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1.
Graph drawing research has been mostly oriented toward two-dimensional drawings. This paper describes an investigation of
fundamental aspects of three-dimensional graph drawing. In particular we give three results concerning the space required
for three-dimensional drawings.
We show how to produce a grid drawing of an arbitraryn-vertex graph with all vertices located at integer grid points, in ann×2n×2n grid, such that no pair of edges cross. This grid size is optimal to within a constant. We also show how to convert an orthogonal
two-dimensional drawing in anH×V integer grid to a three-dimensional drawing with
volume. Using this technique we show, for example, that three-dimensional drawings of binary trees can be computed with volume
. We give an algorithm for producing drawings of rooted trees in which thez-coordinate of a node represents the depth of the node in the tree; our algorithm minimizes thefootprint of the drawing, that is, the size of the projection in thexy plane.
Finally, we list significant unsolved problems in algorithms for three-dimensional graph drawing.
This work was performed as part of the Information Visualization Group(IVG) at the University of Newcastle. The IVG is supported
in part by IBM Toronto Laboratory. 相似文献
2.
Joviša Žunić 《Journal of Mathematical Imaging and Vision》2004,21(3):199-204
A digital disc is defined as the set of all integer points inside of a given real disc. In this paper we show that there are no more than
different (up to translations) digital discs consisting of n points. 相似文献
3.
4.
The aim of our research is to demonstrate the role of attractive intermolecular potential energy on normal pressure tensor
of confined molecular fluids inside nanoslit pores of two structureless purely repulsive parallel walls in xy plane at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a uniform density. To achieve this we have derived
the perturbation theory version of the normal pressure tensor of confined inhomogeneous fluids in nanoslit pores:
$ P_{ZZ} = kT\rho \left( {Z_{1} } \right) + \pi kT\rho \left( {Z_{1} } \right)\int\limits_{ - d}^{0} {\rho \left( {Z_{2} } \right)} Z_{2}^{2} g_{Z,H} (d){\text{d}}Z_{2} - \frac{1}{2}\iint {\int\limits_{0}^{2\pi } {\phi^{\prime } \left( {\vec{r}_{2} } \right)\rho \left( {Z_{1} } \right)\rho \left( {Z_{2} } \right)g_{Z,H} (r_{2} )} }{\frac{{Z_{2}^{2} }}{{(R_{2}^{2} + Z_{2}^{2} )^{{\frac{1}{2}}} }}}R_{2} {\text{d}}R_{2} {\text{d}}Z_{2} {\text{d}}\Uptheta ;\quad \left| {\overset{\lower0.5em\hbox{$ P_{ZZ} = kT\rho \left( {Z_{1} } \right) + \pi kT\rho \left( {Z_{1} } \right)\int\limits_{ - d}^{0} {\rho \left( {Z_{2} } \right)} Z_{2}^{2} g_{Z,H} (d){\text{d}}Z_{2} - \frac{1}{2}\iint {\int\limits_{0}^{2\pi } {\phi^{\prime } \left( {\vec{r}_{2} } \right)\rho \left( {Z_{1} } \right)\rho \left( {Z_{2} } \right)g_{Z,H} (r_{2} )} }{\frac{{Z_{2}^{2} }}{{(R_{2}^{2} + Z_{2}^{2} )^{{\frac{1}{2}}} }}}R_{2} {\text{d}}R_{2} {\text{d}}Z_{2} {\text{d}}\Uptheta ;\quad \left| {\overset{\lower0.5em\hbox{ 相似文献
5.
