Showing papers by "Mihai Patrascu published in 2012"

A New Infinity of Distance Oracles for Sparse Graphs

Abstract: Given a weighted undirected graph, our basic goal is to represent all pair wise distances using much less than quadratic space, such that we can estimate the distance between query vertices in constant time. We will study the inherent trade-off between space of the representation and the stretch (multiplicative approximation disallowing underestimates) of the estimates when the input graph is sparse with $m=\wt O(n)$ edges. In this paper, for any fixed positive integers $k$ and $\ell$, we obtain stretches $\alpha=2k+1\pm\frac{2}{\ell}= 2k+1-\frac{2}{\ell}, 2k+1+\frac{2}{\ell}$, using space $S(\alpha, m) = \wt O(m^{1+2/(\alpha+1)})$. The query time is $O(k+\ell)=O(1)$. For integer stretches, this coincides with the previous bounds (odd stretches with $\ell=1$ and even stretches with $\ell=2$). The infinity of fractional stretches between consecutive integers are all new (even though $\ell$ is fixed as a constant independent of the input, the number of integers $\ell$ is still countably infinite). % We will argue that the new fractional points are not just arbitrary, but that they, at least for fixed stretches below $3$, provide a complete picture of the inherent trade-off between stretch and space in $m$. Consider any fixed stretch $\alpha

Necklaces, Convolutions, and X+Y

Abstract: We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p even, and p = \infty. For p even, we reduce the problem to standard convolution, while for p = \infty and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n^2) time, whereas the obvious algorithms for these problems run in \Theta(n^2) time.

Using hashing to solve the dictionary problem

Abstract: We consider the dictionary problem in external memory and improve the update time of the well-known buffer tree by roughly a logarithmic factor. For any λ ≥ max{lg lg n, logM/B (n/B)}, we can support updates in time O(λ/B) and queries in sublogarithmic time, O(logλn). We also present a lower bound in the cell-probe model showing that our data structure is optimal.In the RAM, hash tables have been use to solve the dictionary problem faster than binary search for more than half a century. By contrast, our data structure is the first to beat the comparison barrier in external memory. Ours is also the first data structure to depart convincingly from the indivisibility paradigm.

Flexible, electrostatic microfluidic actuators based on thin film fabrication

Abstract: Active microfluidic components are required in many lab-on-a-chip applications, for handling tiny amounts of liquids on the platform. The work presented here reveals an electrostatic microfluidic actuators that can be fabricated with thin film process technology, and is based on flexible, biocompatible materials. We have successfully characterized microvalves for both the static and dynamic mode, using non-polar fluids. An extension of the proposed fabrication method enables actuation of fluids of any polarity, as the electrostatic field no longer crosses the transport fluid

Picture-Hanging Puzzles

Abstract: We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.