scholarly journals Algorithms for Necklace Maps

2015 ◽  
Vol 25 (01) ◽  
pp. 15-36 ◽  
Author(s):  
Bettina Speckmann ◽  
Kevin Verbeek
Keyword(s):  
Exact Algorithm ◽  
Dimensional Problem ◽  
Order Problem ◽  
Fixed Parameter Tractable ◽  
One Dimensional ◽  
Algorithmic Question ◽  
Fixed Parameter ◽  
Fixed Order ◽  
The One ◽  
Decision Version

Necklace maps visualize quantitative data associated with regions by placing scaled symbols, usually disks, without overlap on a closed curve (the necklace) surrounding the map regions. Each region is projected onto an interval on the necklace that contains its symbol. In this paper we address the algorithmic question how to maximize symbol sizes while keeping symbols disjoint and inside their intervals. For that we reduce the problem to a one-dimensional problem which we solve efficiently. Solutions to the one-dimensional problem provide a very good approximation for the original necklace map problem. We consider two variants: Fixed-Order, where an order for the symbols on the necklace is given, and Any-Order where any symbol order is possible. The Fixed-Order problem can be solved in O(n log n) time. We show that the Any-Order problem is NP-hard for certain types of intervals and give an exact algorithm for the decision version. This algorithm is fixed-parameter tractable in the thickness K of the input. Our algorithm runs in O(n log n + n2K4K) time which can be improved to O(n log n + nK2K) time using a heuristic. We implemented our algorithm and evaluated it experimentally.

Oscillating flames: multiple-scale analysis

2009 ◽  
Vol 465 (2107) ◽  
pp. 2089-2110 ◽  
Author(s):  
Luc Bauwens ◽  
C. Regis L. Bauwens ◽  
Ida Wierzba
Keyword(s):  
Approximate Solution ◽  
Reaction Front ◽  
Burning Velocity ◽  
Propagation Speed ◽  
Multiple Scale ◽  
Expansion Ratio ◽  
Scale Analysis ◽  
Dimensional Problem ◽  
One Dimensional ◽  
The One

A complete multiple-scale solution is constructed for the one-dimensional problem of an oscillating flame in a tube, ignited at a closed end, with the second end open. The flame front moves into the unburnt mixture at a constant burning velocity relative to the mixture ahead, and the heat release is constant. The solution is based upon the assumption that the propagation speed multiplied by the expansion ratio is small compared with the speed of sound. This approximate solution is compared with a numerical solution for the same physical model, assuming a propagation speed of arbitrary magnitude, and the results are close enough to confirm the validity of the approximate solution. Because ignition takes place at the closed end, the effect of thermal expansion is to push the column of fluid in the tube towards the open end. Acoustics set in motion by the impulsive start of the column of fluid play a crucial role in the oscillation. The analytical solution also captures the subsequent interaction between acoustics and the reaction front, the effect of which does not appear to be as significant as that of the impulsive start, however.

The one-dimensional problem of the nonpiston type displacement of a gas by water, taking account of capillary forces

1972 ◽  
Vol 7 (1) ◽  
pp. 62-66
Author(s):  
G. A. Osipova ◽  
G. V. Rassokhin ◽  
G. P. Tsybul'skii
Keyword(s):  
Capillary Forces ◽  
Dimensional Problem ◽  
One Dimensional ◽  
The One

Diffraction of Electrons by Two and Three Mutually Orthogonal Standing Light Waves

1975 ◽  
Vol 53 (2) ◽  
pp. 157-164 ◽  
Author(s):  
F. Ehlotzky
Keyword(s):  
Electron Scattering ◽  
Light Wave ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Standing Light Wave ◽  
Light Waves ◽  
The One ◽  
Rectangular Lattices

The one-dimensional problem of electron scattering by a standing light wave, known as the Kapitza–Dirac effect, is shown to be easily extendable to two and three dimensions, thus showing all characteristics of diffraction of electrons by simple two- and three-dimensional rectangular lattices.

scholarly journals Maximum Exact Satisfiability: NP-completeness Proofs and Exact Algorithms

2004 ◽  
Vol 11 (19) ◽  
Author(s):  
Bolette Ammitzbøll Madsen ◽  
Peter Rossmanith
Keyword(s):  
Time Complexity ◽  
Exact Algorithm ◽  
Conjunctive Normal Form ◽  
Exact Algorithms ◽  
Fixed Parameter Tractable ◽  
Complete Problem ◽  
Satisfiability Problems ◽  
Fixed Parameter ◽  
Np Complete ◽  
Open Question

Inspired by the Maximum Satisfiability and Exact Satisfiability problems we present two Maximum Exact Satisfiability problems. The first problem called Maximum Exact Satisfiability is: given a formula in conjunctive normal form and an integer k, is there an assignment to all variables in the formula such that at least k clauses have exactly one true literal. The second problem called Restricted Maximum Exact Satisfiability has the further restriction that no clause is allowed to have more than one true literal. Both problems are proved NP-complete restricted to the versions where each clause contains at most two literals. In fact Maximum Exact Satisfiability is a generalisation of the well-known NP-complete problem MaxCut. We present an exact algorithm for Maximum Exact Satisfiability where each clause contains at most two literals with time complexity O(poly(L) . 2^{m/4}), where m is the number of clauses and L is the length of the formula. For the second version we give an algorithm with time complexity O(poly(L) . 1.324718^n) , where n is the number of variables. We note that when restricted to the versions where each clause contains exactly two literals and there are no negations both problems are fixed parameter tractable. It is an open question if this is also the case for the general problems.

A Comparison of Diffusion Flame Stability in One and Two Spatial Dimensions Near Cold, Inert Surfaces

2001 ◽  
Author(s):  
Robert Vance ◽  
Indrek S. Wichman
Keyword(s):  
Two Dimensions ◽  
Low Flow ◽  
Dimensional System ◽  
Flame Stability ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Cellular Flames ◽  
The One ◽  
Inert Surfaces

Abstract A linear stability analysis is performed on two simplified models representing a one-dimensional flame between oxidizer and fuel reservoirs and a two-dimensional “edge-flame” between the same reservoirs but above a cold, inert wall. Comparison of the eigenvalue spectra for both models is performed to discern the validity of extending the results from the one-dimensional problem to the two-dimensional problem. Of primary interest is the influence on flame stability of thermal-diffusive imbalances, i.e. non-unity Lewis numbers. Flame oscillations are observed when Le > 1, and cellular flames are witnessed when Le < 1. It is found that when Le > 1 the characteristics of flame behavior are consistent between the two models. Furthermore, when Le < 1, the models are found to be in good agreement with respect to the magnitude of the critical wave numbers. Results from the coarse mesh analysis of the two-dimensional system are presented and compared to the one-dimensional eigenvalue spectra. Additionally, an examination of low reactant convection is undertaken. It is concluded that for low flow rates the behavior in one and two dimensions are similar qualitatively and quantitatively.

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