114 research outputs found
Multi-Party Protocols, Information Complexity and Privacy
We introduce the new measure of Public Information Complexity (PIC), as a tool for the study of multi-party computation protocols, and of quantities such as their communication complexity, or the amount of randomness they require in the context of information-theoretic private computations. We are able to use this measure directly in the natural asynchronous message-passing peer-to-peer model and show a number of interesting properties and applications of our new notion:
the Public Information Complexity is a lower bound on the Communication Complexity and an upper bound on the Information Complexity; the difference between the Public Information Complexity and the Information Complexity provides a lower bound on the amount of randomness used in a protocol; any communication protocol can be compressed to its Public Information Cost; an explicit calculation of the zero-error Public Information Complexity of the k-party, n-bit Parity function, where a player outputs the bit-wise parity of the inputs. The latter result establishes that the amount of randomness needed for a private protocol that computes this function is Omega(n)
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
On Regularity Lemma and Barriers in Streaming and Dynamic Matching
We present a new approach for finding matchings in dense graphs by building
on Szemer\'edi's celebrated Regularity Lemma. This allows us to obtain
non-trivial albeit slight improvements over longstanding bounds for matchings
in streaming and dynamic graphs. In particular, we establish the following
results for -vertex graphs:
* A deterministic single-pass streaming algorithm that finds
a -approximate matching in bits of space. This constitutes
the first single-pass algorithm for this problem in sublinear space that
improves over the -approximation of the greedy algorithm.
* A randomized fully dynamic algorithm that with high probability maintains a
-approximate matching in worst-case update time per each edge
insertion or deletion. The algorithm works even against an adaptive adversary.
This is the first update-time dynamic algorithm with approximation
guarantee arbitrarily close to one.
Given the use of regularity lemma, the improvement obtained by our algorithms
over trivial bounds is only by some factor.
Nevertheless, in each case, they show that the ``right'' answer to the problem
is not what is dictated by the previous bounds.
Finally, in the streaming model, we also present a randomized
-approximation algorithm whose space can be upper bounded by the
density of certain Ruzsa-Szemer\'edi (RS) graphs. While RS graphs by now have
been used extensively to prove streaming lower bounds, ours is the first to use
them as an upper bound tool for designing improved streaming algorithms
Safety and Liveness of Quantitative Automata
The safety-liveness dichotomy is a fundamental concept in formal languages which plays a key role in verification. Recently, this dichotomy has been lifted to quantitative properties, which are arbitrary functions from infinite words to partially-ordered domains. We look into harnessing the dichotomy for the specific classes of quantitative properties expressed by quantitative automata. These automata contain finitely many states and rational-valued transition weights, and their common value functions Inf, Sup, LimInf, LimSup, LimInfAvg, LimSupAvg, and DSum map infinite words into the totally-ordered domain of real numbers. In this automata-theoretic setting, we establish a connection between quantitative safety and topological continuity and provide an alternative characterization of quantitative safety and liveness in terms of their boolean counterparts. For all common value functions, we show how the safety closure of a quantitative automaton can be constructed in PTime, and we provide PSpace-complete checks of whether a given quantitative automaton is safe or live, with the exception of LimInfAvg and LimSupAvg automata, for which the safety check is in ExpSpace. Moreover, for deterministic Sup, LimInf, and LimSup automata, we give PTime decompositions into safe and live automata. These decompositions enable the separation of techniques for safety and liveness verification for quantitative specifications
Musings on Encodings and Expressiveness
This paper proposes a definition of what it means for one system description
language to encode another one, thereby enabling an ordering of system
description languages with respect to expressive power. I compare the proposed
definition with other definitions of encoding and expressiveness found in the
literature, and illustrate it on a case study: comparing the expressive power
of CCS and CSP.Comment: In Proceedings EXPRESS/SOS 2012, arXiv:1208.244
Distance-Preserving Graph Contractions
Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs.
In this paper we propose and study a new framework contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible.
Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance d, the corresponding super-vertices remain at distance at least varphi(d) in the contracted graph, where varphi is a tolerance function bounding the permitted distance distortion.
We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions varphi(x)=x/alpha-beta, where alpha geq 1 and beta geq 0 are arbitrary real-valued parameters.
Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases.
Further we analyze the asymptotic behavior of the size of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results
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
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