ABACUS: frequent pAttern mining-BAsed Community discovery in mUltidimensional networkS Abstract Community discovery in complex networks is the problem of detecting, for each node of the network, its membership to one of more groups of nodes, the communities, that are densely connected, or highly interactive, or, more in general, similar, according to a similarity function. So far, the problem has been widely studied in monodimensional networks, i.e. networks where only one connection between two entities may exist. However, real networks are often multidimensional, i.e., multiple connections between any two nodes may exist, either reflecting different kinds of relationships, or representing different values of the same type of tie. In this context, the problem of community discovery has to be redefined, taking into account multidimensional structure of the graph. We define a new concept of community that groups together nodes sharing memberships to the same monodimensional communities in the different single dimensions. As we show, such communities are meaningful and able to group nodes even if they might not be connected in any of the monodimensional networks. We devise frequent pAttern mining-BAsed Community discoverer in mUltidimensional networkS (ABACUS), an algorithm that is able to extract multidimensional communities based on the extraction of frequent closed itemsets from monodimensional community memberships. Experiments on two different real multidimensional networks confirm the meaningfulness of the introduced concepts, and Responsible editor: Inspired by real-world scenarios such as social networks, technology networks, the Web, biological networks, and so on, in the last years, wide, multidisciplinary, and extensive research has been devoted to the extraction of non trivial knowledge from networks. Predicting future links among the nodes or actors of a network (Bringmann et al. 2010) , detecting and studying the diffusion of information among them (Goyal et al. 2008; Mongiovì et al. 2012) , mining frequent patterns of nodes' behaviors (Benevenuto et al. 2009; Yan and han 2002; Cook et al. 2010) , are only a few examples of tasks in the field of complex network analysis, that includes, among all, physicians, mathematicians, computer scientists, sociologists, economists and biologists. The data at the basis of this field of research is huge, heterogeneous, and semantically rich, and this allows to identify many properties and behaviors of the actors involved in a network. One crucial task at the basis of complex network analysis is community discovery, i.e., the discovery of a group of nodes densely connected, or highly related. Many techniques have been proposed to identify communities in networks (Leskovec et al. 2010b; Fortunato 2010) , allowing the detection of hierarchical connections, influential nodes in communities, or just groups of nodes that share some properties or behaviors. In order to do so, the connections among the nodes of a network were so far posed at the center of investigation, since they play a key role in the study of the network structure, evolution, and behavior.. Nowadays, most of the work done in the literature is limited to a very simplified perspective of such relations, focusing only on whether two nodes are connected or not, and possibly assigning a strength to this connection. In the real world, however, this is not always enough to model all the available information about the interactions between actors, including their multiple preferences, their multifaceted behaviors, and their complex interactions. While multiple types of connections among actors could still be represented into a monodimensional network, by collapsing all connections to one type and potentially affecting a measure of tie strength, a more sophisticated analysis of the network structure, which could maintain information on the semantic differences in how actors are connected, would help all the techniques to provide more meaningful communities.. To this aim, in this paper we deal with multidimensional networks, i.e. networks in which multiple connections may exist between a pair of nodes, reflecting various interactions (i.e., dimensions) between them. Multidimensionality in real networks may be expressed by either different types of connections (two persons may be connected because they are friends, colleagues, they play together in a team, and so on), or different quantitative values of one specific relationship (co-authorship between two authors may occur in several different years, for example). In this paper, we have addressed the problem of multidimensional community discovery. We have given a definition of multidimensional community for which nodes sharing memberships to the same monodimensional communities in the different single dimensions are grouped together. This leads us to define a community extractor combining the use of -A given monodimensional community discovery algorithm (that could also allow for overlapping communities) -Frequent itemset pattern mining to allow merging discovered monodimensional communities into multidimensional ones By browsing over the lattice generated by the frequent closed itemset mining algorithm, it is possible to extract multidimensional communities of different sizes (pattern lengths) and so navigate the complex multidimensional structure of a network, in a way that previous methods could not permit. The proposed method could lead to the development of analytical tools to characterize the redundancy in the dimensions, the impact of new dimensions on the network structure, and more in general to evaluate the interplay between dimensions. For these reasons, we see potential applications in real world problems including characterizing the interplay between mobility and communication dimensions in a place-to-place network (Calabrese et al. 2011) , the similarity between users in a user-mobility profile network (Trasarti et al. 2011) , or in the analysis spreading of infectious diseases (Balcan et al. 2009 ). We leave for future research the analysis of potential applications of ABACUS. Detecting communities in networks has been studied from many angles. Two comprehensive surveys on the