835 research outputs found
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update
Characterization and Efficient Search of Non-Elementary Trapping Sets of LDPC Codes with Applications to Stopping Sets
In this paper, we propose a characterization for non-elementary trapping sets
(NETSs) of low-density parity-check (LDPC) codes. The characterization is based
on viewing a NETS as a hierarchy of embedded graphs starting from an ETS. The
characterization corresponds to an efficient search algorithm that under
certain conditions is exhaustive. As an application of the proposed
characterization/search, we obtain lower and upper bounds on the stopping
distance of LDPC codes.
We examine a large number of regular and irregular LDPC codes, and
demonstrate the efficiency and versatility of our technique in finding lower
and upper bounds on, and in many cases the exact value of, . Finding
, or establishing search-based lower or upper bounds, for many of the
examined codes are out of the reach of any existing algorithm
The Traveling Salesman Problem
This paper presents a self-contained introduction into algorithmic and computational aspects of the traveling salesman problem and of related problems, along with their theoretical prerequisites as seen from the point of view of an operations researcher who wants to solve practical problem instances. Extensive computational results are reported on most of the algorithms described. Optimal solutions are reported for instances with sizes up to several thousand nodes as well as heuristic solutions with provably very high quality for larger instances
Computer Aided Verification
This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book
Fountain Codes under Maximum Likelihood Decoding
This dissertation focuses on fountain codes under maximum likelihood (ML)
decoding. First LT codes are considered under a practical and widely used ML
decoding algorithm known as inactivation decoding. Different analysis
techniques are presented to characterize the decoding complexity. Next an upper
bound to the probability of decoding failure of Raptor codes under ML decoding
is provided. Then, the distance properties of an ensemble of fixed-rate Raptor
codes with linear random outer codes are analyzed. Finally, a novel class of
fountain codes is presented, which consists of a parallel concatenation of a
block code with a linear random fountain code.Comment: PhD Thesi
A STUDY OF ERASURE CORRECTING CODES
This work focus on erasure codes, particularly those that of high performance,
and the related decoding algorithms, especially with low
computational complexity. The work is composed of different pieces,
but the main components are developed within the following two main
themes.
Ideas of message passing are applied to solve the erasures after the
transmission. Efficient matrix-representation of the belief propagation
(BP) decoding algorithm on the BEG is introduced as the recovery
algorithm. Gallager's bit-flipping algorithm are further developed
into the guess and multi-guess algorithms especially for the
application to recover the unsolved erasures after the recovery algorithm.
A novel maximum-likelihood decoding algorithm, the In-place
algorithm, is proposed with a reduced computational complexity. A
further study on the marginal number of correctable erasures by the
In-place algoritinn determines a lower bound of the average number
of correctable erasures. Following the spirit in search of the most likable
codeword based on the received vector, we propose a new branch-evaluation-
search-on-the-code-tree (BESOT) algorithm, which is powerful
enough to approach the ML performance for all linear block
codes.
To maximise the recovery capability of the In-place algorithm in
network transmissions, we propose the product packetisation structure
to reconcile the computational complexity of the In-place algorithm.
Combined with the proposed product packetisation structure,
the computational complexity is less than the quadratic complexity
bound. We then extend this to application of the Rayleigh fading
channel to solve the errors and erasures. By concatenating an outer
code, such as BCH codes, the product-packetised RS codes have the
performance of the hard-decision In-place algorithm significantly better
than that of the soft-decision iterative algorithms on optimally
designed LDPC codes
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