Data-Oblivious Graph Algorithms in Outsourced External Memory

Motivated by privacy preservation for outsourced data, data-oblivious external memory is a computational framework where a client performs computations on data stored at a semi-trusted server in a way that does not reveal her data to the server. This approach facilitates collaboration and reliability over traditional frameworks, and it provides privacy protection, even though the server has full access to the data and he can monitor how it is accessed by the client. The challenge is that even if data is encrypted, the server can learn information based on the client data access pattern; hence, access patterns must also be obfuscated. We investigate privacy-preserving algorithms for outsourced external memory that are based on the use of data-oblivious algorithms, that is, algorithms where each possible sequence of data accesses is independent of the data values. We give new efficient data-oblivious algorithms in the outsourced external memory model for a number of fundamental graph problems. Our results include new data-oblivious external-memory methods for constructing minimum spanning trees, performing various traversals on rooted trees, answering least common ancestor queries on trees, computing biconnected components, and forming open ear decompositions. None of our algorithms make use of constant-time random oracles.Comment: 20 page

On the Expansions in Spin Foam Cosmology

We discuss the expansions used in spin foam cosmology. We point out that already at the one vertex level arbitrarily complicated amplitudes contribute, and discuss the geometric asymptotics of the five simplest ones. We discuss what type of consistency conditions would be required to control the expansion. We show that the factorisation of the amplitude originally considered is best interpreted in topological terms. We then consider the next higher term in the graph expansion. We demonstrate the tension between the truncation to small graphs and going to the homogeneous sector, and conclude that it is necessary to truncate the dynamics as well.Comment: 17 pages, 4 figures, published versio

On the complexity of computing the kk-restricted edge-connectivity of a graph

The \emph{$k$-restricted edge-connectivity} of a graph $G$, denoted by $\lambda_k(G)$, is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least $k$ vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the complexity of computing $\lambda_k(G)$. Very recently, in the parameterized complexity community the notion of \emph{good edge separation} of a graph has been defined, which happens to be essentially the same as the $k$-restricted edge-connectivity. Motivated by the relevance of this invariant from both combinatorial and algorithmic points of view, in this article we initiate a systematic study of its computational complexity, with special emphasis on its parameterized complexity for several choices of the parameters. We provide a number of NP-hardness and W[1]-hardness results, as well as FPT-algorithms.Comment: 16 pages, 4 figure