topic can be found in Fortunato (2010) and Leskovec et al. (2010b) . From one side, a community has been defined as a set of nodes with a high density of links among them, and sparse connections with nodes outside the community. The papers working with this quantitative definition rely on information theory principles (Papadimitriou et al. 2008) or on the notion of modularity (Clauset et al. 2004) , which is a function defined to detect the ratio between intra-and inter-community number of edges. Modularity is widely used in many works, and several algorithms have been proposed to extract high modularity partitioning of a network: one of them is a greedy optimization able to scale up to networks with billions of edges (Blondel et al. 2008) . From another side, communities have been approached looking at the statistical properties of the graph. In Gregory (2009) , a framework for the detection of overlapping communities, i.e. communities allowing the vertices to be in more than one community, is presented. The framework is based on the "split betweenness" concept: vertices and edges are ranked by their betweenness centrality (the portion of shortest path in which they appear) and then split in order to form a transformed network, where classical algorithms can be used to detect communities. The resulting communities are then merged in order to find overlaps. Another class of approaches relies on the propagation in the network of a label (Raghavan et al. 2007) or a particular definition of structure (usually a clique Palla et al. 2005) . The first approach is known for being a quasi linear solution for the problem, the second one allows to find overlapping communities. One algorithm that maximizes quality and quantity measures on its results is InfoMap (Rosvall and Bergstrom 2008) , a random walk-based algorithm. An emerging novel problem definition can be found in Ahn et al. (2010) , in which the authors state that community discovery algorithms should not group nodes but edges, emphasizing the role of the relation residing in a community. Previously described methods have focused on both unweighted or weighted graphs, but still considering the network as a monodimensional entity. Only since recently, multidimensionality has started to be taken into account in network analysis. A few examples of studies are: link prediction in networks with positive and negative links (Leskovec et al. 2010a) or in multidimensional networks (Rossetti et al. 2011 ); a statistical analysis over different kinds of relations in the same network in an online game community (Szell et al. 2010) ; analysis of structural properties of multidimensional networks (Berlingerio et al. 2011a (Berlingerio et al. , 2012 and its applications to multidimensional hub analysis (Berlingerio et al. 2011b) .. From a community discovery point of view, to the best of our knowledge, the main approaches to take into account multiple dimensions are three. In Mucha et al. (2010) the authors extend the definition of modularity to fit the multidimensional case, which they call "multislice". However, no definition of "multidimensional community" is provided, nor the approach characterizes and analyzes the communities found. Instead, the authors use the multidimensional information to extract monodimensional communities. In Tang and Liu (2009) the authors create a machine learning procedure which detects the possible different latent dimensions among the entities in the network and uses them as features for the node classification algorithm. In other words, they use the multidimensional labels of some nodes to infer labels for other nodes, by means of edge clustering. Hence, multidimensionality is only present here in terms of node labels, but the input network is not multidimensional according to our definition (see Sect. 3), and the output is not in the form of multidimensional communities. In Berlingerio et al. (2011) , a possible formulation of community discovery and characterization in multidimensional networks was given. A new measure was introduced to capture the interplay among the dimensions, that makes multidimensional communities emerge even where the connections among nodes reside in different dimensions. In this paper, we approach the problem from a similar angle, but focus on extract-ing communities using frequent itemset mining, and giving a semantic description to each multidimensional community as the subset of dimensions used to characterize it. Resulting multidimensional communities may be different from the ones extracted in Berlingerio et al. (2011) and are navigable using the lattice extracted in the frequent itemset mining process.. Another work that deals with networks containing heterogenous information, but not multiple dimensions, is presented in Sun et al. (2009) , where the authors propose a method to generate net-clusters using links across multi-typed objects. This approach works on heterogeneous networks, i.e. networks where nodes may have different types (e.g. papers or authors), and does not deal with multidimensional networks, i.e. networks where edges may be of different types and two nodes may be connected by multiple edges.. The authors of Cai et al. (2005) studied the problem of community mining in multirelational networks. The problem setting, however, is different: the authors exploit the multi-relational links to evaluate the importance of the relations based on labeled examples, provided by a user as queries. Hence, they do not perform community detection, but rather extract the importance of each dimension for a given node, in the form of a weight.. The idea of applying closed frequent pattern mining to multi-relational data is not new. In Cerf et al. (2009a) , the authors extract all closed n-sets satisfying given piecewise (anti-)monotonic constraints, from n-ary relations. In Nijssen et al. (2011) , the authors presented a framework for constraint-based pattern mining in multi-relational databases, finding patterns not under (anti-)monotonic and closedness constraints, expressed over complex aggregates over multiple relations. However, both the works solve the technical problem of finding the frequent closed patterns, but do not apply this technique