285,123 research outputs found
Exploring Differential Obliviousness
In a recent paper, Chan et al. [SODA \u2719] proposed a relaxation of the notion of (full) memory obliviousness, which was introduced by Goldreich and Ostrovsky [J. ACM \u2796] and extensively researched by cryptographers. The new notion, differential obliviousness, requires that any two neighboring inputs exhibit similar memory access patterns, where the similarity requirement is that of differential privacy. Chan et al. demonstrated that differential obliviousness allows achieving improved efficiency for several algorithmic tasks, including sorting, merging of sorted lists, and range query data structures.
In this work, we continue the exploration of differential obliviousness, focusing on algorithms that do not necessarily examine all their input. This choice is motivated by the fact that the existence of logarithmic overhead ORAM protocols implies that differential obliviousness can yield at most a logarithmic improvement in efficiency for computations that need to examine all their input. In particular, we explore property testing, where we show that differential obliviousness yields an almost linear improvement in overhead in the dense graph model, and at most quadratic improvement in the bounded degree model. We also explore tasks where a non-oblivious algorithm would need to explore different portions of the input, where the latter would depend on the input itself, and where we show that such a behavior can be maintained under differential obliviousness, but not under full obliviousness. Our examples suggest that there would be benefits in further exploring which class of computational tasks are amenable to differential obliviousness
Distance-generalized Core Decomposition
The -core of a graph is defined as the maximal subgraph in which every
vertex is connected to at least other vertices within that subgraph. In
this work we introduce a distance-based generalization of the notion of
-core, which we refer to as the -core, i.e., the maximal subgraph in
which every vertex has at least other vertices at distance within
that subgraph. We study the properties of the -core showing that it
preserves many of the nice features of the classic core decomposition (e.g.,
its connection with the notion of distance-generalized chromatic number) and it
preserves its usefulness to speed-up or approximate distance-generalized
notions of dense structures, such as -club.
Computing the distance-generalized core decomposition over large networks is
intrinsically complex. However, by exploiting clever upper and lower bounds we
can partition the computation in a set of totally independent subcomputations,
opening the door to top-down exploration and to multithreading, and thus
achieving an efficient algorithm
On the Von Neumann Entropy of Graphs
The von Neumann entropy of a graph is a spectral complexity measure that has
recently found applications in complex networks analysis and pattern
recognition. Two variants of the von Neumann entropy exist based on the graph
Laplacian and normalized graph Laplacian, respectively. Due to its
computational complexity, previous works have proposed to approximate the von
Neumann entropy, effectively reducing it to the computation of simple node
degree statistics. Unfortunately, a number of issues surrounding the von
Neumann entropy remain unsolved to date, including the interpretation of this
spectral measure in terms of structural patterns, understanding the relation
between its two variants, and evaluating the quality of the corresponding
approximations.
In this paper we aim to answer these questions by first analysing and
comparing the quadratic approximations of the two variants and then performing
an extensive set of experiments on both synthetic and real-world graphs. We
find that 1) the two entropies lead to the emergence of similar structures, but
with some significant differences; 2) the correlation between them ranges from
weakly positive to strongly negative, depending on the topology of the
underlying graph; 3) the quadratic approximations fail to capture the presence
of non-trivial structural patterns that seem to influence the value of the
exact entropies; 4) the quality of the approximations, as well as which variant
of the von Neumann entropy is better approximated, depends on the topology of
the underlying graph
Algorithmic aspects of disjunctive domination in graphs
For a graph , a set is called a \emph{disjunctive
dominating set} of if for every vertex , is either
adjacent to a vertex of or has at least two vertices in at distance
from it. The cardinality of a minimum disjunctive dominating set of is
called the \emph{disjunctive domination number} of graph , and is denoted by
. The \textsc{Minimum Disjunctive Domination Problem} (MDDP)
is to find a disjunctive dominating set of cardinality .
Given a positive integer and a graph , the \textsc{Disjunctive
Domination Decision Problem} (DDDP) is to decide whether has a disjunctive
dominating set of cardinality at most . In this article, we first propose a
linear time algorithm for MDDP in proper interval graphs. Next we tighten the
NP-completeness of DDDP by showing that it remains NP-complete even in chordal
graphs. We also propose a -approximation
algorithm for MDDP in general graphs and prove that MDDP can not be
approximated within for any unless NP
DTIME. Finally, we show that MDDP is
APX-complete for bipartite graphs with maximum degree
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