5,162 research outputs found
Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs
We introduce a new approach to computing an approximately maximum s-t flow in
a capacitated, undirected graph. This flow is computed by solving a sequence of
electrical flow problems. Each electrical flow is given by the solution of a
system of linear equations in a Laplacian matrix, and thus may be approximately
computed in nearly-linear time.
Using this approach, we develop the fastest known algorithm for computing
approximately maximum s-t flows. For a graph having n vertices and m edges, our
algorithm computes a (1-\epsilon)-approximately maximum s-t flow in time
\tilde{O}(mn^{1/3} \epsilon^{-11/3}). A dual version of our approach computes a
(1+\epsilon)-approximately minimum s-t cut in time
\tilde{O}(m+n^{4/3}\eps^{-8/3}), which is the fastest known algorithm for this
problem as well. Previously, the best dependence on m and n was achieved by the
algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute
approximately maximum s-t flows in time \tilde{O}(m\sqrt{n}\epsilon^{-1}), and
approximately minimum s-t cuts in time \tilde{O}(m+n^{3/2}\epsilon^{-3})
A Modular Order-sorted Equational Generalization Algorithm
Generalization, also called anti-unification, is the dual of unification. Given terms t and t
,
a generalizer is a term t of which t and t are substitution instances. The dual of
a most general unifier (mgu) is that of least general generalizer (lgg). In this work, we
extend the known untyped generalization algorithm to, first, an order-sorted typed setting
with sorts, subsorts, and subtype polymorphism; second, we extend it to work modulo
equational theories, where function symbols can obey any combination of associativity,
commutativity, and identity axioms (including the empty set of such axioms); and third, to
the combination of both, which results in a modular, order-sorted equational generalization
algorithm. Unlike the untyped case, there is in general no single lgg in our framework, due
to order-sortedness or to the equational axioms. Instead, there is a finite, minimal and
complete set of lggs, so that any other generalizer has at least one of them as an instance.
Our generalization algorithms are expressed by means of inference systems for which we
give proofs of correctness. This opens up new applications to partial evaluation, program
synthesis, and theorem proving for typed equational reasoning systems and typed rulebased
languages such as ASF+SDF, Elan, OBJ, Cafe-OBJ, and Maude.
© 2014 Elsevier Inc. All rights reserved.
1.M. Alpuente, S. Escobar, and J. Espert have been partially supported by the EU (FEDER) and the Spanish MEC/MICINN under grant TIN 2010-21062-C02-02, and by Generalitat Valenciana PROMETEO2011/052. J. Meseguer has been supported by NSF Grants CNS 09-04749, and CCF 09-05584.Alpuente Frasnedo, M.; Escobar Román, S.; Espert Real, J.; Meseguer, J. (2014). A Modular Order-sorted Equational Generalization Algorithm. Information and Computation. 235:98-136. https://doi.org/10.1016/j.ic.2014.01.006S9813623
Dualities and identities for entanglement-assisted quantum codes
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from exchanging the original code\u27s information qubits with its ebits. To introduce this notion, we show how entanglement-assisted repetition codes and accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes. We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We also establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and we use these to examine the existence of some EAQEC codes. Finally, we provide upper bounds on the block error probability when transmitting maximal-entanglement EAQEC codes over the depolarizing channel, and we derive variations of the hashing bound for EAQEC codes, which is a lower bound on the maximum rate at which reliable communication over Pauli channels is possible with the use of pre-shared entanglement. © 2013 Springer Science+Business Media New York
Dual-Context Calculi for Modal Logic
We present natural deduction systems and associated modal lambda calculi for
the necessity fragments of the normal modal logics K, T, K4, GL and S4. These
systems are in the dual-context style: they feature two distinct zones of
assumptions, one of which can be thought as modal, and the other as
intuitionistic. We show that these calculi have their roots in in sequent
calculi. We then investigate their metatheory, equip them with a confluent and
strongly normalizing notion of reduction, and show that they coincide with the
usual Hilbert systems up to provability. Finally, we investigate a categorical
semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see
arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089
Constacyclic codes of length over the Galois ring
For prime , represents the Galois ring of order and
characterise , where is any positive integer. In this article, we study
the Type (1) -constacyclic codes of length over the ring
, where , are
nonzero elements and . In first case, when is a
square, we show that any ideal of
is the direct sum of the ideals of
and
. In second, when
is not a square, we show that is a chain
ring whose ideals are , for where .
Also, we prove the dual of the above code is and
present the necessary and sufficient condition for these codes to be
self-orthogonal and self-dual, respectively. Moreover, the Rosenbloom-Tsfasman
(RT) distance, Hamming distance and weight distribution of Type (1)
-constacyclic codes of length are obtained when is
not a square.Comment: This article has 18 pages and ready to submit in a journa
A simple dual ascent algorithm for the multilevel facility location problem
We present a simple dual ascent method for the multilevel facility location problem which finds a solution within times the optimum for the uncapacitated case and within times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual technique. \u
Combinatorial Alphabet-Dependent Bounds for Locally Recoverable Codes
Locally recoverable (LRC) codes have recently been a focus point of research
in coding theory due to their theoretical appeal and applications in
distributed storage systems. In an LRC code, any erased symbol of a codeword
can be recovered by accessing only a small number of other symbols. For LRC
codes over a small alphabet (such as binary), the optimal rate-distance
trade-off is unknown. We present several new combinatorial bounds on LRC codes
including the locality-aware sphere packing and Plotkin bounds. We also develop
an approach to linear programming (LP) bounds on LRC codes. The resulting LP
bound gives better estimates in examples than the other upper bounds known in
the literature. Further, we provide the tightest known upper bound on the rate
of linear LRC codes with a given relative distance, an improvement over the
previous best known bounds.Comment: To appear in IEEE Transactions on Information Theor
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