Optimal Surface Parameterization Using Inverse Curvature Map Abstract-Mesh parameterization is a fundamental technique in computer graphics. The major goals during mesh parameterization are to minimize both the angle distortion and the area distortion. Angle distortion can be eliminated by the use of conformal mapping, in principle. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. First, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. On a mesh, the vertex curvature is related to edge lengths by the curvature map. Our result shows the map is invertible, i.e., the edge lengths can be computed from the curvature (by integration). Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Second, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Third, we give an energy form to measure the area distortions, and show that it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum; it has a natural physical interpretation. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced. The theoretical framework of the inverse curvature map can be applied to further study discrete conformal mappings. S URFACE parameterization is the process of mapping a surface to a planar region, and it has broad applications in graphics. Parameterizations introduce distortions between the original surface and its planar image, which can be separated into angle distortion and area distortion [1] , [2] , [3] . In theory, angle distortion can be eliminated completely by conformal mapping, but it is impossible for conformal mappings to further eliminate area distortion completely, except for developable surfaces.. For a given surface, we can define infinitely many different conformal mappings with different area distortions, as shown in Fig. 1 . The central problem of the optimal parameterization can be stated as follows:. How can we find the best conformal mapping that has the least area distortion?. In this paper, we present a set of theoretical tools as well as practical algorithms to tackle this problem. Most of the previous works minimize some energy forms which measure both angle distortion and area distortion. In this work, we take the approach similar to those in [5] and [20] to separate these two criteria. As shown in Fig. 1 , we only minimize the area distortion within the conformal mappings, which eliminate the angle distortions.. We address the angle distortion by using the discrete conformal mappings based on circle packing. The given mesh is covered by circles, each of which is centered at a vertex as shown in Fig. 3 . A circle centered at a vertex is tangent to or intersects with another circle centered at its neighbor vertex. We approximate the conformal mapping by varying the radii while preserving the intersection angles among the circles (see Section 2) .. With circle packing, we can establish the mapping from the configuration of radii to the configuration of curvatures, the so-called curvature map K:. We show that the curvature map is bijective in the conformal mapping. We give an analytical formula for the inverse curvature map by explicitly computing its Jacobian, which is revealed as a dynamic Poisson equation (see Section 2) . Therefore, we can easily compute the radii from the prescribed curvatures.. Discrete conformal parameterization can be treated as finding a configuration of radii such that all curvatures are zeros except for those at the boundaries and cone singularities. All curvature configurations corresponding to parameterizations form an affine subspace, which we call the admissible curvature space. Area distortions can be measured by various energy forms defined on the configurations of radii. Optimal conformal parameterization is equivalent to minimizing the specific energy in the admissible curvature space, and therefore, it is a nonlinear optimization problem with linear constraints.. Energies with good properties, such as differentiability, unique global minimum, simple forms of gradient, and Hessian, are highly preferred in practice. We discovered an energy form that met all the requirements (see Section 3.5). Furthermore, a simple curvature flow algorithm with free boundary conditions is guaranteed to converge to the global minimum.. The pipeline of optimal parameterization system is as follows:. 1. mesh preparation (Section 3.1), 2. computing the initial circle packing metric (Section 3.2), 3. selecting the singular vertex set (Section 3.3), 4. computing the optimal circle packing metric (Sections 3.4 and 3.5), and 5. isometric embedding (Section 3.6). The theoretical results of inverse curvature map are explained in Section 2. Each step of the algorithm pipeline is elucidated in Section 3. The experimental results are demonstrated in Section 4. We conclude our work and point out the future direction in Section 5. Detailed theoretical proofs are presented in the Appendix. In this work, we introduce a set of rigorous theoretical tools and practical algorithms to solve the optimal conformal parameterization problem.. Inverse curvature map represents the exact analytical relation between area distortion and curvature as a dynamic Poisson system. This enables us to find the conformal parameterization with the least area distortion using nonlinear optimization techniques with linear constraints. The explicit conditions for the optima are deduced from the variational principle. A special energy form to measure the area distortion, called the curvature entropy, is investigated. It has a unique global minimum and can be optimized using curvature diffusion algorithm with free boundaries. Our experiments on complex meshes support our theoretical discoveries.. The inverse curvature map theorem is deduced for meshes with euclidean geometry, i.e., the mesh is formed by gluing euclidean triangles. We believe that the inverse curvature map holds for meshes with hyperbolic and spherical geometry, and leads to novel hyperbolic and spherical parameterization algorithms. Both of them play important roles for shape analysis and geometric modeling.. Although we solved the optimal parameterization problem for a given singular vertex set, it remains a challenging problem to determine the optimal vertex set. The common belief for choosing singular vertices is to pick the critical points of the area distortion function, as our algorithm in Section 3.3 does. In the future, we will apply our theoretic tools to continue the exploration along this direction. Parameterization methods have become a fundamental tool in graphics, and a significant amount of research has focused on it. Here, we briefly overview the most related works and refer readers to [1] , [2] , and [3] for wider surveys.. A common approach for parameterization is to minimize a certain energy to control the distortion. Lévy et al. [4] defined an energy to approximate the Cauchy-Riemann equation; Desbrun et al. [5] optimized Dirichlet energy. Variations of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6] , [7] , [8] , [9] , [10] , and [11] . More general energy forms can be found in [12] , [13] , [14] , [15] , [16] , and [17] . Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applied in [17] and [18] to absorb distortions introduced by the convex boundary conditions. Alternatively, [4] and [5] provided parameterizations which require to fix only a few vertices in the parametric domain. Karni et al. [19] discussed the design of geometrically complex boundary conditions with constraints. Zayer et al. [20] applied discrete tensorial quasiharmonic maps to improve the boundary and reduce the distortion.. One of the most prominent characteristics of conformal mapping is that it preserves angles. Angle-based flattening (ABF) method [21] utilizes this property to produce highquality conformal mappings. It derives the discrete conformal mapping by minimizing the ABF energy which is defined as differences between the corner angles of faces on the original mesh and their images on the parameter plane. During the process, the boundary evolves freely to further reduce the distortion. Recently, the method has been improved by several derivative works [22] , [23] in terms of speed and robustness.. Another characteristic of conformal mapping is that it can map infinitesimal circles to infinitesimal circles and preserve their intersection angles. This inspired the circle packing method in [24] . Circle packings and circle patterns replace infinitesimal circles with finite circles. In the limit of refinement, the continuous conformal maps are recovered [25] . Collins and Stephenson [26] have implemented circle packing in their software CirclePack which only considers combinatorics. The connection between circle packing and smooth surface Ricci flow [27] was discovered in [28] . The discrete Ricci flow method was introduced in [29] for hyperbolic parameterization.. Kharevych et al. [30] provided conformal parameterizations for arbitrary genus types by applying circle patterns based on the variational principle in Bobenko and Springborn [31] . The method in [30] supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. It can further reduce the distortion by incorporating manually selected cone singularities. Ben-Chen et al. [32] introduced a conformal parameterization which automatically determines the locations and target curvatures of the cone singularities.. Our work differs from the previous works in the following aspects. Based on a rigorous theoretic proof, our method can produce a discrete conformal parameterization with least area distortion among all possible parameterizations. Similar to [30] , our method is applicable for meshes with general topologies. Furthermore, the method can be extended for optimizing more general energies with constraints on curvatures and area distortions. (Fig. 8 shows parameterizations with special curvature constraints, such that all boundary curvatures are constant.). In this paper, we will explain our theoretical results and algorithmic implementations with circle packings. Since circle packing and circle patterns are equivalent in theory [33] , [34] , our results can also be explained with the setting of circle patterns (see Appendix B).