to the setting of multidimensional network analysis.. There are other works in the literature that deal with the extraction of knowledge across networks. In Wang et al. (2006) and Zeng et al. (2006) , for example, the authors deal with the problem of finding cross-graph quasi-cliques. This problem can be seen as a sub-problem of the one we deal with in this paper. However, our concept of community is independent from the density of the connections among the nodes. Other two papers (Cerf et al. 2009b; Mougel et al. 2012 ) deal with the extraction of cliques with particular constraints: in the first work, the authors search cliques that remain cliques over time; in the second, cliques with homogeneous node attributes are found. They however do not deal with the community discovery problem.. Based on all the above, we believe that two approaches may be considered really related to our problem formulation, namely Mucha et al. (2010) and Berlingerio et al. (2011) , thus we use these as baselines for comparison in Sect. 5. We performed our experiments following three questions related to our problem: Q1. Quantitative evaluation: given the high number of resulting communities, how can we easily reduce the patterns to select only a set of meaningful ones? Q2. Are there relational dependencies between our concept of communities and structural properties of them? Q3. Qualitative evaluation: among the communities found, are there any relevant ones? Can we reason on the multidimensional density of the connections within the communities?. In order to answer the above questions, we define a simple and easy to compute measure of connectedness within communities. The multidimensional community density (MCD) is then the number of edges in a community normalized by the maximum possible for that community, or, in formula:. where ndim is the number of different dimensions found in the community. Let us answer Q1. The frequent pattern mining literature reports that the problem of finding few relevant patterns to be interpreted, among the many returned, is hard (Bringmann and Zimmermann 2007; Günnemann et al. 2010) . We can overcome this problem in three different ways. First, we can look at the distributions of the MCD (defined above), the support of the patterns, and the size of the itemsets to focus our search towards the communities that we consider relevant, depending on the final application. Figures 7 and 8 report the mentioned distributions (we report the cumulative versions, to be able to use the three measures as straightforward filters). For better comparison, we reported on the y axes the percentage of communities with values of the measures greater than a certain thresholds. However, the absolute number of communities can be used to choose, depending on the application, the best support, size of the itemset, and MCD to select only the relevant communities. Second, more generally speaking, the entire constraint-based frequent pattern mining literature can be applied in our scenario at running stage, to drive the search to fewer, more focused, patterns (Pei et al. 2001; Pei and Han 2000; . For example, we may want only patterns including or excluding a specific dimension, or patterns including dimensions with specific properties (e.g., at least 1,000 authors). To this extent, it is worth noting that MCD is neither (anti-)monotone, nor convertible, nor loose-antimonotone. We leave for further research the definition of meaningful, application-driven constraints, and their effect to the results. Lastly, the authors of Bringmann and Zimmermann (2007) present another methodology for selecting few interesting patterns among many, which is not based on constraints. We believe that this technique may be also used, and we plan to investigate this opportunity in the future. To answer Q2 we check whether MCD is correlated to other structural properties of the nodes of the communities. For example, one possible intuition is that communities with low density may group together nodes that were at the borders of the monodimensional communities. To study this, we computed the closeness centrality for each node and for each dimension, and checked the correlation between the centrality and the density. We did not find any clear sign of direct correlation. We checked also for correlation with PageRank, the degree centrality and the betweeness centrality, for which again we did not have signs of correlation. Based on these results, we believe that MCD is yet another measure to be used to filter the results towards more focused results.. Lastly, in order to answer Q3, we extracted a few communities either minimizing or maximising MCD. In the remainder, we call MCS the number of nodes in a community. As we have stated above, we can use the distributions of size, support and MCD to post-process the results to get only the few interesting ones. We have extracted a few (i.e., 200) communities for each network, and we report in Fig. 9 four of them. Besides the first example, that was found by searching for one of the co-authors of this paper, the other ones were found by examining the results filtered by means of the above mentioned three measures. In particular: Fig. 9b Consider the one in Fig. 9a . We discovered a size-4 community connecting FP, FG, MN and DP with dimensions set {K DD, G I S, S AC, S E B D}. It is interesting to observe that, given its very dense connections, this multidimensional community would have been found also by using the methods proposed in Berlingerio et al. (2011) and Mucha et al. (2010) .. However the method proposed in this paper has the possibility to discover more complex interactions between dimensions. Indeed, the lattice can be used to browse the multidimensional communities by selecting different dimensions sets. To give an example we extracted a size-3 community composed of authors AA, PN, QC, KR and SG with connections in dimensions set {I O LT S, D AT E, I SL P E D}, see Fig. 9b where the different multidimensional memberships are shown. These authors are part of three monodimensional communities, but have not co-authored papers at these three conferences (there are no links connecting them). By adding the dimension I CC D (solid line