A randomized algorithm for Voronoi diagram of line segments on coarse grained multiprocessors

We study the space complexity of refuting unsatisfiable random k-CNFs in the resolution proof system. We prove that for any large enough /spl Delta/, with high probability a random k-CNF over n variables and /spl Delta/n clauses requires resolution clause space of /spl Omega/(n/spl middot//spl Delta//sup -1+/spl epsiv//k-2-/spl epsiv//), for any 0 >/spl radic/n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with /spl Delta/n clauses requires tree-like refutation size of exp(/spl Omega/(n//spl Delta//sup 1+/spl epsiv//1-/spl epsiv//)), for any 0</spl epsiv/<1/2. Our space lower bound is the consequence of three main contributions. 1. We introduce a 2-player matching game on bipartite graphs G to prove that there are no perfect matchings in G. 2. We reduce lower bounds for the clause space of a formula F in resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. 3. We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the pigeonhole principle based on bipartite graphs G. We prove that the clause space for G-PHP can be reduced to the game complexity on G.

Given a set of non-intersecting (except at endpoints) line segments in the plane we want to compute their Voronoi diagram. Although there are several algorithms for this problem in the literature [Yap87, For87, CS89, BDS+92, KMM90] nobody claims to have a correct implementation. This is due to the fact that the algorithms presuppose exact arithmetic and that the Voronoi diagram of segments requires to compute with non-rational algebraic numbers. We report about a detailed study of the numerical precision required for evaluating the geometric test exactly and about first experimental experiences. More specifically, we improve the precision bound implied by classical root separation results by more than two orders of magnitude and we compare the implementation strategies suggested by our theoretical results.KeywordsVoronoi DiagramComputational GeometryAlgebraic NumberPoint SiteFloating Point ArithmeticThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Optimal Mesh Algorithms for the Voronoi Diagram of Line Segments and Motion Planning in the Plane

A finite-difference time-domain (FDTD) analysis is presented to simulate, up to the high-frequency regime, transients on multiconductor transmission lines above dissipative earth. Excitation is in the high-frequency regime if the pulse has significant spectral content up to /spl omega//spl ap//spl omega//sub c/, with /spl omega//sub c/ the frequency where the conduction and displacement current densities have equal magnitude. Thus, /spl omega//sub c/=/spl sigma//sub g///spl epsi//sub 0//spl epsi//sub rg/ with /spl sigma//sub g/ and /spl epsi//sub rg/ the ground conductance and relative permittivity. The FDTD algorithm extends the formulation of Agrawal et al. (1980), by incorporating the theory of D'Amore and Sarto for the interaction of wide-band transients with the earth. Careful experimental testing of the new algorithm is presented. The measurement configuration consists of two parallel wires; first suspended above an aluminum sheet to test the code in the low-frequency regime; then above a block of absorbing material for experiments in the high-frequency regime. The end of one of the conductors is driven with a 2-ns synthesized impulse by a stepped-frequency, automatic network analyzer. Reflected and cross-talk signals are presented in the time domain. These show good agreement with the numerically predicted common mode, differential mode and mode-converted pulses.

We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n+log log(1//spl epsiv/)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2/sup n/ with error bounded by /spl epsiv/. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n//spl epsiv/)). As an application of this depth bound, we show that P. Shor's (1997) factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomial size, in combination with classical polynomial-time pre- and postprocessing. Next, we prove an /spl Omega/(log n) lower bound on the depth complexity of approximations of the QFT with constant error. This implies that the above upper bound is asymptotically tight (for a reasonable range of values of /spl epsiv/). We also give an upper bound of O(n(log n)/sup 2/ log log n) on the circuit size of the exact QFT modulo 2/sup n/, for which the best previous bound was O(n/sup 2/). Finally, based on our circuits for the QFT with power-of-2 moduli, we show that the QFT with respect to an arbitrary modulus m can be approximated with accuracy /spl epsiv/ with circuits of depth O((log log m)(log log 1//spl epsiv/)) and size polynomial in log m+log(1//spl epsiv/).

Given a collection of segments in the plane that intersect pairwise at most k times, regarding the segments as opaque barriers, their upper envelope consists of the portions of the segments visible from point (0,+/spl infin/). We give efficient parallel methods for finding the upper envelope of k-intersecting segments for any integer k/spl ges/0, in the weakest shared memory model, the EREW PRAM. We show that the upper envelope of n k-intersecting segments can be found in 0(log/sup 1+/spl epsiv//n) time using 0(/spl lambda//sub k+1/(n)/log/sup /spl epsiv//n) processors for any /spl epsiv/>0, where /spl lambda//sub k+2/(n)/sup 1/ is the size of the upper envelope. In particular, for line segments we show the following optimal algorithms: the upper envelope of n line segments can be found in O(log n) time using O(n) processors, and if the line segments are nonintersecting and sorted, the envelope can be found in O(log n) time using O(n/log n) processors. We also show that our methods imply a fast sequential result: the upper envelope of n sorted line segments can be found in O(n log log n) time sequentially, which improves the known lowest upper bound O(n log n).

