610 research outputs found
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
Topological conjugations are not constructable
We construct two computable topologically conjugate functions for which no
conjugacy is computable, or even hyperarithmetic, resolving an open question of
Kennedy and Stockman.Comment: 9 pages. v2 adds a note on folklor
The FC-rank of a context-free language
We prove that the finite condensation rank (FC-rank) of the lexicographic
ordering of a context-free language is strictly less than
Computability of probability measures and Martin-Lof randomness over metric spaces
In this paper we investigate algorithmic randomness on more general spaces
than the Cantor space, namely computable metric spaces. To do this, we first
develop a unified framework allowing computations with probability measures. We
show that any computable metric space with a computable probability measure is
isomorphic to the Cantor space in a computable and measure-theoretic sense. We
show that any computable metric space admits a universal uniform randomness
test (without further assumption).Comment: 29 page
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
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