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Generalizing the Wilcoxon rank-sum test for interval data Here we propose an adaption of Wilcoxon's two-sample rank-sum test to interval data. This adaption is interval-valued: it computes the minimum and maximum values of the statistic when we rank the set of all feasible samples (all joint samples compatible with the initial set-valued information). We prove that these bounds can be explicitly computed using a very low computational cost algorithm. Interpreting this generalized test is straightforward: if the obtained interval-valued p-value is on one side of the significance level, we will be able to make a decision (reject/no reject). Otherwise, we will conclude that our information is too vague to lead to a clear decision. Our method is also applicable to quantized data: in the presence of quantized information, the joint sample may contain a high proportion of draws, which can prevent the test from drawing a clear conclusion. According to the usual convention, when there are ties, the ranks for the observations in a tie are taken to be the average of the ranks for those observations. This convention can lead to wrong conclusions. Here, we consider the family of all possible rank permutations, such that a sample containing ties will not just be associated with a single value, but rather with a collection of values for the Wilcoxon's rank-sum statistic, with each one of them being associated with a different p-value. When the impact of quantization is too high to lead to a clear decision, our test provides an interval-valued p-value that includes the chosen significance level. It indicates that there is no clear conclusion according to this test. Two different experiments exemplify the properties of the generalized test: the first one illustrates its ability to avoid wrong decisions in the presence of quantized data. The second one shows the performance of the generalized test when used with interval data. The Wilcoxon rank-sum test [17] , also know as Mann-Whitney U test, is a non-parametric hypothesis test used to check whether or not two independent samples containing n and m elements correspond to the same distribution. It does not require the data normality assumption. It can thus be regarded as an alternative to the two-sample t-test when the normality of the data clearly cannot be assumed. The Wilcoxon rank-sum statistic W is computed as follows: the n + m observations of the two independent samples are combined in a single dataset. The elements of this dataset are sorted from smallest to largest. If there are ties, i.e. duplicated values in the combined dataset, the ranks for the observations in a tie are taken to be the average of the ranks for those observations. The Wilcoxon statistic, W , is calculated as the sum of the ranks for the n observations from the first population. If the null hypothesis of identical population distributions is true, n ranks from the first population are just a random sample from the n + m integers 1, . . . , n + m. Under this null hypothesis, the expectation and variance of W are, respectively: Furthermore, when both sample sizes are sufficiently large (n > 10 and m > 10, by convention), the distribution of the statistic T = W −μ 0 σ 0 is assumed to be Gaussian (with null expectation, and variance equal to 1). Intuitively, if W is much smaller (or larger) than μ 0 , this is evidence that the null hypothesis is false and, in fact, that the considered samples come from distinct populations. Under this assumption, the critical value (or p-value) is calculated according to the following equality:. where w denotes the value of the rank-sum statistic in the sample, and φ denotes the cumulative distribution function of the standard normal. For a specific significance level α ∈ (0, 1), the null hypothesis will be rejected whenever p(w) < α.. In this paper, we explore an alternative to the convention that assigns an average rank to tied values. If there is a high proportion of draws in the sample (which can easily occur in the presence of discrete or quantized data), considering a different (but also compatible) sequence of rank assignments can lead to a completely different final decision. Indeed, the computed rank-sum statistic w, and therefore the associated p-value, p(w), may vary significantly. In this paper, we consider all the possible rank assignments, so that a sample containing ties, will not just be associated with a single value (computed by averaging the ranks of the tied elements), but rather to a collection of values for the Wilcoxon rank sum statistic, with each one of them being associated with a different p-value. The decision process thus becomes more expressive. If all of the computed p-values are below a fixed significance level α, we will reject the null hypothesis. Likewise, if the p-values are all above α, then one would conclude that there is little evidence for the alternative hypothesis to be true (we "accept" the null hypothesis). There is a now a third possible answer: if the set of p-values and the significance level do overlap, we cannot make a decision because our information on the original values that led to that "quantized" data is not sufficiently accurate. In other words, quantization hides the information and thus the problem is no longer decidable.. The procedure described in the previous paragraph can be applied, more generally, to situations where, instead of pointvalued sample observations, we are provided with interval-valued sample observations, with each one containing the true instance.. Interval-valued observations can account for a known defect in the measurement system (see e.g. [16, 18, 14] and references therein), quantization of data, guaranteed estimation ( [9, 8, 10] ) or for imprecision in the model (see e.g. [15, 13] ). We aim to cover the whole collection of samples compatible with our incomplete information in order to calculate the set of feasible rank-sum statistics. A p-value corresponds to each of those rank-sum statistics. Thus, our information about the critical value associated with the true (imprecisely observed) joint sample will be determined by a subset of [0, 1] , where it certainly belongs. When the initial information is not very accurate, the set of possible p-values will be larger. The decision procedure is similar to the previously presented approach. When the whole set is on one side of the α−level, we will be able to make a decision, otherwise we will not. This approach was recently considered by several authors (see, for instance [1] [2] [3] [4] [5] ). Specifically, Denoeux et al. [3] applied this procedure to the Wilcoxon rank-sum test, in the presence of fuzzy data. This encompasses, as a particular case, the case of interval-valued data. The same idea of looking for the least and most favorable cases within the imprecise dataset can be also found in the recent literature (see [1, 6, 7, 12, 11] among others).. One of the main pitfalls in this framework is to compute the bounds of the set of feasible p-values. This task can be relatively computationally expensive, depending on the test we are generalizing. As Denoeux et al. [3] state, a simplistic approach used to solve the problem of determining the maximum and minimum values for the statistic and its corresponding p-value, in this kind of rank-based tests, might be to generate all rank assignments that are compatible with the available incomplete information. However, this approach is intractable, as they point out, due to the potentially exponential number of different rankings, which can reach n! in the empty information limit case. Thus, they propose to use a Monte-Carlo simulation method to approximate the bounds of the Wilcoxon rank-sum statistic. Such a method would require many iterations for large n and m values, but would not provide exact values of the bounds of the set of feasible values for the statistic. In this paper, we try to overcome this issue. We present a procedure to calculate exact values for those bounds. Furthermore, we show that the complexity of such a procedure is comparable to the calculation of the Wilcoxon rank-sum statistic for single-valued data, because each of the bounds just involves calculation of the Wilcoxon rank-sum test for a specific extreme sample. We provide two explicit and computationally simple algorithms to compute those bounds.. We propose two experiments to illustrate the properties of this generalization. The first experiment highlights the ability of the imprecise test to discard wrong decisions due to quantization in precise data. The second experiment shows the behavior of the test when interval-valued data are considered. The Wilcoxon rank-sum test was designed to check whether two independent real valued samples correspond to the same distribution or not. This test must be adjusted to deal with quantized or imprecise data. In this paper, we proposed such an adaption by generalizing the classical test. One of the main features of this new test is that it provides a pair of bounds, p and p, for the p-value (instead of a precise number) leading to a bipolar decision: the answer to such a test can be yes, no or unknown. We proved that the interval [p, p] we computed contains all p-values that should have been obtained by using the conventional test with precise valued data belonging to the set of interval-valued data. This new test also deals with the quantization in a new way. Quantization is perceived as a non-linear modification of the supposed existing true real samples that disturb the test. When the disturbance is too marked, the test is inconclusive. If the test leads to a decision, the test level can be guaranteed.. As a future work, we consider extending this test to fuzzy interval-valued data which would lead to a fuzzy interval of p-values. A very straightforward and exact solution may be obtained by using Algorithms 1 and 2 for each level-cut of the fuzzy interval-valued data, and building the level cut of the p-value fuzzy interval. This solution is tractable only if the membership values are quantized (i.e. can take only a finite number of values). Otherwise, it would lead to a too computationally expensive algorithm. Finding a low computational cost algorithm for extending the generalized Wilcoxon rank-sum test remains an open problem. Computing the generalized Wilcoxon rank-sum test is really easy and can be done at low computational cost compared to the original test that needs to find and process the links. This generalized test just involves two sorts, in case of precise data, and four sorts, in case of imprecise data. Since precise data is just a particular case of imprecise data, here we present the two algorithms needed to process imprecise data. Algorithm 1 computes min W xy while Algorithm 2 computes max W xy . A major issue concerning statistical tests is their robustness with respect to the data model. The Wilcoxon rank-sum test was designed to compare samples taken from pairs of continuous distributions, but it is frequently applied to pairs of samples taken from discrete distributions. In particular, it is often used to deal with quantized information, which involves comparing pairs of integer vectors.. For example, pixel values of a 512 × 512 gray-level image are quantized on 8 bits, i.e. on 256 values. On average, 1024 pixels would thus have the same value. Using a Wilcoxon rank-sum test and ignoring the quantization effect could lead to arbitrary results, and thus wrong decisions: the value of the statistic calculated from quantized data could differ markedly from a calculation based on original (non-quantized) data.. In some applications, e.g. image based medical diagnosis, inferential statistics are often used to compare pairs of regions in the same image or in two different images, and determine whether they are similar or not. Early diagnosis is vital in the treatment of some diseases, which means that we need to detect differences at an early stage. In such cases, accepting the equality hypothesis when it is false must be avoided as much as possible. On the contrary rejecting this hypothesis when it is true may sometimes lead to an unnecessary harmful treatment choice. In fact, physicians would usually try to avoid a wrong decision, and prefer to acquire additional data when the actual data are not fully reliable. Thus, knowing that no-decision can be taken based on the current set of data is a valuable piece of information.. Finally, when the degree of sensor imprecision is known, it can also be valuable to be able to use this information in the test.. We propose two illustrative experiments that highlight the ability of the generalized Wilcoxon rank-sum test to avoid wrong decisions. Those experiments are based on images acquired by a gamma camera (nuclear medicine images). They aim at mimicking real medical situations where the nuclear physician has to compare the distribution of values in two regions of interest in order to decide whether or not a patient has a specific disease.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


