On rainbow tetrahedra in Cayley graphs

Let $\Gamma_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $\Gamma_n$ are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs $G$ of largest degree 6, asymptotic diameter $|V(G)|^{1/3}$ and almost all vertices with degree: {\bf(a)} 6 in $G$; {\bf(b)} 4 in exactly six connected subgraphs of the $(3,6,3,6)$-semi-regular tessellation; and {\bf(c)} 3 in exactly four connected subgraphs of the $\{6,3\}$-regular hexagonal tessellation. These vertices have as closed neighborhoods the union (in a fixed way) of closed neighborhoods in the ten respective resulting tessellations. Generalizing asymptotic results are discussed as well.Comment: 21 pages, 7 figure

Some Ulam's reconstruction problems for quantum states

Provided a complete set of putative $k$-body reductions of a multipartite quantum state, can one determine if a joint state exists? We derive necessary conditions for this to be true. In contrast to what is known as the quantum marginal problem, we consider a setting where the labeling of the subsystems is unknown. The problem can be seen in analogy to Ulam's reconstruction conjecture in graph theory. The conjecture - still unsolved - claims that every graph on at least three vertices can uniquely be reconstructed from the set of its vertex-deleted subgraphs. When considering quantum states, we demonstrate that the non-existence of joint states can, in some cases, already be inferred from a set of marginals having the size of just more than half of the parties. We apply these methods to graph states, where many constraints can be evaluated by knowing the number of stabilizer elements of certain weights that appear in the reductions. This perspective links with constraints that were derived in the context of quantum error-correcting codes and polynomial invariants. Some of these constraints can be interpreted as monogamy-like relations that limit the correlations arising from quantum states. Lastly, we provide an answer to Ulam's reconstruction problem for generic quantum states.Comment: 22 pages, 3 figures, v2: significantly revised final versio

RiffleScrambler - a memory-hard password storing function

We introduce RiffleScrambler: a new family of directed acyclic graphs and a corresponding data-independent memory hard function with password independent memory access. We prove its memory hardness in the random oracle model. RiffleScrambler is similar to Catena -- updates of hashes are determined by a graph (bit-reversal or double-butterfly graph in Catena). The advantage of the RiffleScrambler over Catena is that the underlying graphs are not predefined but are generated per salt, as in Balloon Hashing. Such an approach leads to higher immunity against practical parallel attacks. RiffleScrambler offers better efficiency than Balloon Hashing since the in-degree of the underlying graph is equal to 3 (and is much smaller than in Ballon Hashing). At the same time, because the underlying graph is an instance of a Superconcentrator, our construction achieves the same time-memory trade-offs.Comment: Accepted to ESORICS 201

Cutoff for non-backtracking random walks on sparse random graphs

A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window, and prove that the (suitably rescaled) cutoff profile approaches a remarkably simple, universal shape

GPU cards as a low cost solution for efficient and fast classification of high dimensional gene expression datasets

The days when bioinformatics tools will be so reliable to become a standard aid in routine clinical diagnostics are getting very close. However, it is important to remember that the more complex and advanced bioinformatics tools become, the more performances are required by the computing platforms. Unfortunately, the cost of High Performance Computing (HPC) platforms is still prohibitive for both public and private medical practices. Therefore, to promote and facilitate the use of bioinformatics tools it is important to identify low-cost parallel computing solutions. This paper presents a successful experience in using the parallel processing capabilities of Graphical Processing Units (GPU) to speed up classification of gene expression profiles. Results show that using open source CUDA programming libraries allows to obtain a significant increase in performances and therefore to shorten the gap between advanced bioinformatics tools and real medical practic