1,527 research outputs found
Online unit clustering in higher dimensions
We revisit the online Unit Clustering and Unit Covering problems in higher
dimensions: Given a set of points in a metric space, that arrive one by
one, Unit Clustering asks to partition the points into the minimum number of
clusters (subsets) of diameter at most one; while Unit Covering asks to cover
all points by the minimum number of balls of unit radius. In this paper, we
work in using the norm.
We show that the competitive ratio of any online algorithm (deterministic or
randomized) for Unit Clustering must depend on the dimension . We also give
a randomized online algorithm with competitive ratio for Unit
Clustering}of integer points (i.e., points in , , under norm). We show that the competitive ratio of
any deterministic online algorithm for Unit Covering is at least . This
ratio is the best possible, as it can be attained by a simple deterministic
algorithm that assigns points to a predefined set of unit cubes. We complement
these results with some additional lower bounds for related problems in higher
dimensions.Comment: 15 pages, 4 figures. A preliminary version appeared in the
Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA
2017
Stable Secretaries
We define and study a new variant of the secretary problem. Whereas in the
classic setting multiple secretaries compete for a single position, we study
the case where the secretaries arrive one at a time and are assigned, in an
on-line fashion, to one of multiple positions. Secretaries are ranked according
to talent, as in the original formulation, and in addition positions are ranked
according to attractiveness. To evaluate an online matching mechanism, we use
the notion of blocking pairs from stable matching theory: our goal is to
maximize the number of positions (or secretaries) that do not take part in a
blocking pair. This is compared with a stable matching in which no blocking
pair exists. We consider the case where secretaries arrive randomly, as well as
that of an adversarial arrival order, and provide corresponding upper and lower
bounds.Comment: Accepted for presentation at the 18th ACM conference on Economics and
Computation (EC 2017
A Simple Quantum Neural Net with a Periodic Activation Function
In this paper, we propose a simple neural net that requires only
number of qubits and quantum gates: Here, is the number of input
parameters, and is the number of weights applied to these parameters in the
proposed neural net. We describe the network in terms of a quantum circuit, and
then draw its equivalent classical neural net which involves nodes in
the hidden layer. Then, we show that the network uses a periodic activation
function of cosine values of the linear combinations of the inputs and weights.
The backpropagation is described through the gradient descent, and then iris
and breast cancer datasets are used for the simulations. The numerical results
indicate the network can be used in machine learning problems and it may
provide exponential speedup over the same structured classical neural net.Comment: a discussion session is added. 5 pages, conference paper. To appear
in The 2018 IEEE International Conference on Systems, Man, and Cybernetics
(SMC2018
On the Continuous CNN Problem
In the (discrete) CNN problem, online requests appear as points in
. Each request must be served before the next one is revealed. We
have a server that can serve a request simply by aligning either its or
coordinate with the request. The goal of the online algorithm is to minimize
the total distance traveled by the server to serve all the requests. The
best known competitive ratio for the discrete version is 879 (due to Sitters
and Stougie).
We study the continuous version, in which, the request can move continuously
in and the server must continuously serve the request. A simple
adversarial argument shows that the lower bound on the competitive ratio of any
online algorithm for the continuous CNN problem is 3. Our main contribution is
an online algorithm with competitive ratio . Our
analysis is tight. The continuous version generalizes the discrete orthogonal
CNN problem, in which every request must be or aligned with the
previous request. Therefore, Our result improves upon the previous best
competitive ratio of 9 (due to Iwama and Yonezawa)
Online Exploration of Polygons with Holes
We study online strategies for autonomous mobile robots with vision to
explore unknown polygons with at most h holes. Our main contribution is an
(h+c_0)!-competitive strategy for such polygons under the assumption that each
hole is marked with a special color, where c_0 is a universal constant. The
strategy is based on a new hybrid approach. Furthermore, we give a new lower
bound construction for small h.Comment: 16 pages, 9 figures, submitted to WAOA 201
- …