ERD:

a

fast

and

reliable

tool

for

RNA

design

including

constraints

AbstractThe

function

of

an

RNA

in

cellular

processes

is

directly

related

to

its

structure.

The

free

energy

of

RNA

structure

in

another

important

key

to

its

function

as

only

some

structures

with

a

specific

level

of

free

energy

can

take

part

in

cellular

reactions.

Therefore,

to

perform

a

specific

function,

a

particular

RNA

structure

with

specific

level

of

free

energy

is

required.

For

a

given

RNA

structure,

the

goal

of

the

RNA

design

problem

is

to

design

an

RNA

sequence

that

folds

into

the

given

structure.

To

mimic

the

biological

features

of

RNA

sequences

and

structures,

some

sequence

and

energy

constraints

should

be

considered

in

designing

RNA.

Although

the

level

of

free

energy

is

important,

it

is

not

considered

in

the

available

approaches

for

RNA

design

problem.

Results:

In

this

paper,

we

present

a

new

version

of

our

evolutionary

algorithm

for

RNA

design

problem,

entitled

ERD,

and

extend

it

to

handle

some

sequence

and

energy

constraints.

In

the

sequence

constraints,

one

can

restrict

sequence

positions

to

a

fixed

nucleotide

or

to

a

subset

of

nucleotides.

As

for

the

energy

constraint,

one

can

specify

an

interval

for

the

free

energy

ranges

of

the

designed

sequences.

We

compare

our

algorithm

with

INFO-RNA,

MODENA,

NUPACK,

and

RNAiFold

approaches

for

some

artificial

and

natural

RNA

secondary

structures

and

constraints.The

results

indicate

that

our

algorithm

outperforms

the

other

mentioned

approaches

in

terms

of

accuracy,

speedup,

divergency,

nucleotides

distribution,

and

similarity

to

the

natural

RNA

sequences.

Particularly,

the

designed

RNA

sequences

in

our

method

are

much

more

reliable

and

similar

to

the

natural

counterparts.

The

generated

sequences

are

more

diverse

and

they

have

closer

nucleotides

distribution

to

the

natural

one.

The

ERD

tool

and

web

server

are

freely

available

at

We

have

shown

that

the

ERD

tool

is

a

very

fast

and

successful

approach

to

design

RNA

sequences

which

fold

into

a

given

structure

and

fulfill

some

sequence

and

energy

constraints.

The

core

of

the

algorithm

was

previously

introduced

in

where

we

showed

that

it

proceeds

better

and

faster

than

the

other

existing

approaches.

Here,

we

have

demonstrated

that

the

ERD

tool,

with

additional

constraints

on

the

sequence

and

energy

level,

also

performs

better

and

faster

than

the

other

competitors..

In

addition

to

the

sequence

constraints,

the

energy

ranges

of

the

generated

sequences

over

the

target

structure

can

be

specified

in

ERD

tool.

This

capability

of

ERD

allows

researchers

to

generate

sequences

with

similar

structure

and

a

specific

level

of

energy

value.

In

this

section,

we

briefly

introduce

our

evolutionary

algorithm

for

designing

an

RNA

sequence

that

folds

into

a

given

target

structure

Any

RNA

secondary

structure

can

be

uniquely

decomposed

into

its

structural

components

(stems,

hairpin

loops,

internal

loops,

bulge

loops,

multi-loops,

and

external

loop),

each

having

a

different

length.

By

employing

the

natural

RNA

sequences,

we

first

construct

the

pools

of

RNA

sub-sequences

corresponding

to

different

components

with

different

lengths.

Using

these

pools,

we

then

construct

an

initial

RNA

sequence

which

is

compatible

with

the

given

target

structure.

After

that,

the

target

structure

is

hierarchically

decomposed

into

smaller

sub-structures.

This

decomposition

is

performed

in

positions

where

the

multi-loops

occur.

Finally,

we

use

an

evolutionary

algorithm

to

improve

the

quality

of

the

sub-sequences

corresponding

to

the

decomposed

sub-structures.

The

ERD

tool

has

been

implemented

in

C

programming

language

to

be

consistent

with

the

Vienna

RNA

package

and

to

benefit

from

the

faster

execution

of

the

compiled

code

in

this

language.

The

details

of

the

above-mentioned

steps

are

presented

in

the

following

subsections.

