Approximating Dynamic Time Warping and Edit Distance for a Pair of Point Sequences

We give the first subquadratic-time approximation schemes for dynamic time warping (DTW) and edit distance (ED) of several natural families of point sequences in $\mathbb{R}^d$, for any fixed $d \ge 1$. In particular, our algorithms compute $(1+\varepsilon)$-approximations of DTW and ED in time near-linear for point sequences drawn from k-packed or k-bounded curves, and subquadratic for backbone sequences. Roughly speaking, a curve is $\kappa$-packed if the length of its intersection with any ball of radius $r$ is at most $\kappa \cdot r$, and a curve is $\kappa$-bounded if the sub-curve between two curve points does not go too far from the two points compared to the distance between the two points. In backbone sequences, consecutive points are spaced at approximately equal distances apart, and no two points lie very close together. Recent results suggest that a subquadratic algorithm for DTW or ED is unlikely for an arbitrary pair of point sequences even for $d=1$. Our algorithms work by constructing a small set of rectangular regions that cover the entries of the dynamic programming table commonly used for these distance measures. The weights of entries inside each rectangle are roughly the same, so we are able to use efficient procedures to approximately compute the cheapest paths through these rectangles

Quantum compiling with diffusive sets of gates

Given a set of quantum gates and a target unitary operation, the most elementary task of quantum compiling is the identification of a sequence of the gates that approximates the target unitary to a determined precision $\varepsilon$. Solovay-Kitaev theorem provides an elegant solution which is based on the construction of successively tighter `nets' around the unity comprised by successively longer sequences of gates. The procedure for the construction of the nets, according to this theorem, requires accessibility to the inverse of the gates as well. In this work, we propose a method for constructing nets around unity without this requirement. The algorithmic procedure is applicable to sets of gates which are diffusive enough, in the sense that sequences of moderate length cover the space of unitary matrices in a uniform way. We prove that the number of gates sufficient for reaching a precision $\varepsilon$ scales as $\log (1/\varepsilon )^{\log 3 / log 2}$ while the pre-compilation time is increased as compared to thatof the Solovay-Kitaev algorithm by the exponential factor 3/2.Comment: 6 pages, several corrections in text, figures & bibliograph

CDMA signature sequences with low peak-to-average-power ratio via alternating projection

Several algorithms have been proposed to construct optimal signature sequences that maximize the sum capacity of the uplink in a direct-spread synchronous code division multiple access (CDMA) system. These algorithms produce signatures with real-valued or complex-valued entries that generally have a large peak-to-average power ratio (PAR). This paper presents an alternating projection algorithm that can design optimal signature sequences that satisfy PAR side constraints. This algorithm converges to a fixed point, and these fixed points are partially characterized

Comparing different sampling schemes for approximating the integrals involved in the semi-Bayesian optimal design of choice experiments.

In conjoint choice experiments, the semi-Bayesian D-optimality criterion is often used to compute efficient designs. The traditional way to compute this criterion which involves multi-dimensional integrals over the prior distribution is to use Pseudo-Monte Carlo samples. However, other sampling approaches are available. Examples are the Quasi-Monte Carlo approach (randomized Halton sequences, modified Latin hypercube sampling and extensible shifted lattice points with Baker's transformation), the Gaussian-Hermite quadrature approach and a method using spherical-radial transformations. Not much is known in general about which sampling scheme performs best in constructing efficient choice designs. In this study, we compare the performance of these approaches under various scenarios. We try to identify the most efficient sampling scheme for each situation.Conjoint choice design; Pseudo-Monte Carlo; Quasi-Monte Carlo; Gaussian-Hermite quadrature; Spherical-radial transformation;

Universal Cycles for Minimum Coverings of Pairs by Triples, with Application to 2-Radius Sequences

A new ordering, extending the notion of universal cycles of Chung {\em et al.} (1992), is proposed for the blocks of $k$-uniform set systems. Existence of minimum coverings of pairs by triples that possess such an ordering is established for all orders. Application to the construction of short 2-radius sequences is given, with some new 2-radius sequences found through computer search.Comment: 18 pages, to appear in Mathematics of Computatio