28,731 research outputs found
Winning quick and dirty: the greedy random walk
As a strategy to complete games quickly, we investigate one-dimensional
random walks where the step length increases deterministically upon each return
to the origin. When the step length after the kth return equals k, the
displacement of the walk x grows linearly in time. Asymptotically, the
probability distribution of displacements is a purely exponentially decaying
function of |x|/t. The probability E(t,L) for the walk to escape a bounded
domain of size L at time t decays algebraically in the long time limit, E(t,L)
~ L/t^2. Consequently, the mean escape time ~ L ln L, while ~
L^{2n-1} for n>1. Corresponding results are derived when the step length after
the kth return scales as k^alpha$ for alpha>0.Comment: 7 pages, 6 figures, 2-column revtext4 forma
Greedy adaptive walks on a correlated fitness landscape
We study adaptation of a haploid asexual population on a fitness landscape
defined over binary genotype sequences of length . We consider greedy
adaptive walks in which the population moves to the fittest among all single
mutant neighbors of the current genotype until a local fitness maximum is
reached. The landscape is of the rough mount Fuji type, which means that the
fitness value assigned to a sequence is the sum of a random and a deterministic
component. The random components are independent and identically distributed
random variables, and the deterministic component varies linearly with the
distance to a reference sequence. The deterministic fitness gradient is a
parameter that interpolates between the limits of an uncorrelated random
landscape () and an effectively additive landscape ().
When the random fitness component is chosen from the Gumbel distribution,
explicit expressions for the distribution of the number of steps taken by the
greedy walk are obtained, and it is shown that the walk length varies
non-monotonically with the strength of the fitness gradient when the starting
point is sufficiently close to the reference sequence. Asymptotic results for
general distributions of the random fitness component are obtained using
extreme value theory, and it is found that the walk length attains a
non-trivial limit for , different from its values for and
, if is scaled with in an appropriate combination.Comment: minor change
Absorbing random-walk centrality: Theory and algorithms
We study a new notion of graph centrality based on absorbing random walks.
Given a graph and a set of query nodes , we aim to
identify the most central nodes in with respect to . Specifically,
we consider central nodes to be absorbing for random walks that start at the
query nodes . The goal is to find the set of central nodes that
minimizes the expected length of a random walk until absorption. The proposed
measure, which we call absorbing random-walk centrality, favors diverse
sets, as it is beneficial to place the absorbing nodes in different parts
of the graph so as to "intercept" random walks that start from different query
nodes.
Although similar problem definitions have been considered in the literature,
e.g., in information-retrieval settings where the goal is to diversify
web-search results, in this paper we study the problem formally and prove some
of its properties. We show that the problem is NP-hard, while the objective
function is monotone and supermodular, implying that a greedy algorithm
provides solutions with an approximation guarantee. On the other hand, the
greedy algorithm involves expensive matrix operations that make it prohibitive
to employ on large datasets. To confront this challenge, we develop more
efficient algorithms based on spectral clustering and on personalized PageRank.Comment: 11 pages, 11 figures, short paper to appear at ICDM 201
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An average-case analysis of bin packing with uniformly distributed item sizes
We analyze the one-dimensional bin-packing problem under the assumption that bins have unit capacity, and that items to be packed are drawn from a uniform distribution on [0,1]. Building on some recent work by Frederickson, we give an algorithm which uses n/2+0(n^½) bins on the average to pack n items. (Knodel has achieved a similar result.) The analysis involves the use of a certain 1-dimensional random walk. We then show that even an optimum packing under this distribution uses n/2+0(n^1/2) bins on the average, so our algorithm is asymptotically optimal, up to constant factors on the amount of wasted space. Finally, following Frederickson, we show that two well-known greedy bin-packing algorithms use no more bins than our algorithm; thus their behavior is also in asymptotically optimal in this sense
Optimal control for diffusions on graphs
Starting from a unit mass on a vertex of a graph, we investigate the minimum
number of "\emph{controlled diffusion}" steps needed to transport a constant
mass outside of the ball of radius . In a step of a controlled diffusion
process we may select any vertex with positive mass and topple its mass equally
to its neighbors. Our initial motivation comes from the maximum overhang
question in one dimension, but the more general case arises from optimal mass
transport problems.
On we show that steps are necessary and
sufficient to transport the mass. We also give sharp bounds on the comb graph
and -ary trees. Furthermore, we consider graphs where simple random walk has
positive speed and entropy and which satisfy Shannon's theorem, and show that
the minimum number of controlled diffusion steps is , where is the Avez asymptotic entropy and is the speed
of random walk. As examples, we give precise results on Galton-Watson trees and
the product of trees .Comment: 32 pages, 2 figure
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