O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A simple and fast algorithm for computing exponentials of power series As was initially shown by Brent, exponentials of truncated power series can be computed using a constant number of polynomial multiplications. This note gives a relatively simple algorithm with a low constant factor.

O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O


Photon Induced Entanglement in Atom-Cavity Systems We study the evolution of quantum entanglement in double cavity systems. The entanglement of cavity atoms induced by entangled pair of photons is investigated. Both entanglement sudden death and entanglement sudden birth phenomena are shown to be existed and analyzed in detail. We also propose a strategy to enhance the entanglement between the atom in one cavity and the photon in another cavity by using quantum Zeno effect. Entanglement plays a key role in quantum computation and quantum information processing [1] . Due to the interactions with the environment in preparation and transmission, the entangled states usually become mixed ones that are no longer maximally entangled. It is of great importance to known and control the evolution of entanglement in quantum systems. The entanglement evolution under the influence of local decoherence has been studied by many authors recently [2, 3, 4, 5, 6, 7] . For a bipartite system with one subsystem undergoing an arbitrary noisy channel, elegant relations have been obtained between the concurrence of the initial and final states [2, 3] . In [4] the authors investigated the time evolution of entanglement of a bipartite qubit system undergoing various modes of decoherence. It is found that although it takes infinite time to complete the decoherence locally, the global entanglement may vanish in finite time, a phenomenon so called entanglement sudden death (ESD). Such phenomena have been studied further in various systems with different entanglement measures and purposes [8] . Experimental evidences of ESD have been also reported for optical setups and atomic ensembles [9] .. In this paper we study double cavity systems with a two level atom in each cavity. In stead the case that the atoms are initially entangled [4] , we consider that the atoms are initially spatially separated and in a separable state. We investigate the evolution of quantum entanglement when a pair of entangled photons are introduced into the cavities. It is found that when two atoms are initially in the ground state, there exits a kind of entanglement transfer between atoms and photons. When two atoms were initially in exited state, then there exit both entanglement sudden death and entanglement sudden birth (ESB) phenomena between the two atoms, and between the atom in one cavity and the photon in another cavity. For both initial conditions we find that the entanglement between the atom in one cavity and the photon in another cavity is rather small in general. We show that their entanglement can be enhanced in terms of quantum Zeno effect.. We use concurrence as the measure to characterize the quantum entanglement of a two-qubit state ρ [10] ,. where λ i are the eigenvalues, in decreasing order, of the matrix ρ(σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), ρ * denotes the complex conjugation of ρ and σ y is the Pauli matrix. If a density matrix ρ only contains nonzero elements along the main diagonal and anti-diagonal such as. then its concurrence is verified to be of the form. For qubit-qutrit systems, there is no analytical formula of concurrence in general. We use negativity [11, 12] as the measure of quantum entanglement, which also gives rise to a necessary and sufficient criterion for separability of qubit-qubit or qubit-qutrit states. The negativity N(ρ) of a state ρ is defined by. where λ min is the smallest eigenvalue of ρ Tx , T x stands for the partial transpose with respect to the subsystem x.. 2 Photon-induced entanglement in double cavities Figure 1 : The schematic diagram of the model in this paper.. We consider a model consisting of two two-level atoms A and B, each interacting with a single-model near-resonant cavity field, denoted a and b respectively, see Fig.1 . It is assumed that each atom-cavity system is isolated and that the cavities are initially in a entangled state while the atoms are, different from the X states [4] , in a separable excited/ground state. The dynamics of the model is characterized by the double J-C Hamiltonian. where ν is the field frequency, ω is the transition frequency between the excited and the ground states of the atoms, g is the coupling constant between the cavity field and the atoms, a and a + (resp. b and b + ) are the field annihilation and creation operators associated with the atom A's (resp. B's) cavity, and σ ± are the spin-flip operators. The eigenstates of the Hamiltonian (4) are products of the eigenstates of the separate J-C system [13] . For simplicity, in the following we consider the case of zero detuning, ω = ν. We denote | ↑> (resp. | ↓>) the excited (resp. ground) state of the atoms.

O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Fast finite difference Poisson solvers on heterogeneous architectures In this paper we propose and evaluate a set of new strategies for the solution of three dimensional separable elliptic problems on CPU-GPU platforms. The numerical solution of the system of linear equations arising when discretizing those operators often represents the most time consuming part of larger simulation codes tackling a variety of physical situations. Incompressible fluid flows, electromagnetic problems, heat transfer and solid mechanic simulations are just a few examples of application areas that require efficient solution strategies for this class of problems. GPU computing has emerged as an attractive alternative to conventional CPUs for many scientific applications. High speedups over CPU implementations have been reported and this trend is expected to continue in the future with improved programming support and tighter CPU-GPU integration. These speedups by no means imply that CPU performance is no longer critical. The conventional CPU-control-GPU-compute pattern used in many applications wastes much of CPU's computational power. Our proposed parallel implementation of a classical cyclic reduction algorithm to tackle the large linear systems arising from the discretized form of the elliptic problem at hand, schedules computing on both the GPU and the CPUs in a cooperative way. The experimental result demonstrates the effectiveness of this approach. The era of single-threaded processors has come to an end due to the limitation of the CMOS technology and in response, most hardware manufactures are designing and developing multi-core processors and specialized hardware accelerators such as GPUs [6, 16, 17] . As a consequence, applications can only improve their performance if they are able to exploit the available parallelism of the new architectures.. In this paper we study the implementation of a fast solver based on a block cyclic reduction algorithm to tackle the linear systems that arise when discretizing a three dimensional separable elliptic problem with standard finite difference. A clear example of the importance of dealing efficiently with three dimensional elliptic systems is found in the numerical simulation of incompressible fluid flows. Indeed, the most time consuming part of almost any incompressible unsteady Navier Stokes solver (i.e., incompressible fluid dynamic simulation codes) is related to the solution of a pressure Poisson equation at each time step (see for instance [12] ). The achievement of a satisfactory computational efficiency to tackle this class of elliptic partial differential equations is therefore a key issue when simulating unsteady fluid flow processes (turbulent flows for instance).. Other authors have addressed topics which are somehow related to the present contribution. [5] analyzes the performance of a block tridiagonal benchmark on GPUs. This is the first known implementation of a block tridiagonal solver in CUDA but the pattern of the block matrices they analyzed differ from our target problem. The sub-matrix element rank (m) was assumed to be small (m = 5). In our case both m and the arithmetic intensity of problem are higher.. For distributed multicore clusters, the BCYCLIC algorithm developed by Hirshman et al. [3] is able to solve linear problems with dense tridiagonal blocks. Our target algorithm, the BLKTRI code [13] is not well-suited for dense blocks but it is the most popular approach for solving block tridiagonal matrices which arise from separable elliptic partial differential equations.. Many authors have studied the implementation of scalar tridiagonal solver on GPUs [8, 7, 1, 2, 4] . D. Göddeke et al. [8] proposed an efficient implementation of the Cyclic Reduction (CR) algorithm, which is used as a line smoother in a multigrid solver. Yao Zhang et al. [7] proposed some hybrid algorithms that combine CR with other tridiagonal solvers such as Parallel Cyclic Reduction (PCR) or Recursive Doubling (RD). More recently, H. Kim et al. [4] have analyzed other hybrid algorithms and found that a combination of PCR and Thomas gave the best overall performance.. The rest of this paper is structured as follows. Section 2 introduces the extended block cyclic reduction algorithm used by the BLKTRI solver. Section 3 gives a brief description of the standard algorithms for solving scalar tridiagonal systems. In Section 4 we detail the mapping of the BLKTRI solver on multicore and GPUs and analyze their performance and then in Section 5 we extend our discussion to 3D problems. Finally, Section 6 concludes summarizing the most relevant contributions. The efficient solution of block tridiagonal linear systems is of crucial importance since it is the major bottleneck of several large scale simulation codes dealing with time-dependent elliptic partial differential equations. We have analyzed the performance of different parallel implementation that exploit homogeneous multicores and GPUs systems, as well as heterogeneous multicores-GPUs platforms.. On multicore, a coarse grain approach based on the Thomas algorithm is the best option for solving the intermediate scalar tridiagonal systems that arise on both 2D and 3D problems. In contracts, on GPUs it is much better a fine grain alternative based on using the PCR algorithm.. As expected, an heterogeneous approach that combines both implementations is the best option on 2D problems. For 3D problems, we have shown that this is also the best choice despite the 3D problem has both higher arithmetic intensity and higher parallelism than the 2D case. Indeed, the homogeneous GPUs implementation outperform the multicore counterpart even for medium size 3D problems. However, the heterogeneous approach benefits from CPUs-GPUs overlapping an is able to achieve an additional 15% performance gain.