Summary form only given. We present the reflectance spectra of conducting polyaniline film protonated with sulfuric acid (PANI-H/sub 2/SO/sub 4/) and of newly processed film protonated with camphor sulfonic acid (PANI-CSA) over a wide spectral range (0.006 eV-6 eV) at room temperature. The reflectance of PANI-H/sub 2/SO/sub 4/reveals the signatures of the intraband excitations in the infrared (IR), including increasing reflectance approaching to far-IR and a reflectance minimum around 1.7 eV. However, the corresponding optical conductivity (/spl sigma/(/spl omega/)) and the real part of dielectric function (/spl epsiv//sub 1/(/spl omega/)) are far from those of a metal. Instead, the optical spectra indicate that the charge carriers are localized with considerable oscillator strength within the gap. On the other hand, the optical spectra of PANI- CSA are characterized by distinct metal-like signatures arising from the combination of the Drude-like free carrier excitations in the far-IR and the existence of plasma resonance around 1.2 eV. These results indicate that the new processing of polyaniline with functionalized sulfonic acid (with surfactants as counterions) has greatly improved the homogeneity of the material and has resulted in a significant decrease in the extent of molecular scale disorder, thereby leading to the delocalization of the charge carriers in PANI-CSA. However, the deviation of /spl sigma/(/spl omega/) and /spl epsiv//sub 1/(/spl omega/) from the normal Drude metallic behaviors below 0.2 eV indicates that disorder induced localization still plays an important role in PANI-CSA. We present a quantitative analysis of /spl sigma/(/spl omega/) and /spl spl epsiv//sub 1/(/spl omega/) of PANI-CSA in terms of the localization modified Drude model, implying that PANI-CSA is a disordered metal on the metal-insulator boundary.

Summary form only given. The single-particle theory of electron multipactor is now well understood. The multipactor parameter space is spanned by the normalized magnetic and electric field parameters. In particular, for given normalized magnetic field /spl Omega/ multipactor can errupt within a certain range of electric field amplitudes /spl epsi//sub 1/</spl epsi/</spl epsi//sub 2/; to each value /spl epsi/ corresponds a different locked-phase /spl phi/. Once a multipactor has been launched, it is very important to know the saturation level; the latter determines the cavity power losses when the multipactor is a parasitic side effect, or the saturated current when a multipactor is deliberately triggered for the production of short period high current pulsed beams. The collective space charge effects must be included to address multipactor saturation. One proposed approach employs a single charge sheet and yields saturation when the secondary emission coefficient /spl delta/ equals one. Setting /spl delta/=1 fixes the impact parameters and therefore the value of the driving electric field /spl epsi/; the external RF field adjusts to that value through the nonlinear effect of the induced multipactor current on the cavity LC circuit. A limitation of the method is that one must start near the appropriate /spl epsi/ since the range of the voltage self-adjustment is usually small. A most general approach introduced here is based on multilayered space charge cloud involving electrons emitted at slightly different times. There is a continuous distribution of impact energies and interaction phases between the leading and the trailing edge of the bunch.

A dynamic algorithm can maintain the solution of a given problem under insertions and deletions of input objects. In this paper we propose a general scheme to obtain dynamic algorithms which is based on the abstract setting introduced by Clarkson and Shor. This scheme uses a novel data structure that combines the conflict graph and the history structure used by incremental algorithms. The randomized analysis of the dynamic algorithms assumes a probabilistic model of the update sequence, in which each currently present input object is equally likely to have been added by the previous insertion or to be deleted by the next deletion. We apply our general technique to obtain new and efficient algorithms for dynamically maintaining arrangements of line segments, lower envelopes of triangles, convex hulls and Voronoi diagrams of points in any dimension, and Voronoi diagrams of line segments in a plane.KeywordsConvex HullVoronoi DiagramPriority QueueDynamic AlgorithmIncremental AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present an O((log log N)/sup 2/) -time algorithm for computing the distance transform of an N /spl times/ N binary image. Our algorithm is designed for the common concurrent read concurrent write parallel random access machine (CRCW PRAM) and requires O(N/sup 2+/spl epsi///log log N) processors, for any /spl epsi/ such that 0 < /spl epsi/ < 1. Our algorithm is based on a novel deterministic sampling scheme and can be used for computing distance transforms for a very general class of distance functions. We also present a scalable version of our algorithm when the number of processors is available p/sup 2+/spl epsi///log log p for some p < N. In this case, our algorithm runs in O((N/sup 2//p/sup 2/)+(N/p) log log p + (log log p)/sup 2/) time. This scalable algorithm is more practical since usually the number of available processors is much less than the size of the image.