We investigate the arithmetic formula complexity of the elementary symmetric polynomials Skn{S^k_n} . We show that every multilinear homogeneous formula computing Skn{S^k_n} has size at least kW(logk)n{k^{\Omega(\log k)}n} , and that product-depth d multilinear homogeneous formulas for Skn{S^k_n} have size at least 2W(k1/d)n{2^{\Omega(k^{1/d})}n} . Since Sn2n{S^{n}_{2n}} has a multilinear formula of size O(n
2), we obtain a superpolynomial separation between multilinear and multilinear homogeneous formulas. We also show that Skn{S^k_n} can be computed by homogeneous formulas of size kO(logk)n{k^{O(\log k)}n} , answering a question of Nisan and Wigderson. Finally, we present a superpolynomial separation between monotone and non-monotone
formulas in the noncommutative setting, answering a question of Nisan. 相似文献
6.
We generalize the recent relative loss bounds for on-line algorithms where the additional loss of the algorithm on the whole sequence of examples over the loss of the best expert is bounded. The generalization allows the sequence to be partitioned into segments, and the goal is to bound the additional loss of the algorithm over the sum of the losses of the best experts for each segment. This is to model situations in which the examples change and different experts are best for certain segments of the sequence of examples. In the single segment case, the additional loss is proportional to log n, where n is the number of experts and the constant of proportionality depends on the loss function. Our algorithms do not produce the best partition; however the loss bound shows that our predictions are close to those of the best partition. When the number of segments is k+1 and the sequence is of length &ell, we can bound the additional loss of our algorithm over the best partition by
. For the case when the loss per trial is bounded by one, we obtain an algorithm whose additional loss over the loss of the best partition is independent of the length of the sequence. The additional loss becomes
, where L is the loss of the best partitionwith k+1 segments. Our algorithms for tracking the predictions of the best expert aresimple adaptations of Vovk's original algorithm for the single best expert case. As in the original algorithms, we keep one weight per expert, and spend O(1) time per weight in each trial. 相似文献
7.
We consider distributed broadcasting in radio networks, modeled as undirected graphs, whose nodes have no information on the
topology of the network, nor even on their immediate neighborhood. For randomized broadcasting, we give an algorithm working
in expected time
in n-node radio networks of diameter D, which is optimal, as it matches the lower bounds of Alon et al. [1] and Kushilevitz and Mansour [16]. Our algorithm improves
the best previously known randomized broadcasting algorithm of Bar-Yehuda, Goldreich and Itai [3], running in expected time
. (In fact, our result holds also in the setting of n-node directed radio networks of radius D.) For deterministic broadcasting, we show the lower bound
on broadcasting time in n-node radio networks of diameter D. This implies previously known lower bounds of Bar-Yehuda, Goldreich and Itai [3] and Bruschi and Del Pinto [5], and is sharper
than any of them in many cases. We also give an algorithm working in time
, thus shrinking - for the first time - the gap between the upper and the lower bound on deterministic broadcasting time to
a logarithmic factor.
Received: 1 August 2003, Accepted: 18 March 2005, Published online: 15 June 2005
Dariusz R. Kowalski: This work was done during the stay of Dariusz Kowalski at the Research Chair in Distributed Computing
of the Université du Québec en Outaouais, as a postdoctoral fellow. Research supported in part by KBN grant 4T11C04425.
Andrzej Pelc: Research of Andrzej Pelc was supported in part by NSERC discovery grant and by the Research Chair in Distributed
Computing of the Université du Québec en Outaouais. 相似文献
8.
D. Gangopadhyay 《Gravitation and Cosmology》2010,16(3):231-238
A lagrangian for a k-essence field is constructed for a constant scalar potential, and its form is determined when the scale factor is very small
as compared to the present epoch but very large as compared to the inflationary epoch. This means that one is already in an
expanding and flat universe. The form is similar to that of an oscillator with time-dependent frequency. Expansion is naturally
built into the theory with the existence of growing classical solutions of the scale factor. The formalism allows one to estimate
the temperature fluctuations of the background radiation at these early stages (as compared to the present epoch) of the Universe.
If the temperature is T
a
at time t
a
and T
b
at time t
b
(t
b
> t
a
), then, for small times, the probability evolution for the logarithm of the inverse temperature can be estimated as
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