51,805 research outputs found
IST Austria Technical Report
We consider the problem of developing automated techniques to aid the average-case complexity analysis of programs. Several classical textbook algorithms have quite efficient average-case complexity, whereas the corresponding worst-case bounds are either inefficient (e.g., QUICK-SORT), or completely ineffective (e.g., COUPONCOLLECTOR). Since the main focus of average-case analysis is to obtain efficient bounds, we consider bounds that are either logarithmic,
linear, or almost-linear (O(log n), O(n), O(n · log n),
respectively, where n represents the size of the input). Our main contribution is a sound approach for deriving such average-case bounds for randomized recursive programs. Our approach is efficient (a simple linear-time algorithm), and it is based on (a) the analysis of recurrence relations induced by randomized algorithms, and (b) a guess-and-check technique. Our approach can infer the asymptotically optimal average-case bounds for classical randomized algorithms, including RANDOMIZED-SEARCH, QUICKSORT, QUICK-SELECT, COUPON-COLLECTOR, where the worstcase
bounds are either inefficient (such as linear as compared to logarithmic of average-case, or quadratic as compared to linear or almost-linear of average-case), or ineffective. We have implemented our approach, and the experimental results show that we obtain the bounds efficiently for various classical algorithms
On dynamic breadth-first search in external-memory
We provide the first non-trivial result on dynamic breadth-first search (BFS) in external-memory: For general sparse undirected graphs of initially nodes and O(n) edges and monotone update sequences of either edge insertions or edge deletions, we prove an amortized high-probability bound of O(n/B^{2/3}+\sort(n)\cdot \log B) I/Os per update. In contrast, the currently best approach for static BFS on sparse undirected graphs requires \Omega(n/B^{1/2}+\sort(n)) I/Os. 1998 ACM Subject Classification: F.2.2. Key words and phrases: External Memory, Dynamic Graph Algorithms, BFS, Randomization
Qualitative Analysis of Universes with Varying Alpha
Assuming a Friedmann universe which evolves with a power-law scale factor,
, we analyse the phase space of the system of equations that describes
a time-varying fine structure 'constant', , in the
Bekenstein-Sandvik-Barrow-Magueijo generalisation of general relativity. We
have classified all the possible behaviours of in ever-expanding
universes with different and find new exact solutions for . We
find the attractors points in the phase space for all . In general, will be a non-decreasing function of time that increases logarithmically in
time during a period when the expansion is dust dominated (), but
becomes constant when . This includes the case of negative-curvature
domination (). also tends rapidly to a constant when the
expansion scale factor increases exponentially. A general set of conditions is
established for to become asymptotically constant at late times in an
expanding universe.Comment: 26 pages, 6 figure
A periodic elastic medium in which periodicity is relevant
We analyze, in both (1+1)- and (2+1)- dimensions, a periodic elastic medium
in which the periodicity is such that at long distances the behavior is always
in the random-substrate universality class. This contrasts with the models with
an additive periodic potential in which, according to the field theoretic
analysis of Bouchaud and Georges and more recently of Emig and Nattermann, the
random manifold class dominates at long distances in (1+1)- and
(2+1)-dimensions. The models we use are random-bond Ising interfaces in
hypercubic lattices. The exchange constants are random in a slab of size
and these coupling constants are periodically repeated
along either {10} or {11} (in (1+1)-dimensions) and {100} or {111} (in
(2+1)-dimensions). Exact ground-state calculations confirm scaling arguments
which predict that the surface roughness behaves as: and , with in
-dimensions and; and , with in -dimensions.Comment: Submitted to Phys. Rev.
On the Mixing of Diffusing Particles
We study how the order of N independent random walks in one dimension evolves
with time. Our focus is statistical properties of the inversion number m,
defined as the number of pairs that are out of sort with respect to the initial
configuration. In the steady-state, the distribution of the inversion number is
Gaussian with the average ~N^2/4 and the standard deviation sigma N^{3/2}/6.
The survival probability, S_m(t), which measures the likelihood that the
inversion number remains below m until time t, decays algebraically in the
long-time limit, S_m t^{-beta_m}. Interestingly, there is a spectrum of
N(N-1)/2 distinct exponents beta_m(N). We also find that the kinetics of
first-passage in a circular cone provides a good approximation for these
exponents. When N is large, the first-passage exponents are a universal
function of a single scaling variable, beta_m(N)--> beta(z) with
z=(m-)/sigma. In the cone approximation, the scaling function is a root of a
transcendental equation involving the parabolic cylinder equation, D_{2
beta}(-z)=0, and surprisingly, numerical simulations show this prediction to be
exact.Comment: 9 pages, 6 figures, 2 table
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