4,372 research outputs found
On Succinct Representations of Binary Trees
We observe that a standard transformation between \emph{ordinal} trees
(arbitrary rooted trees with ordered children) and binary trees leads to
interesting succinct binary tree representations. There are four symmetric
versions of these transformations. Via these transformations we get four
succinct representations of -node binary trees that use bits and support (among other operations) navigation, inorder
numbering, one of pre- or post-order numbering, subtree size and lowest common
ancestor (LCA) queries. The ability to support inorder numbering is crucial for
the well-known range-minimum query (RMQ) problem on an array of ordered
values. While this functionality, and more, is also supported in time
using bits by Davoodi et al.'s (\emph{Phil. Trans. Royal Soc. A}
\textbf{372} (2014)) extension of a representation by Farzan and Munro
(\emph{Algorithmica} \textbf{6} (2014)), their \emph{redundancy}, or the
term, is much larger, and their approach may not be suitable for practical
implementations.
One of these transformations is related to the Zaks' sequence (S.~Zaks,
\emph{Theor. Comput. Sci.} \textbf{10} (1980)) for encoding binary trees, and
we thus provide the first succinct binary tree representation based on Zaks'
sequence. Another of these transformations is equivalent to Fischer and Heun's
(\emph{SIAM J. Comput.} \textbf{40} (2011)) \minheap\ structure for this
problem. Yet another variant allows an encoding of the Cartesian tree of to
be constructed from using only bits of working space.Comment: Journal version of part of COCOON 2012 pape
Banks-Zaks fixed point analysis in momentum subtraction schemes
We analyse the critical exponents relating to the quark mass anomalous
dimension and beta-function at the Banks-Zaks fixed point in Quantum
Chromodynamics (QCD) in a variety of representations for the quark in the
momentum subtraction (MOM) schemes of Celmaster and Gonsalves. For a specific
range of values of the number of quark flavours, estimates of the exponents
appear to be scheme independent. Using the recent five loop modified minimal
subtraction (MSbar) scheme quark mass anomalous dimension and estimates of the
fixed point location we estimate the associated exponent as 0.263-0.268 for the
SU(3) colour group and 12 flavours when the quarks are in the fundamental
representation.Comment: 33 latex pages, 25 tables, anc directory contains txt file with
electronic version of renormalization group function
The conformal window in QCD and supersymmetric QCD
In both QCD and supersymmetric QCD (SQCD) with N_f flavors there are
conformal windows where the theory is asymptotically free in the ultraviolet
while the infrared physics is governed by a non-trivial fixed-point. In SQCD,
the lower N_f boundary of the conformal window, below which the theory is
confining is well understood thanks to duality. In QCD there is just a
sufficient condition for confinement based on superconvergence. Studying the
Banks-Zaks expansion and analyzing the conditions for the perturbative coupling
to have a causal analyticity structure, it is shown that the infrared
fixed-point in QCD is perturbative in the entire conformal window. This finding
suggests that there can be no analog of duality in QCD. On the other hand in
SQCD the infrared region is found to be strongly coupled in the lower part of
the conformal window, in agreement with duality. Nevertheless, we show that it
is possible to interpolate between the Banks-Zaks expansions in the electric
and magnetic theories, for quantities that can be calculated perturbatively in
both. This interpolation is explicitly demonstrated for the critical exponent
that controls the rate at which a generic physical quantity approaches the
fixed-point.Comment: Journal versio
Fixing the conformal window in QCD
A physical characterization of Landau singularities is emphasized, which
should trace the lower boundary N_f^* of the conformal window in QCD and
supersymmetric QCD. A natural way to disentangle ``perturbative'' from
``non-perturbative'' contributions to amplitudes below N_f^* is suggested.
Assuming an infrared fixed point persists in the perturbative part of the QCD
coupling even below N_f^* leads to the condition \gamma(N_f^*)=1, where \gamma
is the critical exponent. Using the Banks-Zaks expansion, one gets 4<N_f^*<6.
