48,353 research outputs found
Replacement Paths via Row Minima of Concise Matrices
Matrix is {\em -concise} if the finite entries of each column of
consist of or less intervals of identical numbers. We give an -time
algorithm to compute the row minima of any -concise matrix.
Our algorithm yields the first -time reductions from the
replacement-paths problem on an -node -edge undirected graph
(respectively, directed acyclic graph) to the single-source shortest-paths
problem on an -node -edge undirected graph (respectively, directed
acyclic graph). That is, we prove that the replacement-paths problem is no
harder than the single-source shortest-paths problem on undirected graphs and
directed acyclic graphs. Moreover, our linear-time reductions lead to the first
-time algorithms for the replacement-paths problem on the following
classes of -node -edge graphs (1) undirected graphs in the word-RAM model
of computation, (2) undirected planar graphs, (3) undirected minor-closed
graphs, and (4) directed acyclic graphs.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete
Mathematic
Observables and Microscopic Entropy of Higher Spin Black Holes
In the context of recently proposed holographic dualities between higher spin
theories in AdS3 and 1+1-dimensional CFTs with W-symmetry algebras, we revisit
the definition of higher spin black hole thermodynamics and the dictionary
between bulk fields and dual CFT operators. We build a canonical formalism
based on three ingredients: a gauge-invariant definition of conserved charges
and chemical potentials in the presence of higher spin black holes, a canonical
definition of entropy in the bulk, and a bulk-to-boundary dictionary aligned
with the asymptotic symmetry algebra. We show that our canonical formalism
shares the same formal structure as the so-called holomorphic formalism, but
differs in the definition of charges and chemical potentials and in the
bulk-to-boundary dictionary. Most importantly, we show that it admits a
consistent CFT interpretation. We discuss the spin-2 and spin-3 cases in detail
and generalize our construction to theories based on the hs[\lambda] algebra,
and on the sl(N,R) algebra for any choice of sl(2,R) embedding.Comment: 47 pages, references added, published versio
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