1,407 research outputs found
Optimal prefix codes for pairs of geometrically-distributed random variables
Optimal prefix codes are studied for pairs of independent, integer-valued
symbols emitted by a source with a geometric probability distribution of
parameter , . By encoding pairs of symbols, it is possible to
reduce the redundancy penalty of symbol-by-symbol encoding, while preserving
the simplicity of the encoding and decoding procedures typical of Golomb codes
and their variants. It is shown that optimal codes for these so-called
two-dimensional geometric distributions are \emph{singular}, in the sense that
a prefix code that is optimal for one value of the parameter cannot be
optimal for any other value of . This is in sharp contrast to the
one-dimensional case, where codes are optimal for positive-length intervals of
the parameter . Thus, in the two-dimensional case, it is infeasible to give
a compact characterization of optimal codes for all values of the parameter
, as was done in the one-dimensional case. Instead, optimal codes are
characterized for a discrete sequence of values of that provide good
coverage of the unit interval. Specifically, optimal prefix codes are described
for (), covering the range , and
(), covering the range . The described codes produce the expected
reduction in redundancy with respect to the one-dimensional case, while
maintaining low complexity coding operations.Comment: To appear in IEEE Transactions on Information Theor
Synthesis of Stochastic Flow Networks
A stochastic flow network is a directed graph with incoming edges (inputs)
and outgoing edges (outputs), tokens enter through the input edges, travel
stochastically in the network, and can exit the network through the output
edges. Each node in the network is a splitter, namely, a token can enter a node
through an incoming edge and exit on one of the output edges according to a
predefined probability distribution. Stochastic flow networks can be easily
implemented by DNA-based chemical reactions, with promising applications in
molecular computing and stochastic computing. In this paper, we address a
fundamental synthesis question: Given a finite set of possible splitters and an
arbitrary rational probability distribution, design a stochastic flow network,
such that every token that enters the input edge will exit the outputs with the
prescribed probability distribution.
The problem of probability transformation dates back to von Neumann's 1951
work and was followed, among others, by Knuth and Yao in 1976. Most existing
works have been focusing on the "simulation" of target distributions. In this
paper, we design optimal-sized stochastic flow networks for "synthesizing"
target distributions. It shows that when each splitter has two outgoing edges
and is unbiased, an arbitrary rational probability \frac{a}{b} with a\leq b\leq
2^n can be realized by a stochastic flow network of size n that is optimal.
Compared to the other stochastic systems, feedback (cycles in networks)
strongly improves the expressibility of stochastic flow networks.Comment: 2 columns, 15 page
Source Coding for Quasiarithmetic Penalties
Huffman coding finds a prefix code that minimizes mean codeword length for a
given probability distribution over a finite number of items. Campbell
generalized the Huffman problem to a family of problems in which the goal is to
minimize not mean codeword length but rather a generalized mean known as a
quasiarithmetic or quasilinear mean. Such generalized means have a number of
diverse applications, including applications in queueing. Several
quasiarithmetic-mean problems have novel simple redundancy bounds in terms of a
generalized entropy. A related property involves the existence of optimal
codes: For ``well-behaved'' cost functions, optimal codes always exist for
(possibly infinite-alphabet) sources having finite generalized entropy. Solving
finite instances of such problems is done by generalizing an algorithm for
finding length-limited binary codes to a new algorithm for finding optimal
binary codes for any quasiarithmetic mean with a convex cost function. This
algorithm can be performed using quadratic time and linear space, and can be
extended to other penalty functions, some of which are solvable with similar
space and time complexity, and others of which are solvable with slightly
greater complexity. This reduces the computational complexity of a problem
involving minimum delay in a queue, allows combinations of previously
considered problems to be optimized, and greatly expands the space of problems
solvable in quadratic time and linear space. The algorithm can be extended for
purposes such as breaking ties among possibly different optimal codes, as with
bottom-merge Huffman coding.Comment: 22 pages, 3 figures, submitted to IEEE Trans. Inform. Theory, revised
per suggestions of reader
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