This result is incompatible with the existence of an analogue of Seiberg
duality in QCD. The presence of a negative ultraviolet fixed point is required
both in QCD and in supersymmetric QCD to preserve causality within the
conformal window. Evidence for the existence of such a fixed point in QCD is
provided.Comment: 10 pages, 1 figure, extended version of a talk given at the
QCDNET2000 meeting, Paris, September 11-14 2000; main new material added is
evidence for negative ultraviolet fixed point in QC
Conformal window and Landau singularities
A physical characterization of Landau singularities is emphasized, which
should trace the lower boundary N_f^* of the conformal window in QCD and
supersymmetric QCD. A natural way to disentangle ``perturbative'' from
``non-perturbative'' contributions below N_f^* is suggested. Assuming an
infrared fixed point is present in the perturbative part of the QCD coupling
even in some range below N_f^* leads to the condition gamma(N_f^*)=1, where
gamma is the critical exponent. This result is incompatible with the existence
of an analogue of Seiberg free dual magnetic phase in QCD. Using the Banks-Zaks
expansion, one gets
4<N_f^*<6. The low value of N_f^* gives some justification to the infrared
finite coupling approach to power corrections, and suggests a way to compute
their normalization from perturbative input. If the perturbative series are
still asymptotic in the negative coupling region, the presence of a negative
ultraviolet fixed point is required both in QCD and in supersymmetric QCD to
preserve causality within the conformal window. Some evidence for such a fixed
point in QCD is provided through a modified Banks-Zaks expansion. Conformal
window amplitudes, which contain power contributions, are shown to remain
generically finite in the N_f=-\infty one-loop limit in simple models with
infrared finite perturbative coupling.Comment: 35 pages, 1 figure, JHEP style. A new section added to point out the
results give some justification to the infrared finite coupling approach to
power corrections, and suggest a way to compute their normalization from
perturbative inpu
On the intersection of tolerance and cocomparability graphs.
It has been conjectured by Golumbic and Monma in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. Since cocomparability graphs can be efficiently recognized, a positive answer to this conjecture in the general case would enable us to efficiently distinguish between tolerance and bounded tolerance graphs, although it is NP-complete to recognize each of these classes of graphs separately. The conjecture has been proved under some – rather strong – structural assumptions on the input graph; in particular, it has been proved for complements of trees, and later extended to complements of bipartite graphs, and these are the only known results so far. Furthermore, it is known that the intersection of tolerance and cocomparability graphs is contained in the class of trapezoid graphs. In this article we prove that the above conjecture is true for every graph G, whose tolerance representation satisfies a slight assumption; note here that this assumption concerns only the given tolerance representation R of G, rather than any structural property of G. This assumption on the representation is guaranteed by a wide variety of graph classes; for example, our results immediately imply the correctness of the conjecture for complements of triangle-free graphs (which also implies the above-mentioned correctness for complements of bipartite graphs). Our proofs are algorithmic, in the sense that, given a tolerance representation R of a graph G, we describe an algorithm to transform R into a bounded tolerance representation R ∗ of G. Furthermore, we conjecture that any minimal tolerance graph G that is not a bounded tolerance graph, has a tolerance representation with exactly one unbounded vertex. Our results imply the non-trivial result that, in order to prove the conjecture of Golumbic and Monma, it suffices to prove our conjecture. In addition, there already exists evidence in the literature that our conjecture is true
Bounds on SCFTs from Conformal Perturbation Theory
The operator product expansion (OPE) in 4d (super)conformal field theory is
of broad interest, for both formal and phenomenological applications. In this
paper, we use conformal perturbation theory to study the OPE of nearly-free
fields coupled to SCFTs. Under fairly general assumptions, we show that the OPE
of a chiral operator of dimension with its complex
conjugate always contains an operator of dimension less than . Our
bounds apply to Banks-Zaks fixed points and their generalizations, as we
illustrate using several examples.Comment: 36 pages; v2: typos fixed, minor change
Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part II
Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT
graph (i.e. the vertex intersection graph) of the paths P, and EPT(P) is the
EPT graph (i.e. the edge intersection graph) of P. These graphs have been
extensively studied in the literature. Given two (edge) intersecting paths in a
graph, their split vertices is the set of vertices having degree at least 3 in
their union. A pair of (edge) intersecting paths is termed non-splitting if
they do not have split vertices (namely if their union is a path). We define
the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed
the ENPT graph, as the graph having a vertex for each path in P, and an edge
between every pair of vertices representing two paths that are both
edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a
tree T and a set of paths P of T such that G=ENPT(P), and we say that is
a representation of G.
Our goal is to characterize the representation of chordless ENPT cycles
(holes). To achieve this goal, we first assume that the EPT graph induced by
the vertices of an ENPT hole is given. In [2] we introduce three assumptions
(P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we
define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize
the representations of ENPT holes that satisfy (P1), (P2), (P3).
In this work, we continue our work by relaxing these three assumptions one by
one. We characterize the representations of ENPT holes satisfying (P3) by
providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also
show that there does not exist a polynomial-time algorithm to solve
HamiltonianPairRec, unless P=NP
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