Computing Permutation Encodings

We consider some encodings of permutations of the first N natural numbers, discuss some relations among them and how one can be computed from others. We show a short proof of an existing efficient algorithm for one encoding, and present two new efficient algorithms for encoding permutations. One of these algorithms is constructed as the inverse of an existing algorithm for decoding, making it the first efficient permutation encoding algorithm obtained that way

Efficient digital-to-analog encoding

An important issue in analog circuit design is the problem of digital-to-analog conversion, i.e., the encoding of Boolean variables into a single analog value which contains enough information to reconstruct the values of the Boolean variables. A natural question is: what is the complexity of implementing the digital-to-analog encoding function? That question was answered by Wegener (see Inform. Processing Lett., vol.60, no.1, p.49-52, 1995), who proved matching lower and upper bounds on the size of the circuit for the encoding function. In particular, it was proven that [(3n-1)/2] 2-input arithmetic gates are necessary and sufficient for implementing the encoding function of n Boolean variables. However, the proof of the upper bound is not constructive. In this paper, we present an explicit construction of a digital-to-analog encoder that is optimal in the number of 2-input arithmetic gates. In addition, we present an efficient analog-to-digital decoding algorithm. Namely, given the encoded analog value, our decoding algorithm reconstructs the original Boolean values. Our construction is suboptimal in that it uses constants of maximum size n log n bits; the nonconstructive proof uses constants of maximum size 2n+[log n] bits

Efficient Universal Noiseless Source Codes

Although the existence of universal noiseless variable-rate codes for the class of discrete stationary ergodic sources has previously been established, very few practical universal encoding methods are available. Efficient implementable universal source coding techniques are discussed in this paper. Results are presented on source codes for which a small value of the maximum redundancy is achieved with a relatively short block length. A constructive proof of the existence of universal noiseless codes for discrete stationary sources is first presented. The proof is shown to provide a method for obtaining efficient universal noiseless variable-rate codes for various classes of sources. For memoryless sources, upper and lower bounds are obtained for the minimax redundancy as a function of the block length of the code. Several techniques for constructing universal noiseless source codes for memoryless sources are presented and their redundancies are compared with the bounds. Consideration is given to possible applications to data compression for certain nonstationary sources

Efficient superdense coding in the presence of non-Markovian noise

Many quantum information tasks rely on entanglement, which is used as a resource, for example, to enable efficient and secure communication. Typically, noise, accompanied by loss of entanglement, reduces the efficiency of quantum protocols. We develop and demonstrate experimentally a superdense coding scheme with noise, where the decrease of entanglement in Alice's encoding state does not reduce the efficiency of the information transmission. Having almost fully dephased classical two-photon polarization state at the time of encoding with concurrence $0.163\pm0.007$, we reach values of mutual information close to $1.52\pm 0.02$ ($1.89\pm 0.05$) with 3-state (4-state) encoding. This high efficiency relies both on non-Markovian features, that Bob exploits just before his Bell-state measurement, and on very high visibility ($99.6\%\pm0.1\%$) of the Hong-Ou-Mandel interference within the experimental set-up. Our proof-of-principle results with measurements on mutual information pave the way for exploiting non-Markovianity to improve the efficiency and security of quantum information processing tasks.Comment: 6 pages, 4 figures. V2: Minor change

Twenty-Five Comparators is Optimal when Sorting Nine Inputs (and Twenty-Nine for Ten)

This paper describes a computer-assisted non-existence proof of nine-input sorting networks consisting of 24 comparators, hence showing that the 25-comparator sorting network found by Floyd in 1964 is optimal. As a corollary, we obtain that the 29-comparator network found by Waksman in 1969 is optimal when sorting ten inputs. This closes the two smallest open instances of the optimal size sorting network problem, which have been open since the results of Floyd and Knuth from 1966 proving optimality for sorting networks of up to eight inputs. The proof involves a combination of two methodologies: one based on exploiting the abundance of symmetries in sorting networks, and the other, based on an encoding of the problem to that of satisfiability of propositional logic. We illustrate that, while each of these can single handed solve smaller instances of the problem, it is their combination which leads to an efficient solution for nine inputs.Comment: 18 page