B I I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Structural tendencies - Effects of adaptive evolution of complex (chaotic) systems We describe systems using Kauffman and similar networks. They are directed functioning networks consisting of finite number of nodes with finite number of discrete states evaluated in synchronous mode of discrete time. In this paper we introduce the notion and phenomenon of 'structural tendencies'. Along the way we expand Kauffman networks, which were a synonym of Boolean networks, to more than two signal variants and we find a phenomenon during network growth which we interpret as 'complexity threshold'. For simulation we define a simplified algorithm which allows us to omit the problem of periodic attractors. We estimate that living and human designed systems are chaotic (in Kauffman sense) which can be named -complex. Such systems grow in adaptive evolution. These two simple assumptions lead to certain statistical effects i.e. structural tendencies observed in classic biology but still not explained and not investigated on theoretical way. E.g. terminal modifications or terminal predominance of additions where terminal means: near system outputs. We introduce more than two equally probable variants of signal, therefore our networks generally are not Boolean networks. They grow randomly by additions and removals of nodes imposed on Darwinian elimination. Fitness is defined on external outputs of system. During growth of the system we observe a phase transition to chaos (threshold of complexity) in damage spreading. Above this threshold we identify mechanisms of structural tendencies which we investigate in simulation for a few different networks types, including scale-free BA networks. In this short and condensed article we are going to introduce structural tendencies which are effects of random adaptive evolution of complex systems. Terminal modification and terminal predominance of addition are the main examples of these tendencies. The former one means that a random change of a system near system outputs has a higher probability of being accepted by adaptive condition than a change far from system outputs. The latter one means that near system outputs more additions of new nodes to network are accepted than removals of nodes in this place. These tendencies are an equivalent of Naef's 'Terminal Modifications' 28 and Weismann's 'Terminal Additions' 38 in the evolutionary biology. Now these classic regularities are forgotten together with comparative embryology 39 only due to lack of their explanation 18 . Therefore our investigations should have influence on biology 16 . Also growth of the network is one of such structural tendencies and there is no need to assume it.. Complexity has as many descriptions and definitions (e.g. 31,17 or more recent 23,5 ) as different aspects and meanings. We define the complexity threshold during system growth as the phase transition to chaos. Over this threshold we observe in a simulation the mechanisms of these structural tendencies. We use the term 'chaos' as Kauffman 26 does. Systems exhibit chaotic behaviour when after a small disturbance of system the damage statistically explodes and reaches a high level in equilibrium. Damage is the difference between node output states in the disturbed and undisturbed systems.. We estimate that the typical living or human-designed system grows under adaptive condition and is chaotic. We treat the gene regulatory network as a strange exception.. To describe such a system we use Kauffman's network 24,26 and similar ones but not in the autonomous case. We do not use cellular automata ordered on lattice but a randomly structured network. It is a directed network where each node is influenced by K other nodes or external inputs, and influences k other nodes or external outputs. We study K = 2 or K = 3. The distribution of number k depends on network type, k may be fixed k = K or flexible in range starting from k = 0. Each node influences others by signals which are deterministic functions of the node's input signals. The whole network has m external input signals (in this paper they are constant) and m external output signals (in first step m = 0). System's fitness is defined as the number b of output signals which agree with an arbitrarily defined sequence of m ideal signals. In the network evolution changes are made randomly but they form adaptive evolution only if they do not reduce system fitness b.. Kauffman places living systems in phase transition between order and chaos 26 but he uses the Boolean network -the case of Kauffman network with two variants of signal. He uses the assumption that the two variants of signal have equal probability. We denote the number of equally probable variants of signal as s. Note that using this parameter s we know that these s variants are equally probable and we will not repeat it every time. For more precision we assume in this all paper that the 'internal homogeneity' 26 P is the smallest. Only for s = 2 the typical (K = 2) big random system can exhibit ordered (opposite to chaotic) behaviour. If variants of signal are not equally probable (i.e. we cannot use description s), chaos can be avoided also for more than two variants.. The mostly used case s = 2 is also often not a realistic simplification. Typically real alternatives, especially when there are two alternatives, are not equally proba- ble. This expected inequality is described using probability p for one alternative 9,3 . We estimate that the typical occurrence of two unequally probable alternatives in descriptions of living and humane designed systems is an effect of our concentration on one of the possibilities which is special for us, and of collecting all remaining as second alternatives. Such a view needs another description than using p for statistical mechanism because there are more than two real alternatives.. In the first step of our way we introduce s > 2 and we show that the parameter s cannot be substituted by others (e.g. K -see fig. 1 ). Typically s = 2 is used for genome modelling (e.g. Ref. 25, 34) . For genetic regulatory network where 1 is interpreted as active and 0 as inactive it seems adequate and gives results close to experimental data 37,32,33 . However, when we are going to describe certain properties coded by genes or their mechanisms assessed using fitness we should remark that there are 4 nucleotide or 20 amino acids or other unclear spectra of alternatives, not only 2, therefore s = 20 or s = 4 seems much more adequate. Such strong simplification (s = 2) can only be used if it does not significantly changed results, but we show, that it would do it. Stability or order in gene regulatory network results in properties assessed by natural selection in a long process where assumption of s = 2 is probably not adequate. Such a case described well by s > 2 differs from one described by p and 1 − p in the statistical mechanism and its result. E.g. for extreme p and small K > 2 order is expected 9,3 but not for s > 2. For s > 2 a system should be chaotic which our coefficient w = k(s − 1)/s of damage propagation shows easily. However, to be chaotic a system also needs to be big enough. In the second step our complexity threshold shows how big it should be. This depends on the network type. Our complexity threshold can be easily detected in the reality and in the simulation.. In the last, third step we investigate structural tendencies in the adaptive evolution of chaotic system (above the complexity threshold on system size axis, with s > 2). In our model we define fitness b, which is needed in adaptive evolution, only on the system external outputs, not on all the node states as in Kauffman model. Therefore our system in the third step is not an autonomous one (m > 0, in simulation we use typically m = 64). In the investigation of complexity threshold in second step we also prefer such a form of the system to be appropriate for the third step. Only in the first step the considered system is autonomous (m = 0). In our fitness there are no local extremes. To keep fitness not maximal but high and constant the ideal vector of output signals is changed accordingly.. For simulation we use our simplified algorithm which allows us to obtain one particular vector of output signals instead of a periodic attractor. It gives correct answers for our statistical questions and practically allows to investigate the emerging structural tendencies. The way to a mechanism of recapitulation is long and contains (in opposite direction): tendency of terminal predominance of addition, which needs tendency of terminal modifications. We have observed them in simulation in a few different network types as effects of adaptive condition of network growth, over a certain complexity threshold. This threshold is observed the in damage size distribution on network external outputs. To model damage spreading we use functioning, directed networks, e.g. Kauffman networks, but we use more than two equally probable signal variants, therefore they are no longer the Boolean networks. Such an assumption (argued using interpretation) places the considered systems in the chaotic area, far from 'order' and 'phase transition to order', in addition the complexity threshold guards this assumption. In comparison to the Kauffman model we introduce two new elements. Firstly, we allow s > 2 (more than two equally probable signal variants), which keeps our system in the chaotic area (unlike 'internal homogeneity' P ). Secondly, we introduce external outputs which we use for differentiation of system body areas and as 'essential variables' for fitness definition. We find 'structural stability', important for adaptive evolution, in our 'small change tendency' which causes differences in elimination between various areas of network body. These differences lead to 'structural tendencies'. The famous Kauffman conclusion from his model 'We shall find grounds for thinking that the ordered regime near the transition to chaos is favored by, attained by, and sustained by natural selection' seems not applicable to our model. Because our definition of fitness uses output signals, we omit the problem of local optima, which are absent here and the problem of 'complexity catastrophe' expected by Kauffman. Simplifications in 'fitness landscape' and especially in algorithm in the aspect of attractors allow us to investigate a large, new and interesting area of 'structural tendencies' which is waiting for a more mathematical description. In case of addition, links g and h and function of node are drawn. Node j is drawn instead of link h for ss. Removal only needs a draw of a node to remove. For node removing in ss and sf the outgoing links, which were added after addition of this node to the network, are moved to the start node of link g. Node added on link i remains a k = 0 node while removing, therefore for adaptive growth with removing the sf network pattern for addition is supplemented by drawing node j additionally (se network). For K > 2 additional inputs are constructed like the right ones. The ak network is maintained as aa but there is only c output signal. (1) and small change tendency (2) for s = 4. When network is complex the main right peak of P (L) is out of range of acceptable cases for higher fitness b. (1). P (L) distribution of change size on the system outputs from simulations as the effect of random changes in the networks. The base curve a -for complex network with feedbacks. It has two peaks and exactly zero in-between. The same network on complexity threshold -curve b, and below it, when network was still small -curve c. Size of the left peak strongly depends on network type, here aa network with small left peak. For networks without feedbacks lw (most ordered) and lx -curves d and e, the space between peaks has a small but visibly non-zero value. (2) . Calculated distribution of probability P (a|L, b) of acceptance by adaptive condition test, for three higher values of fitness b. For interesting higher b the P (a|L, b) is significantly different from zero only for very small change size (L < 20 for b = 40). To the right of the indicated points P (a|L, b) becomes exactly zero (L = 48 for b = 40). Cone of influence in an extremely simple network lw without feedbacks. Black node divides the node set into three parts: earlier and later nodes, which form the cone, and the restindependent nodes. If the black node changes its state, then it becomes a source of damage, which flows up to the outputs(grey nodes). If w > 1 usually it is also a cone (of damage). The depth of change initiation is the height of the cone and suggests the average change size L. It is connected with functional order and place description as early, late and terminal as shown. 2.Depth definition used in aa simulation and its comparison to the shortest way to outputs used for sf, ss and levels number used for lw, lx. It is a structural and sequential approximation of the functional order. (2) for Kauffman networks with feedbacks (se and ss), containing k = 0 and k = 1. Here D is the shortest distance to outputs and can be large, but with small probabilities, therefore the boundaries of these probabilities are shown.

