319 research outputs found
Space-Efficient Data Structures for Lattices
A lattice is a partially-ordered set in which every pair of elements has a
unique meet (greatest lower bound) and join (least upper bound). We present new
data structures for lattices that are simple, efficient, and nearly optimal in
terms of space complexity.
Our first data structure can answer partial order queries in constant time
and find the meet or join of two elements in time, where is
the number of elements in the lattice. It occupies bits of
space, which is only a factor from the -bit
lower bound for storing lattices. The preprocessing time is . This
structure admits a simple space-time tradeoff so that, for any , the data structure supports meet and join queries in
time, occupies bits of space, and can be
constructed in time.
Our second data structure uses bits of space and supports
meet and join in time, where is the maximum
degree of any element in the transitive reduction graph of the lattice. This
structure is much faster for lattices with low-degree elements.
This paper also identifies an error in a long-standing solution to the
problem of representing lattices. We discuss the issue with this previous work.Comment: Accepted in SWAT 202
A Simple Algorithm for Minimum Cuts in Near-Linear Time
We consider the minimum cut problem in undirected, weighted graphs. We give a
simple algorithm to find a minimum cut that -respects (cuts two edges of) a
spanning tree of a graph . This procedure can be used in place of the
complicated subroutine given in Karger's near-linear time minimum cut algorithm
(J. ACM, 2000). We give a self-contained version of Karger's algorithm with the
new procedure, which is easy to state and relatively simple to implement. It
produces a minimum cut on an -edge, -vertex graph in time
with high probability, matching the complexity of Karger's approach.Comment: To appear in SWAT 202
Improved Time and Space Bounds for Dynamic Range Mode
Given an array A of n elements, we wish to support queries for the most frequent and least frequent element in a subrange [l, r] of A. We also wish to support updates that change a particular element at index i or insert/ delete an element at index i. For the range mode problem, our data structure supports all operations in O(n^{2/3}) deterministic time using only O(n) space. This improves two results by Chan et al. [Timothy M. Chan et al., 2014]: a linear space data structure supporting update and query operations in O~(n^{3/4}) time and an O(n^{4/3}) space data structure supporting update and query operations in O~(n^{2/3}) time. For the range least frequent problem, we address two variations. In the first, we are allowed to answer with an element of A that may not appear in the query range, and in the second, the returned element must be present in the query range. For the first variation, we develop a data structure that supports queries in O~(n^{2/3}) time, updates in O(n^{2/3}) time, and occupies O(n) space. For the second variation, we develop a Monte Carlo data structure that supports queries in O(n^{2/3}) time, updates in O~(n^{2/3}) time, and occupies O~(n) space, but requires that updates are made independently of the results of previous queries. The Monte Carlo data structure is also capable of answering k-frequency queries; that is, the problem of finding an element of given frequency in the specified query range. Previously, no dynamic data structures were known for least frequent element or k-frequency queries
On Approximate Range Mode and Range Selection
For any epsilon in (0,1), a (1+epsilon)-approximate range mode query asks for the position of an element whose frequency in the query range is at most a factor (1+epsilon) smaller than the true mode. For this problem, we design a data structure occupying O(n/epsilon) bits of space to answer queries in O(lg(1/epsilon)) time. This is an encoding data structure which does not require access to the input sequence; the space cost of this structure is asymptotically optimal for constant epsilon as we also prove a matching lower bound. Furthermore, our solution improves the previous best result of Greve et al. (Cell Probe Lower Bounds and Approximations for Range Mode, ICALP\u2710) by saving the space cost by a factor of lg n while achieving the same query time. In dynamic settings, we design an O(n)-word data structure that answers queries in O(lg n /lg lg n) time and supports insertions and deletions in O(lg n) time, for any constant epsilon in (0,1); the bounds for non-constant epsilon = o(1) are also given in the paper. This is the first result on dynamic approximate range mode; it can also be used to obtain the first static data structure for approximate 3-sided range mode queries in two dimensions.
