Towards Optimal Degree-distributions for Left-perfect Matchings in Random Bipartite Graphs

Consider a random bipartite multigraph $G$ with $n$ left nodes and $m \geq n \geq 2$ right nodes. Each left node $x$ has $d_x \geq 1$ random right neighbors. The average left degree $\Delta$ is fixed, $\Delta \geq 2$. We ask whether for the probability that $G$ has a left-perfect matching it is advantageous not to fix $d_x$ for each left node $x$ but rather choose it at random according to some (cleverly chosen) distribution. We show the following, provided that the degrees of the left nodes are independent: If $\Delta$ is an integer then it is optimal to use a fixed degree of $\Delta$ for all left nodes. If $\Delta$ is non-integral then an optimal degree-distribution has the property that each left node $x$ has two possible degrees, \floor{\Delta} and \ceil{\Delta}, with probability $p_x$ and $1-p_x$, respectively, where $p_x$ is from the closed interval $[0,1]$ and the average over all $p_x$ equals \ceil{\Delta}-\Delta. Furthermore, if $n=c\cdot m$ and $\Delta>2$ is constant, then each distribution of the left degrees that meets the conditions above determines the same threshold $c^*(\Delta)$ that has the following property as $n$ goes to infinity: If $c<c^*(\Delta)$ then there exists a left-perfect matching with high probability. If $c>c^*(\Delta)$ then there exists no left-perfect matching with high probability. The threshold $c^*(\Delta)$ is the same as the known threshold for offline $k$-ary cuckoo hashing for integral or non-integral $k=\Delta$

Mercury BLAST dictionaries: analysis and performance measurement

This report describes a hashing scheme for a dictionary of short bit strings. The scheme, which we call near-perfect hashing, was designed as part of the construction of Mercury BLAST, an FPGA-based accelerator for the BLAST family of biosequence comparison algorithms. Near-perfect hashing is a heuristic variant of the well-known displacement hashing approach to building perfect hash functions. It uses a family of hash functions composed from linear transformations on bit vectors and lookups in small precomputed tables, both of which are especially appropriate for implementation in ardware logic. We show empirically that for inputs derived from genomic DNA sequences, our scheme obtains a good tradeoff between the size of the hash table and the time required to ompute it from a set of input strings, while generating few or no collisions between keys in the table. One of the building blocks of our scheme is the H_3 family of hash functions, which are linear transformations on bit vectors. We show that the uniformity of hashing performed with randomly chosen linear transformations depends critically on their rank, and that randomly chosen transformations have a high probability of having the maximum possible uniformity. A simple test is sufficient to ensure that a randomly chosen H3 hash function will not cause an unexpectedly large number of collisions. Moreover, if two such functions are chosen independently at random, the second function is unlikely to hash together two keys that were hashed together by the first. Hashing schemes based on H3 hash functions therefore tend to distribute their inputs more uniformly than would be expected under a simple uniform hashing model, and schemes using pairs of these functions are more uniform than would be assumed for a pair of independent hash functions

Long-range big quantum-data transmission

We introduce an alternative type of quantum repeater for long-range quantum communication with improved scaling with the distance. We show that by employing hashing, a deterministic entanglement distillation protocol with one-way communication, one obtains a scalable scheme that allows one to reach arbitrary distances, with constant overhead in resources per repeater station, and ultrahigh rates. In practical terms, we show that also with moderate resources of a few hundred qubits at each repeater station, one can reach intercontinental distances. At the same time, a measurement-based implementation allows one to tolerate high loss, but also operational and memory errors of the order of several percent per qubit. This opens the way for long-distance communication of big quantum data.Comment: revised manuscript including new result

Simple proof of confidentiality for private quantum channels in noisy environments

Complete security proofs for quantum communication protocols can be notoriously involved, which convolutes their verification, and obfuscates the key physical insights the security finally relies on. In such cases, for the majority of the community, the utility of such proofs may be restricted. Here we provide a simple proof of confidentiality for parallel quantum channels established via entanglement distillation based on hashing, in the presence of noise, and a malicious eavesdropper who is restricted only by the laws of quantum mechanics. The direct contribution lies in improving the linear confidentiality levels of recurrence-type entanglement distillation protocols to exponential levels for hashing protocols. The proof directly exploits the security relevant physical properties: measurement-based quantum computation with resource states and the separation of Bell-pairs from an eavesdropper. The proof also holds for situations where Eve has full control over the input states, and obtains all information about the operations and noise applied by the parties. The resulting state after hashing is private, i.e., disentangled from the eavesdropper. Moreover, the noise regimes for entanglement distillation and confidentiality do not coincide: Confidentiality can be guaranteed even in situation where entanglement distillation fails. We extend our results to multiparty situations which are of special interest for secure quantum networks.Comment: 5 + 11 pages, 0 + 4 figures, A. Pirker and M. Zwerger contributed equally to this work, replaced with accepted versio

Finding succinct ordered minimal perfect hashing functions

An ordered minimal perfect hash table is one in which no collisions occur among a predefined set of keys, no space is unused, and the data are placed in the table in order. A new method for creating ordered minimal perfect hashing functions is presented. The method presented is based on a method developed by Fox, Heath, Daoud, and Chen, but it creates hash functions with representation space requirements closer to the theoretical lower bound. The method presented requires approximately 10% less space to represent generated hash functions, and is easier to implement than Fox et al's. However, a higher time complexity makes it practical for small sets only (&lt; 1000)

Interpolation of recurrence and hashing entanglement distillation protocols

We construct new entanglement distillation protocols by interpolating between the recurrence and hashing protocols. This leads to asymptotic two-way distillation protocols, resulting in an improvement of the distillation rate for all mixed Bell diagonal entangled states, even for the ones with very high fidelity. We also present a method how entanglement-assisted distillation protocols can be converted into non-entanglement-assisted protocols with the same yield