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Bibliometric evolution: Is the journal of the association for information science and technology transforming into a specialty Journal? Applying a recently developed method for measuring the level of specialization over time for a selection of library and information science (LIS)-core journals seems to reveal that Journal of the Association for Information Science and Technology (JASIST) is slowly transforming into a specialty journal. The transformation seems to originate from a growing interest in bibliometric topics. This is evident from a longitudinal study (1990-2012) of the bibliometric coupling strength between Scientometrics and other LIS-core journals (including JASIST). The cause of this gradual transformation is discussed, and possible explanations are analyzed. These results support the earlier findings by Schneider (2009) . They indicate that it is not our measure of specialization that is flawed but that JASIST over time has shifted its focus more toward bibliometrics, thus becoming gradually more and more specialized. The cause of this transformation would be interesting to investigate further. Is it mainly caused by editorial self-preferences or is it a symptom of something deeper going on in our discipline?. The editor, Blaise Cronin, is a highly respected bibliometrician, widely recognized for his many important contributions to bibliometrics, and winner of the prestigious Derek de Solla Price Medal. It is thus quite natural to suspect that his interest in bibliometrics is what has caused the gradual transformation. However, Cronin took over the editorship of JASIST on January 1, 2009. As seen in Figures 1 and 2 , the gradual transformation started years before that date.. Perhaps the transformation is caused by a change in what might be termed "practical interests." In his famous dissertation, Merton (1938 Merton ( /1970 showed how the problems that 17th-century British scientists chose to work on were influenced by a series of economic and military concerns confronting their society. The core domains of LIS have traditionally been: bibliometrics, information seeking and retrieval (IS&R), knowledge organization (KO), and information retrieval (IR) systems design. Within each of these domains, information scientists have worked for decades on developing new theories and methods for improving our understanding of various information phenomena and for achieving better utilization of different types of information systems. Information scientists still do! Yet, as described by Hjørland (2012) , the Google era has, to some extent, reintroduced the idea of a quick "technological fix"-that many of the old research problems that have traditionally occupied information scientists may be solved easily by new technology. The domain of KO is, for instance, challenged at the practical level by libraries ceasing to classify their own books. Hjørland (2012) gives examples from Denmark where at least two major libraries have ceased classifying their own collections. Instead, they are relying on classifications from external sources and on user tagging. He noted that:. Many library directors expect that, in the future, large scanning projects (such as that which is being conducted by Google) may enable full text searches to be carried out of all available content. For this reason, they may consider it a waste of resources to classify or index books. (p. 300) Thus, if Hjørland is right, traditional LIS domains such as IS&R, KO, and IR systems design are challenged at the practical level. Of course, this also may affect the research level of LIS where researchers also work on solving practical problems. The bibliometric domain is not challenged in the same way as are the three other traditional LIS domains. On the contrary, in recent years, a massive interest in the practical aspects of bibliometrics for research evaluation has emerged. A number of countries (including Denmark) have developed national university rankings based on various bibliometric indicators. Because the problems that 17th-century British scientists chose to work on were influenced by a series of economic and military concerns confronting their society, practical interest in bibliometrics also may have affected the foci of attention in LIS, and thus may be part of the explanation for the bibliometric transformation of JASIST. The problem with that explanation is, of course, that this transformation is seemingly only affecting JASIST. Why are the other LIS journals not "going bibliometric?". Other explanations also may be worth pursuing. Migration of topics might be a place to start. It is quite obvious that a topic such as IR, which was previously a core information science topic, has now more or less migrated to the computer science field.. Our initial idea was just to develop a new method for measuring the process of specialization over time. We discovered a possible flaw in the method when we applied it to a selection of LIS journals. Initially, this bothered us, but after having looked deeper into it, we conclude that it is not the method that is flawed. Moreover, we believe to have discovered an interesting paradox, perhaps worth pursuing further. As can seen in Figure 1 , the degree of re-citations in Scientometrics varies from 0.13 to 0.28 and is thus considerably higher than the degrees shown in Figure 2 . A high level of self-dependency is to be expected. During the last decade, JASIST is by far the journal showing the strongest bibliographic coupling strength with Scientometrics. The four library automation journals are exhibiting a very weak bibliographic coupling strength with Scientometrics, although there may be a tendency during the last few years toward a slight increase. The increase is, however, too small to be considered statistically significant at the .05 level. To test the bibliometric host idea, we have compared year by year the strength of bibliographic coupling between Scientometrics and all other journals in our sample . A bibliographic coupling is defined as a 100% match between a reference in Scientometrics in a specific year and a reference in another journal the same year. Consequently, spelling errors, typing errors, variations of spelling, and the like should be considered a possible source of bias; however, as these irregularities are expected to be evenly distributed across the data set, bias is unlikely. Data registered are name of journal, publication year, cited references in the journal, and the number of instances for every reference. Some of the references appear more than once; consequently, the number of bibliographic couplings depends on the total number of instances and not just the number of unique references. The bibliographic coupling strength between two journals in this case is calculated between two identical publication years. The bibliographic coupling strength between the publications in a given year (y) in a journal (j) and the publications in another journal (s) in the same publication year (y) is calculated as follows:. Strength of bibliographic coupling = number shared references j,y,s,y /total number of references j,y. The Journal of Information Science can serve as an example. In 2010, the Journal of Information Science contained 2,279 references, of which 106 were citations appearing in Scientometrics the same year, resulting in a share of re-citations of 106/2,279 = 0.055.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Characterization of co-blockers for simple perfect matchings in a convex geometric graph Consider the complete convex geometric graph on 2m vertices, CGG(2m), i.e., the set of all boundary edges and diagonals of a planar convex 2m-gon P . In [3] , the smallest sets of edges that meet all the simple perfect matchings (SPMs) in CGG(2m) (called "blockers") are characterized, and it is shown that all these sets are caterpillar graphs with a special structure, and that their total number is m · 2 m−1 . In this paper we characterize the coblockers for SPMs in CGG(2m), that is, the smallest sets of edges that meet all the blockers. We show that the co-blockers are exactly those perfect matchings M in CGG(2m) where all edges are of odd order, and two edges of M that emanate from two adjacent vertices of P never cross. In particular, while the number of SPMs and the number of blockers grow exponentially with m, the number of co-blockers grows super-exponentially. In this paper we consider convex geometric graphs (i.e., graphs whose vertices are points in convex position in the plane, and whose edges are segments connecting pairs of vertices), and in particular, the complete convex geometric graph on 2m vertices, denoted by CGG(2m).