Another problem we consider is approximate range selection. For any alpha in (0,1/2), an alpha-approximate range selection query asks for the position of an element whose rank in the query range is in [k - alpha s, k + alpha s], where k is a rank given by the query and s is the size of the query range. When alpha is a constant, we design an O(n)-bit encoding data structure that can answer queries in constant time and prove this space cost is asymptotically optimal. The previous best result by Krizanc et al. (Range Mode and Range Median Queries on Lists and Trees, Nordic Journal of Computing, 2005) uses O(n lg n) bits, or O(n) words, to achieve constant approximation for range median only. Thus we not only improve the space cost, but also provide support for any arbitrary k given at query time. We also analyse our solutions for non-constant alpha
Non-equilibrium dynamics in an interacting nanoparticle system
Non-equilibrium dynamics in an interacting Fe-C nanoparticle sample,
exhibiting a low temperature spin glass like phase, has been studied by low
frequency ac-susceptibility and magnetic relaxation experiments. The
non-equilibrium behavior shows characteristic spin glass features, but some
qualitative differences exist. The nature of these differences is discussed.Comment: 7 pages, 11 figure
Short range ferromagnetism and spin glass state in
Dynamic magnetic properties of are
reported. The system appears to attain local ferromagnetic order at
K. Below this temperature the low field
magnetization becomes history dependent, i.e. the zero field cooled (ZFC) and
field cooled (FC) magnetization deviate from each other and closely logarithmic
relaxation appears at our experimental time scales (0.3- sec). The zero
field cooled magnetization has a maximum at K,
whereas the field cooled magnetization continues to increase, although less
sharply, also below this temperature. Surprisingly, the dynamics of the system
shows non-equilibrium spin glass (SG) features not only below the maximum in
the ZFC magnetization, but also in the temperature region between this maximum
and . The aging and temperature cycling experiments show only
quantitative differences in the dynamic behavior above and below the maximum in
the ZFC-magnetization; similarly, memory effects are observed in both
temperature regions. We attribute the high temperature behavior to the
existence of clusters of short range ferromagnetic order below
; the configuration evolves into a conventional spin glass
state at temperatures below .Comment: REVTeX style; 8 pages, 8 figure
Recommended from our members
Child mask mandates for COVID-19: a systematic review.
BACKGROUND: Mask mandates for children during the COVID-19 pandemic varied in different locations. A risk-benefit analysis of this intervention has not yet been performed. In this study, we performed a systematic review to assess research on the effectiveness of mask wearing in children. METHODS: We performed database searches up to February 2023. The studies were screened by title and abstract, and included studies were further screened as full-text references. A risk-of-bias analysis was performed by two independent reviewers and adjudicated by a third reviewer. RESULTS: We screened 597 studies and included 22 in the final analysis. There were no randomised controlled trials in children assessing the benefits of mask wearing to reduce SARS-CoV-2 infection or transmission. The six observational studies reporting an association between child masking and lower infection rate or antibody seropositivity had critical (n=5) or serious (n=1) risk of bias; all six were potentially confounded by important differences between masked and unmasked groups and two were shown to have non-significant results when reanalysed. Sixteen other observational studies found no association between mask wearing and infection or transmission. CONCLUSIONS: Real-world effectiveness of child mask mandates against SARS-CoV-2 transmission or infection has not been demonstrated with high-quality evidence. The current body of scientific data does not support masking children for protection against COVID-19
Static chaos and scaling behaviour in the spin-glass phase
We discuss the problem of static chaos in spin glasses. In the case of
magnetic field perturbations, we propose a scaling theory for the spin-glass
phase. Using the mean-field approach we argue that some pure states are
suppressed by the magnetic field and their free energy cost is determined by
the finite-temperature fixed point exponents. In this framework, numerical
results suggest that mean-field chaos exponents are probably exact in finite
dimensions. If we use the droplet approach, numerical results suggest that the
zero-temperature fixed point exponent is very close to
. In both approaches is the lower critical dimension in
agreement with recent numerical simulations.Comment: 28 pages + 6 figures, LateX, figures uuencoded at the end of fil
Mean-field theory of temperature cycling experiments in spin-glasses
We study analytically the effect of temperature cyclings in mean-field
spin-glasses. In accordance with real experiments, we obtain a strong
reinitialization of the dynamics on decreasing the temperature combined with
memory effects when the original high temperature is restored. The same
calculation applied to mean-field models of structural glasses shows no such
reinitialization, again in accordance with experiments. In this context, we
derive some relations between experimentally accessible quantities and propose
new experimental protocols. Finally, we briefly discuss the effect of field
cyclings during isothermal aging.Comment: Some misprints corrected, references updated, final version to apper
in PR
- …