349 research outputs found
Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search
By applying Grover's quantum search algorithm to the lattice algorithms of
Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and
Stehl\'{e}, we obtain improved asymptotic quantum results for solving the
shortest vector problem. With quantum computers we can provably find a shortest
vector in time , improving upon the classical time
complexity of of Pujol and Stehl\'{e} and the of Micciancio and Voulgaris, while heuristically we expect to find a
shortest vector in time , improving upon the classical time
complexity of of Wang et al. These quantum complexities
will be an important guide for the selection of parameters for post-quantum
cryptosystems based on the hardness of the shortest vector problem.Comment: 19 page
A 4/3 Approximation for 2-Vertex-Connectivity
The 2-Vertex-Connected Spanning Subgraph problem (2VCSS) is among the most basic NP-hard (Survivable) Network Design problems: we are given an (unweighted) undirected graph G. Our goal is to find a subgraph S of G with the minimum number of edges which is 2-vertex-connected, namely S remains connected after the deletion of an arbitrary node. 2VCSS is well-studied in terms of approximation algorithms, and the current best (polynomial-time) approximation factor is 10/7 by Heeger and Vygen [SIDMA\u2717] (improving on earlier results by Khuller and Vishkin [STOC\u2792] and Garg, Vempala and Singla [SODA\u2793]).
Here we present an improved 4/3 approximation. Our main technical ingredient is an approximation preserving reduction to a conveniently structured subset of instances which are "almost" 3-vertex-connected. The latter reduction might be helpful in future work
On the combinatorics of suffix arrays
We prove several combinatorial properties of suffix arrays, including a
characterization of suffix arrays through a bijection with a certain
well-defined class of permutations. Our approach is based on the
characterization of Burrows-Wheeler arrays given in [1], that we apply by
reducing suffix sorting to cyclic shift sorting through the use of an
additional sentinel symbol. We show that the characterization of suffix arrays
for a special case of binary alphabet given in [2] easily follows from our
characterization. Based on our results, we also provide simple proofs for the
enumeration results for suffix arrays, obtained in [3]. Our approach to
characterizing suffix arrays is the first that exploits their relationship with
Burrows-Wheeler permutations
Optimal Dynamic Distributed MIS
Finding a maximal independent set (MIS) in a graph is a cornerstone task in
distributed computing. The local nature of an MIS allows for fast solutions in
a static distributed setting, which are logarithmic in the number of nodes or
in their degrees. The result trivially applies for the dynamic distributed
model, in which edges or nodes may be inserted or deleted. In this paper, we
take a different approach which exploits locality to the extreme, and show how
to update an MIS in a dynamic distributed setting, either \emph{synchronous} or
\emph{asynchronous}, with only \emph{a single adjustment} and in a single
round, in expectation. These strong guarantees hold for the \emph{complete
fully dynamic} setting: Insertions and deletions, of edges as well as nodes,
gracefully and abruptly. This strongly separates the static and dynamic
distributed models, as super-constant lower bounds exist for computing an MIS
in the former.
Our results are obtained by a novel analysis of the surprisingly simple
solution of carefully simulating the greedy \emph{sequential} MIS algorithm
with a random ordering of the nodes. As such, our algorithm has a direct
application as a -approximation algorithm for correlation clustering. This
adds to the important toolbox of distributed graph decompositions, which are
widely used as crucial building blocks in distributed computing.
Finally, our algorithm enjoys a useful \emph{history-independence} property,
meaning the output is independent of the history of topology changes that
constructed that graph. This means the output cannot be chosen, or even biased,
by the adversary in case its goal is to prevent us from optimizing some
objective function.Comment: 19 pages including appendix and reference
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