854 research outputs found
Locating regions in a sequence under density constraints
Several biological problems require the identification of regions in a
sequence where some feature occurs within a target density range: examples
including the location of GC-rich regions, identification of CpG islands, and
sequence matching. Mathematically, this corresponds to searching a string of 0s
and 1s for a substring whose relative proportion of 1s lies between given lower
and upper bounds. We consider the algorithmic problem of locating the longest
such substring, as well as other related problems (such as finding the shortest
substring or a maximal set of disjoint substrings). For locating the longest
such substring, we develop an algorithm that runs in O(n) time, improving upon
the previous best-known O(n log n) result. For the related problems we develop
O(n log log n) algorithms, again improving upon the best-known O(n log n)
results. Practical testing verifies that our new algorithms enjoy significantly
smaller time and memory footprints, and can process sequences that are orders
of magnitude longer as a result.Comment: 17 pages, 8 figures; v2: minor revisions, additional explanations; to
appear in SIAM Journal on Computin
Enhanced Symmetries in Multiparameter Flux Vacua
We give a construction of type IIB flux vacua with discrete R-symmetries and
vanishing superpotential for hypersurfaces in weighted projective space with
any number of moduli. We find that the existence of such vacua for a given
space depends on properties of the modular group, and for Fermat models can be
determined solely by the weights of the projective space. The periods of the
geometry do not in general have arithmetic properties, but live in a vector
space whose properties are vital to the construction.Comment: 32 pages, LaTeX. v2: references adde
Faster Geometric Algorithms via Dynamic Determinant Computation
The computation of determinants or their signs is the core procedure in many
important geometric algorithms, such as convex hull, volume and point location.
As the dimension of the computation space grows, a higher percentage of the
total computation time is consumed by these computations. In this paper we
study the sequences of determinants that appear in geometric algorithms. The
computation of a single determinant is accelerated by using the information
from the previous computations in that sequence.
We propose two dynamic determinant algorithms with quadratic arithmetic
complexity when employed in convex hull and volume computations, and with
linear arithmetic complexity when used in point location problems. We implement
the proposed algorithms and perform an extensive experimental analysis. On one
hand, our analysis serves as a performance study of state-of-the-art
determinant algorithms and implementations. On the other hand, we demonstrate
the supremacy of our methods over state-of-the-art implementations of
determinant and geometric algorithms. Our experimental results include a 20 and
78 times speed-up in volume and point location computations in dimension 6 and
11 respectively.Comment: 29 pages, 8 figures, 3 table
Prefix Codes for Power Laws with Countable Support
In prefix coding over an infinite alphabet, methods that consider specific
distributions generally consider those that decline more quickly than a power
law (e.g., Golomb coding). Particular power-law distributions, however, model
many random variables encountered in practice. For such random variables,
compression performance is judged via estimates of expected bits per input
symbol. This correspondence introduces a family of prefix codes with an eye
towards near-optimal coding of known distributions. Compression performance is
precisely estimated for well-known probability distributions using these codes
and using previously known prefix codes. One application of these near-optimal
codes is an improved representation of rational numbers.Comment: 5 pages, 2 tables, submitted to Transactions on Information Theor
Meeting in a Polygon by Anonymous Oblivious Robots
The Meeting problem for searchers in a polygon (possibly with
holes) consists in making the searchers move within , according to a
distributed algorithm, in such a way that at least two of them eventually come
to see each other, regardless of their initial positions. The polygon is
initially unknown to the searchers, and its edges obstruct both movement and
vision. Depending on the shape of , we minimize the number of searchers
for which the Meeting problem is solvable. Specifically, if has a
rotational symmetry of order (where corresponds to no
rotational symmetry), we prove that searchers are sufficient, and
the bound is tight. Furthermore, we give an improved algorithm that optimally
solves the Meeting problem with searchers in all polygons whose
barycenter is not in a hole (which includes the polygons with no holes). Our
algorithms can be implemented in a variety of standard models of mobile robots
operating in Look-Compute-Move cycles. For instance, if the searchers have
memory but are anonymous, asynchronous, and have no agreement on a coordinate
system or a notion of clockwise direction, then our algorithms work even if the
initial memory contents of the searchers are arbitrary and possibly misleading.
Moreover, oblivious searchers can execute our algorithms as well, encoding
information by carefully positioning themselves within the polygon. This code
is computable with basic arithmetic operations, and each searcher can
geometrically construct its own destination point at each cycle using only a
compass. We stress that such memoryless searchers may be located anywhere in
the polygon when the execution begins, and hence the information they initially
encode is arbitrary. Our algorithms use a self-stabilizing map construction
subroutine which is of independent interest.Comment: 37 pages, 9 figure
Nonlinear control synthesis by convex optimization
A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state-space and flows along the system trajectories toward the equilibrium. The new criterion has a remarkable convexity property, which in this note is used for controller synthesis via convex optimization. Recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used
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