accept exactly the unary regular languages, but can be exponentially more
succinct than finite-state automata. We complete the complexity landscape for
udpda by showing that emptiness (and thus universality) is P-hard, equivalence
and compressed membership problems are P-complete, and inclusion is
coNP-complete. Our upper bounds are based on a translation theorem between
udpda and straight-line programs over the binary alphabet (SLPs). We show that
the characteristic sequence of any udpda can be represented as a pair of
SLPs---one for the prefix, one for the lasso---that have size linear in the
size of the udpda and can be computed in polynomial time. Hence, decision
problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP
can be converted in logarithmic space into a udpda, and this forms the basis
for our lower bound proofs. We show coNP-hardness of the ordered matching
problem for SLPs, from which we derive coNP-hardness for inclusion. In
addition, we complete the complexity landscape for unary nondeterministic
pushdown automata by showing that the universality problem is -hard, using a new class of integer expressions. Our techniques have
applications beyond udpda. We show that our results imply -completeness for a natural fragment of Presburger arithmetic and coNP lower
bounds for compressed matching problems with one-character wildcards
Stability and Complexity of Minimising Probabilistic Automata
We consider the state-minimisation problem for weighted and probabilistic
automata. We provide a numerically stable polynomial-time minimisation
algorithm for weighted automata, with guaranteed bounds on the numerical error
when run with floating-point arithmetic. Our algorithm can also be used for
"lossy" minimisation with bounded error. We show an application in image
compression. In the second part of the paper we study the complexity of the
minimisation problem for probabilistic automata. We prove that the problem is
NP-hard and in PSPACE, improving a recent EXPTIME-result.Comment: This is the full version of an ICALP'14 pape
Word-based compression in full-text retrieval systems
Ankara : Department of Industrial Engineering and the Institute of Engineering and Sciences of Bilkent University, 1995.Thesis (Master's) -- Bilkent University, 1995.Includes bibliographical references leaves 44-49.Large space requirement of a full-text retrieval system can be reduced significantly
by data compression. In this study, the problem of compressing the
main text of a full-text retrieval system is addressed and performance of several
coding techniques for compressing the text database is compared. Experiments
show that statistical techniques, such as arithmetic coding and Huffman coding,
give the best compression among the implemented; and using a semi-static
word-based model, the space needed to store English text is less than one third
of the original requirement.Selçuk, Ali AydınM.S
On the van der Waerden numbers w(2;3,t)
We present results and conjectures on the van der Waerden numbers w(2;3,t)
and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed
the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39,
where for t <= 30 we conjecture these lower bounds to be exact. The lower
bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we
present an improved conjecture. We also investigate regularities in the good
partitions (certificates) to better understand the lower bounds.
Motivated by such reglarities, we introduce *palindromic van der Waerden
numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers
w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good
partitions), defined as reading the same from both ends. Different from the
situation for ordinary van der Waerden numbers, these "numbers" need actually
to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide
lower bounds, which we conjecture to be exact, for t <= 35.
All computations are based on SAT solving, and we discuss the various
relations between SAT solving and Ramsey theory. Especially we introduce a
novel (open-source) SAT solver, the tawSolver, which performs best on the SAT
instances studied here, and which is actually the original DLL-solver, but with
an efficient implementation and a modern heuristic typical for look-ahead
solvers (applying the theory developed in the SAT handbook article of the
second author).Comment: Second version 25 pages, updates of numerical data, improved
formulations, and extended discussions on SAT. Third version 42 pages, with
SAT solver data (especially for new SAT solver) and improved representation.
Fourth version 47 pages, with updates and added explanation
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