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Bringing clinical pharmacogenomics information to pharmacists: A qualitative study of information needs and resource requirements Introduction: As key experts in supporting medication-decision making, pharmacists are well-positioned to support the incorporation of pharmacogenomics into clinical care. However, there has been little study to date of pharmacists' information needs regarding pharmacogenomics. Understanding those needs is critical to design information resources that help pharmacists effectively apply pharmacogenomics information. Objectives: We sought to understand the pharmacogenomics information needs and resource requirements of pharmacists. Methods: We conducted qualitative inquiries with 14 pharmacists representing 6 clinical environments, and used the results of those inquiries to develop a model of pharmacists' pharmacogenomics information needs and resource requirements. Results: The inquiries identified 36 pharmacogenomics-specific and pharmacogenomics-related information needs that fit into four information needs themes: background information, patient information, medication information, and guidance information. The results of the inquiries informed a model of pharmacists' pharmacogenomics resource requirements, with 3 themes: structure of the resource, perceptions of the resource, and perceptions of the information. Conclusion: Responses suggest that pharmacists anticipate an imminently growing role for pharmacogenomics in their practice. Participants value information from trust-worthy resources like FDA product labels, but struggle to find relevant information quickly in labels. Specific information needs include clinically relevant guidance about genotypes, phenotypes, and how to care for their patients with known genotypes. Information resources supporting the goal of incorporating complicated genetic information into medication decision-making goals should be well-designed and trustworthy. Pharmacogenomics research has identified over 130 gene variants that can alter the safety and/or effectiveness of drug treatments [1] . Some hospitals have begun using genotype information to inform drug and dose selection for several drugs including warfarin [2, 3] and clopidogrel [4, 5] . As genotyping becomes more common, pharmacists and physicians will encounter more pharmacogenomic information. To use this information to reduce the risks of adverse events and improve treatment effectiveness, clinicians will need information that will help them interpret complex interactions between patient genotypes, resulting phenotypes, and medications [6, 7] . Despite the critical role that pharmacists play in medication decision making, relatively little attention has been paid to date regarding their pharmacogenomicsrelated information needs and preferences. We present a study of pharmacists' information needs and resource requirements for pharmacogenomics-based decision making. This work will inform the development of information resources designed to help pharmacists meet the information challenges presented by pharmacogenomics. We are currently integrating the results of this study into the design and implementation of a clinically-oriented pharmacogenomics information resource that contains pharmacogenomics statements annotated from structured product labels. A laboratorybased task analysis [38, 43] will assess the utility of the tool in helping pharmacists locate and use information to make clinical decisions. As medication experts with a key role in medication decisionmaking, pharmacists are poised to lead the incorporation of pharmacogenomics information into clinical care. For a small set of drugs, the benefits of pharmacogenomics are clear [4, 5, 25, [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] and the pace of research in this field has been rapid [56] . Genotyping We identified perceived identified information needs, both general and specific to pharmacogenomics, relevant to pharmacists' use of pharmacogenomics information. These needs encompass both the information needed and the requirements for compendia and other resources containing these details. To our knowledge, this is the first study to explicitly identify pharmacogenomics-related information needs of pharmacists, and to discuss these needs in the context of preferences and requirements regarding pharmacy information resources. Perception of the information Easy to use 14 "I like it because it's easy. I usually go to certain ones that I'm familiar enough with, using them online, or in a book, or on my phone, so that I know where to retrieve the information quickly when I need it." -Pharmacist 10 "Up-To-Date TM , you can actually use to treat a patient. Like I have one patient who is going through alcohol withdrawal and you can kind of really look it up and figure out that they need. . .I think Up-To-Date TM is a lot more useful clinically." -Pharmacist 14 "I think it's quick. I think it's easy. And I think it's concise. All things that you want, when looking for information of this type. " -Pharmacist 10