All algorithms below are optimal alphabet-independent parallel CRCW PRAM algorithms. In one dimension: Given a pattern string of length m for the string-matching problem, we design an algorithm that computes a deterministic sample of a sufficiently long substring in constant time. This problem used to be a bottleneck in the pattern preprocessing for one- and two-dimensional pattern matching. The best previous time bound was O(log/sup 2/ m/log log m). We use this algorithm to obtain the following results. 1. Improving the preprocessing of the constant-time text search algorithm from O(log/sup 2/ m/log log m) to n(log log m), which is now best possible. 2. A constant-time deterministic string-matching algorithm in the case that the text length n satisfies n=/spl Omega/(m/sup 1+/spl epsiv//) for a constant /spl epsiv/>0. 3. A simple probabilistic string-matching algorithm that has constant time with high probability for random input. 4. A constant expected time Las-Vegas algorithm for computing the period of the pattern and all witnesses and thus string matching itself, solving the main open problem remaining in string matching. >

We introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete-time random process. As a result, we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability. By carefully designing these graphs we can construct for any given rate R and any given real number /spl epsiv/ a family of linear codes of rate R which can be encoded in time proportional to ln(1//spl epsiv/) times their block length n. Furthermore, a codeword can be recovered with high probability from a portion of its entries of length (1+/spl epsiv/)Rn or more. The recovery algorithm also runs in time proportional to n ln(1//spl epsiv/). Our algorithms have been implemented and work well in practice; various implementation issues are discussed.

Differential displacement due to a gravitational force gradient, acting on two superconducting test masses which are elastically suspended from the instrument frame, is sensed by a superconducting quantum interference device (SQUID), coupled to the masses by a current flowing in the sensing circuits. The SQUID transducer absorbs energy from the masses, and, hence, it reacts with a random force (fluctuation-dissipation principle). This force increases if the coupling current grows. Additionally, the transducer outputs an amplification noise that is independent of the coupling. This output noise is equivalent to a force, applied to the masses, which decreases with stronger coupling, Hence, the strength of the total equivalent gravity gradient noise has a minimum with respect to the coupling. The minimum is found to be about 1.5/spl epsiv//sub A//spl omega//sub 0//sup 2//(ml/sup 2/), where /spl epsiv//sub A/ means the equivalent energy sensitivity, in the white noise limit, of the SQUID junctions; /spl omega//sub 0/ is the natural frequency of the mass suspension; m is the test mass, and l is the base length,.

Robust multivariable controllers for stable infinite-dimensional systems in the Callier-Desoer (CD) algebra are discussed. In particular, the following robust regulation problem is solved. Given reference and disturbance signals, which are linear combinations of signals of the form t/sup j/ sin(/spl omega//sub k/t+/spl phi//sub k/), j/spl ges/0, k=0, /spl middot//spl middot//spl middot/, n, find a low-order finite-dimensional controller so that the outputs asymptotically track the reference signals, asymptotically reject the disturbance signals, and the closed-loop system is stable and robust with respect to a class of perturbations in the plant. The proposed controller consists of a positive scalar gain /spl epsi/ and certain polynomial matrices K/sub k/(s) for k=0, /spl middot//spl middot//spl middot/, n. The main result of the paper shows that the matrices K/sub k/(s) must satisfy certain stability conditions involving the values of the plant transfer function only at the reference and disturbance signal frequencies /spl omega//sub k/ for k=0, /spl middot//spl middot//spl middot/, n. The controller has unstable poles on the imaginary axis. The behavior of these poles as a function of the scalar parameter /spl epsi/ in the form of a Puiseux series is given.

We consider the problem of testing bipartiteness in the adjacency matrix model. The best known algorithm, due to Alon and Krivelevich, distinguishes between bipartite graphs and graphs that are /spl epsi/-far from bipartite using 0(1//spl epsi//sup 2/) queries. We show that this is optimal for non-adaptive algorithms, up to polylogarithmic factors. We also show a lower bound of /spl Omega/(1//spl epsi//sup 3/2/) for adaptive algorithms.