199,079 research outputs found
Sharp Bounds on Davenport-Schinzel Sequences of Every Order
One of the longest-standing open problems in computational geometry is to
bound the lower envelope of univariate functions, each pair of which
crosses at most times, for some fixed . This problem is known to be
equivalent to bounding the length of an order- Davenport-Schinzel sequence,
namely a sequence over an -letter alphabet that avoids alternating
subsequences of the form with length
. These sequences were introduced by Davenport and Schinzel in 1965 to
model a certain problem in differential equations and have since been applied
to bounding the running times of geometric algorithms, data structures, and the
combinatorial complexity of geometric arrangements.
Let be the maximum length of an order- DS sequence over
letters. What is asymptotically? This question has been answered
satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and
Nivasch) when is even or . However, since the work of Agarwal,
Sharir, and Shor in the mid-1980s there has been a persistent gap in our
understanding of the odd orders.
In this work we effectively close the problem by establishing sharp bounds on
Davenport-Schinzel sequences of every order . Our results reveal that,
contrary to one's intuition, behaves essentially like
when is odd. This refutes conjectures due to Alon et al.
(2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the
Symposium on Computational Geometry, 201
Operational State Complexity of Deterministic Unranked Tree Automata
We consider the state complexity of basic operations on tree languages
recognized by deterministic unranked tree automata. For the operations of union
and intersection the upper and lower bounds of both weakly and strongly
deterministic tree automata are obtained. For tree concatenation we establish a
tight upper bound that is of a different order than the known state complexity
of concatenation of regular string languages. We show that (n+1) (
(m+1)2^n-2^(n-1) )-1 vertical states are sufficient, and necessary in the worst
case, to recognize the concatenation of tree languages recognized by (strongly
or weakly) deterministic automata with, respectively, m and n vertical states.Comment: In Proceedings DCFS 2010, arXiv:1008.127
Optimization of Information Rate Upper and Lower Bounds for Channels with Memory
We consider the problem of minimizing upper bounds and maximizing lower
bounds on information rates of stationary and ergodic discrete-time channels
with memory. The channels we consider can have a finite number of states, such
as partial response channels, or they can have an infinite state-space, such as
time-varying fading channels. We optimize recently-proposed information rate
bounds for such channels, which make use of auxiliary finite-state machine
channels (FSMCs). Our main contribution in this paper is to provide iterative
expectation-maximization (EM) type algorithms to optimize the parameters of the
auxiliary FSMC to tighten these bounds. We provide an explicit, iterative
algorithm that improves the upper bound at each iteration. We also provide an
effective method for iteratively optimizing the lower bound. To demonstrate the
effectiveness of our algorithms, we provide several examples of partial
response and fading channels, where the proposed optimization techniques
significantly tighten the initial upper and lower bounds. Finally, we compare
our results with an improved variation of the \emph{simplex} local optimization
algorithm, called \emph{Soblex}. This comparison shows that our proposed
algorithms are superior to the Soblex method, both in terms of robustness in
finding the tightest bounds and in computational efficiency. Interestingly,
from a channel coding/decoding perspective, optimizing the lower bound is
related to increasing the achievable mismatched information rate, i.e., the
information rate of a communication system where the decoder at the receiver is
matched to the auxiliary channel, and not to the original channel.Comment: Submitted to IEEE Transactions on Information Theory, November 24,
200
Compressive Mining: Fast and Optimal Data Mining in the Compressed Domain
Real-world data typically contain repeated and periodic patterns. This
suggests that they can be effectively represented and compressed using only a
few coefficients of an appropriate basis (e.g., Fourier, Wavelets, etc.).
However, distance estimation when the data are represented using different sets
of coefficients is still a largely unexplored area. This work studies the
optimization problems related to obtaining the \emph{tightest} lower/upper
bound on Euclidean distances when each data object is potentially compressed
using a different set of orthonormal coefficients. Our technique leads to
tighter distance estimates, which translates into more accurate search,
learning and mining operations \textit{directly} in the compressed domain.
We formulate the problem of estimating lower/upper distance bounds as an
optimization problem. We establish the properties of optimal solutions, and
leverage the theoretical analysis to develop a fast algorithm to obtain an
\emph{exact} solution to the problem. The suggested solution provides the
tightest estimation of the -norm or the correlation. We show that typical
data-analysis operations, such as k-NN search or k-Means clustering, can
operate more accurately using the proposed compression and distance
reconstruction technique. We compare it with many other prevalent compression
and reconstruction techniques, including random projections and PCA-based
techniques. We highlight a surprising result, namely that when the data are
highly sparse in some basis, our technique may even outperform PCA-based
compression.
The contributions of this work are generic as our methodology is applicable
to any sequential or high-dimensional data as well as to any orthogonal data
transformation used for the underlying data compression scheme.Comment: 25 pages, 20 figures, accepted in VLD
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