O O B O O B O O O O B I O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O B B I O B O O O O O O O O O O O B O O B O O O O O O O B B I O O O O B O O O O O O O B O O O O O O O O O O O O O O O O B B B I O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B I O O O O O O O B B I O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The zero exemplar distance problem Given two genomes with duplicate genes, Zero Exemplar Distance is the problem of deciding whether the two genomes can be reduced to the same genome without duplicate genes by deleting all but one copy of each gene in each genome. Blin, Fertin, Sikora, and Vialette recently proved that Zero Exemplar Distance for monochromosomal genomes is NP-hard even if each gene appears at most two times in each genome, thereby settling an important open question on genome rearrangement in the exemplar model. In this paper, we give a very simple alternative proof of this result. We also study the problem Zero Exemplar Distance for multichromosomal genomes without gene order, and prove the analogous result that it is also NP-hard even if each gene appears at most two times in each genome. For the positive direction, we show that both variants of Zero Exemplar Distance admit polynomial-time algorithms if each gene appears exactly once in one genome and at least once in the other genome. In addition, we present a polynomial-time algorithm for the related problem Exemplar Longest Common Subsequence in the special case that each mandatory symbol appears exactly once in one input sequence and at least once in the other input sequence. This answers an open question of Bonizzoni et al. We also show that Zero Exemplar Distance for multichromosomal genomes without gene order is fixed-parameter tractable if the parameter is the maximum number of chromosomes in each genome. Given two genomes with duplicate genes, Genome Rearrangement with Gene Families [12] is the problem of deleting all but one copy of each gene in each genome, so as to minimize some rearrangement distance between the two reduced genomes. The minimum rearrangement distance thus attained is called the exemplar distance between the two genomes. For example, each of the following two monochromosomal genomes G 1 : −4 +1 +2 +3 −5 +1 +2 +3 −6 G 2 : −1 −4 +1 +2 −5 +3 −2 −6 +3 * Supported in part by NSF grant DBI-0743670. A preliminary version of this paper (including Theorem 1, Theorem 2, and a weaker version of Theorem 4) appeared in Proceedings of the 8th Annual RECOMB Satellite Workshop on Comparative Genomics (RECOMB-CG 2010) [11] .. has at most two copies of each gene, and each of the following two reduced genomes has exactly one copy of each gene. Recall that in the study of genome rearrangement, a gene is usually represented by a signed integer: the absolute value of the integer (the unsigned integer) denotes the gene family to which the gene belongs; the sign of the integer denotes the orientation of the gene in its chromosome. Then a chromosome is a sequence of signed integers, and a genome is a collection of chromosomes.. Genome Rearrangement with Gene Families is not a single problem but a whole class of related problems, because the choice of rearrangement distance is not unique. This choice becomes irrelevant, however, when we ask the fundamental question: Is the distance zero? In the example above, the two reduced genomes G ′ 1 and G ′ 2 are identical, thus the exemplar distance between the two original genomes G 1 and G 2 is zero for any reasonable choice of rearrangement distance.. In this paper, we study the most basic version of the problem Genome Rearrangement with Gene Families: Given two sequences of signed integers, Zero Exemplar Distance (for monochromosomal genomes) is the problem of deciding whether the two sequences have a common subsequence including each unsigned integer exactly once in either positive or negative form.. Due to its generic nature, the problem Zero Exemplar Distance has been extensively studied by several groups of researchers [5, 4, 2] focusing on different rearrangement distances, and, not surprisingly, has acquired several different names. Except for trivial distinctions, Zero Exemplar Distance is essentially the same problem as Zero Exemplar Conserved Interval Distance [5] , Exemplar Longest Common Subsequence (deciding whether a feasible solution exists) [4] , and Zero Exemplar Breakpoint Distance [2] .. It is easy to check that if only one of the two genomes has duplicate genes, then Zero Exemplar Distance can be solved in linear time: we simply need to decide whether the genome without duplicates is a subsequence of the genome with duplicates. In sharp contrast, if both genomes contain duplicate genes, then even if each gene appears at most three times in each genome, the problem Zero Exemplar Distance is already NP-hard, as shown independently in three papers [5, 4, 2] In Section 2, we give a very simple alternative proof of this theorem. Both the previous proof of Theorem 1 [3] and our alternative proof depend crucially on the order of the genes in the chromosomes. One may naturally wonder whether the complexity of Zero Exemplar Distance would change if gene order is not known. Note that genome rearrangement distances such as the syntenic distance [8] can be defined in the absence of gene order. Now model each chromosome as a set of unsigned integers instead of a sequence of signed integers. Then Zero Exemplar Distance for multichromosomal genomes without gene order is the following problem: Given two collections G 1 and G 2 of subsets of the same ground set S of unsigned integers, decide whether both G 1 and G 2 can be reduced, by deleting elements from subsets and deleting subsets from collections, to the same collection G ′ of subsets of S such that each unsigned integer in S is contained in exactly one subset in G ′ , i.e., G ′ is a partition of S. For example, S : {1, 2, 3, 4, 5}. In Section 3, we prove the following theorem analogous to Theorem 1: As decision problems, both variants of Zero Exemplar Distance, for monochromosomal genomes and for multichromosomal genomes without gene order, are in NP. Thus, following the NP-hardness results in Theorem 1 and Theorem 2, these two decision problems are both NPcomplete. Moreover, the NP-hardness results in Theorem 1 and Theorem 2 imply that unless NP = P, the corresponding minimization problems of computing the exemplar distance between two genomes do not admit any approximation. We refer to [5, 6, 4, 2, 1] for related results.. The problem Zero Exemplar Distance for monochromosomal genomes, as mentioned earlier, has been studied under several different names. Given two sequences A and B over an alphabet Σ = Σ 1 ∪Σ 2 , where Σ 1 is a set of mandatory symbols and Σ 2 is a set of optional symbols, Exemplar Longest Common Subsequence [4] is the problem of finding a longest common subsequence of A and B that contains all mandatory symbols in Σ 1 . For example, if Σ 1 = {1, 2, 3} and Σ 2 = {4, 5}, then C = 124355 is an exemplar longest common subsequence of the two sequences A = 12423545 and B = 1142443555.. Due to the strict requirement on mandatory symbols, Exemplar Longest Common Subsequence does not always have a feasible solution. It is not difficult to see that simply deciding whether a feasible solution to Exemplar Longest Common Subsequence exists for two sequences A and B is the same as the problem Zero Exemplar Distance for two monochromosomal genomes A ′ and B ′ obtained from A and B by deleting all optional symbols. Recall that the problem Zero Exemplar Distance for monochromosomal genomes becomes trivial when only one of the two genomes has duplicate genes. For the equivalent problem of deciding whether a feasible solution to Exemplar Longest Common Subsequence exists, Bonizzoni et al. [4] showed another tractable special case: If each mandatory symbol appears a total of at most three times in A and B, then there is a polynomial-time algorithm, based on 2SAT, that decides whether A and B have a common subsequence containing all mandatory symbols. This algorithm does not solve the maximization problem, however, and the following question was left open: Question 2 (Bonizzoni et al. [4] ). Is there a polynomial-time algorithm for Exemplar Longest Common Subsequence in the special case that each mandatory symbol appears a total of at most three times in the two input sequences?. Without loss of generality, we assume that each input sequence contains each symbol in the alphabet at least once. If each mandatory symbol appears a total of at most three times in the two input sequences, then it must appear exactly once in one sequence, and at least once in the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O