139,213 research outputs found
Succinct Representations of Permutations and Functions
We investigate the problem of succinctly representing an arbitrary
permutation, \pi, on {0,...,n-1} so that \pi^k(i) can be computed quickly for
any i and any (positive or negative) integer power k. A representation taking
(1+\epsilon) n lg n + O(1) bits suffices to compute arbitrary powers in
constant time, for any positive constant \epsilon <= 1. A representation taking
the optimal \ceil{\lg n!} + o(n) bits can be used to compute arbitrary powers
in O(lg n / lg lg n) time.
We then consider the more general problem of succinctly representing an
arbitrary function, f: [n] \rightarrow [n] so that f^k(i) can be computed
quickly for any i and any integer power k. We give a representation that takes
(1+\epsilon) n lg n + O(1) bits, for any positive constant \epsilon <= 1, and
computes arbitrary positive powers in constant time. It can also be used to
compute f^k(i), for any negative integer k, in optimal O(1+|f^k(i)|) time.
We place emphasis on the redundancy, or the space beyond the
information-theoretic lower bound that the data structure uses in order to
support operations efficiently. A number of lower bounds have recently been
shown on the redundancy of data structures. These lower bounds confirm the
space-time optimality of some of our solutions. Furthermore, the redundancy of
one of our structures "surpasses" a recent lower bound by Golynski [Golynski,
SODA 2009], thus demonstrating the limitations of this lower bound.Comment: Preliminary versions of these results have appeared in the
Proceedings of ICALP 2003 and 2004. However, all results in this version are
improved over the earlier conference versio
Computing Shapley Values for Mean Width in 3-D
The Shapley value is a common tool in game theory to evaluate the importance
of a player in a cooperative setting. In a geometric context, it provides a way
to measure the contribution of a geometric object in a set towards some
function on the set. Recently, Cabello and Chan (SoCG 2019) presented
algorithms for computing Shapley values for a number of functions for point
sets in the plane. More formally, a coalition game consists of a set of players
and a characteristic function with . Let be a uniformly random permutation of , and be
the set of players in that appear before player in the permutation
. The Shapley value of the game is defined to be . More intuitively,
the Shapley value represents the impact of player 's appearance over all
insertion orders. We present an algorithm to compute Shapley values in 3-D,
where we treat points as players and use the mean width of the convex hull as
the characteristic function. Our algorithm runs in time and
space. Our approach is based on a new data structure for a variant of
the dynamic convolution problem , where we want to answer
dynamically. Our data structure supports updating at position ,
incrementing and decrementing and rotating by . We present a data
structure that supports operations in time and
space. Moreover, the same approach can be used to compute the Shapley values
for the mean volume of the convex hull projection onto a uniformly random -subspace in time and space for a point set in
-dimensional space ()
Simple Order-Isomorphic Matching Index with Expected Compact Space
In this paper, we present a novel indexing method for the order-isomorphic pattern matching problem (also known as order-preserving pattern matching, or consecutive permutation matching), in which two equal-length strings are defined to match when X[i] < X[j] iff Y[i] < Y[j] for 0 ? i,j < |X|. We observe an interesting relation between the order-isomorphic matching and the insertion process of a binary search tree, based on which we propose a data structure which not only has a concise structure comprised of only two wavelet trees but also provides a surprisingly simple searching algorithm. In the average case analysis, the proposed method requires ?(R(T)) bits, and it is capable of answering a count query in ?(R(P)) time, and reporting an occurrence in ?(lg |T|) time, where T and P are the text and the pattern string, respectively; for a string X, R(X) is the total time taken for the construction of the binary search tree by successively inserting the keys X[|X|-1],?,X[0] at the root, and its expected value is ?(|X|lg?) where ? is the alphabet size. Furthermore, the proposed method can be viewed as a generalization of some other methods including several heuristics and restricted versions described in previous studies in the literature
Implications of the Daya Bay observation of \theta_{13} on the leptonic flavor mixing structure and CP violation
The Daya Bay Collaboration has recently reported its first \bar{\nu}_e \to
\bar{\nu}_e oscillation result which points to \theta_{13} \simeq 8.8^\circ \pm
0.8^\circ (best-fit \pm 1\sigma range) or \theta_{13} \neq 0^\circ at the
5.2\sigma level. The fact that this smallest neutrino mixing angle is not
strongly suppressed motivates us to look into the underlying structure of
lepton flavor mixing and CP violation. Two phenomenological strategies are
outlined: (1) the lepton flavor mixing matrix U consists of a constant leading
term U_0 and a small perturbation term \Delta U; and (2) the mixing angles of U
are associated with the lepton mass ratios. Some typical patterns of U_0 are
reexamined by constraining their respective perturbations with current
experimental data. We illustrate a few possible ways to minimally correct U_0
in order to fit the observed values of three mixing angles. We point out that
the structure of U may exhibit an approximate \mu-\tau permutation symmetry in
modulus, and reiterate the geometrical description of CP violation in terms of
the leptonic unitarity triangles. The salient features of nine distinct
parametrizations of U are summarized, and its Wolfenstein-like expansion is
presented by taking U_0 to be the democratic mixing pattern.Comment: RevTeX 25 pages, 1 figure, minor changes, version for publicatio
Accelerating Permutation Testing in Voxel-wise Analysis through Subspace Tracking: A new plugin for SnPM
Permutation testing is a non-parametric method for obtaining the max null
distribution used to compute corrected -values that provide strong control
of false positives. In neuroimaging, however, the computational burden of
running such an algorithm can be significant. We find that by viewing the
permutation testing procedure as the construction of a very large permutation
testing matrix, , one can exploit structural properties derived from the
data and the test statistics to reduce the runtime under certain conditions. In
particular, we see that is low-rank plus a low-variance residual. This
makes a good candidate for low-rank matrix completion, where only a very
small number of entries of ( of all entries in our experiments)
have to be computed to obtain a good estimate. Based on this observation, we
present RapidPT, an algorithm that efficiently recovers the max null
distribution commonly obtained through regular permutation testing in
voxel-wise analysis. We present an extensive validation on a synthetic dataset
and four varying sized datasets against two baselines: Statistical
NonParametric Mapping (SnPM13) and a standard permutation testing
implementation (referred as NaivePT). We find that RapidPT achieves its best
runtime performance on medium sized datasets (), with
speedups of 1.5x - 38x (vs. SnPM13) and 20x-1000x (vs. NaivePT). For larger
datasets () RapidPT outperforms NaivePT (6x - 200x) on all
datasets, and provides large speedups over SnPM13 when more than 10000
permutations (2x - 15x) are needed. The implementation is a standalone toolbox
and also integrated within SnPM13, able to leverage multi-core architectures
when available.Comment: 36 pages, 16 figure
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