circle), we are able to extract a size-4 community composed of the first three authors. This fourth dimension includes a paper co-authored by the three authors, which resulted in a I CC D-monodimensional community formed by the three nodes. Interestingly, through this dimension we are able to specialize the previously discovered five authors community. Note that by using the methods proposed in and Mucha et al. (2010) it would not be possible to discover how the I CC D dimension could specialize the community, and so its semantic meaning. This is due to the fact that more information is included in the results w.r.t. the mentioned works.. Similar results can be obtained by applying ABACUS to the Query Log dataset. Finding a community of words in this network means finding a set of words typically used together in queries that lead to good or bad results. A set of words found together only in dimension one is a set of words that, used together in a query, lead to very specific results (users clicked on the first result). Words found together only in dimension five, on the other hand, lead to lower ranked results. If we find words together in different dimensions, it may mean that either the concept the users were looking for is only representable by words used in conjunction, or that they need more terms to be disambiguated. An example of the first kind is shown in Fig. 9c , where, maximising the MCD, we were able to detect a highly connected multidimensional community where the words Machu, Picchu, and unbelievable are connected in three different dimensions of the dataset. An example of the second kind, on the other hand, is shown in Fig. 9d . In this example, we can observe that, if we consider all the dimensions, we obtain a set of words belonging to the same multidimensional community with a strong intrinsic semantic correlation (i.e. Pablo, Picasso, Neruda-besides sharing their first name, there exists an edition of a book from Neruda with a Picasso painting on the cover), removing, then, the most specific dimension (Bin 1-i.e. click on the first returned result) we include words that make the concept broader. Also in this case, the methods proposed in Berlingerio et al. (2011) and Mucha et al. (2010) do not allow to investigate the effect of the different dimensions on the specialization of the communities and, thus, the intrinsic semantic correlation among different words.

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Computer simulation of electron energy state spin-splitting in nanoscale InAs/GaAs semiconductor quantum rings In this paper, we model and computationally investigate the effect of spin-orbit interaction on the electron energy spectra for nanoscale semiconductor quantum rings. Our threedimensional mathematical model considers the effective one-electron band Hamiltonian, the energy-and position-dependent electron effective mass approximation, and the spin-dependent Ben Daniel-Duke boundary conditions. The nonlinear iterative method is applied to solve the corresponding nonlinear eigenvalue problem, which converges monotonically for all energy states. Physically, it is found that the spin-dependent boundary conditions lead to a spin-splitting of the electron energy states with non-zero angular momentum in nanoscale InAs/GaAs quantum rings. The spin-splitting is strongly dependent upon the dimension of the explored quantum ring and is dominated by the inner radius, the base radius, and the height of the quantum ring. Under zero magnetic fields, the spin-splitting energy is decreased when the radius is increased. Meanwhile, it is greater than that of the InAs/GaAs quantum dot and demonstrates an experimentally measurable quantity (up to 2 meV) for relatively small semiconductor quantum rings. Nanoscale semiconductor quantum structures have recently been of great interest [1] [2] [3] [4] . Among these structures, quantum rings are known to possess interesting physical properties and have potential applications to the field of nanoelectronics and optics [5] [6] [7] [8] [9] [10] . Electron spin plays an important role in the manipulation of energy states and modifies the intrinsic property of nanoscale quantum rings. In semiconductor spintronics, the spin-dependent electron confinement and the spin-orbit interaction impact the energy and electronic properties of semiconductor nanostructures [11] [12] [13] [14] [15] [16] [17] [18] [19] . Different theoretical models have been proposed to study the spin-orbit interaction for various quantum structures [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] . Three-dimensional (3D) modeling and computer simulation will accurately estimate effects of geometry and spin-orbit interactions on the energy state in nanoscale semiconductor quantum rings.. In this paper, we computationally explore the effect of spin-orbit interaction on the electron energy states in nanoscale semiconductor quantum rings. The effective one-band Hamiltonian approximation with the spin-dependent Ben Daniel-Duke boundary conditions is firstly formulated, where a hard-wall 3D confinement potential (i.e., a barrier with finite height) is considered. The hard-wall confinement is physically induced by a realistic discontinuity of the conduction bands at the edge of a semiconductor quantum ring. To efficiently solve the corresponding nonlinear Schrödinger equation, the nonlinear iterative method [24] [25] [26] is applied to calculate the energy states and then wavefunctions are solved numerically. Based upon the monotone property of the electron's effective mass with respect to each energy state, it can be shown that the numerical algorithm converges to each energy state monotonically. This solution methodology was developed in our recent work for the simulation of semiconductor quantum nanostructures [23] [24] [25] [26] [27] . A mathematical proof of the convergence property of the nonlinear iterative method in semiconductor quantum dot simulation has also been reported recently [28] . It is robust and cost-effective in the numerical simulation of the spin-dependent semiconductor quantum ring. Due to a significant spin-orbit interaction in the nonsimply connected torus topology, an experimentally measurable spinsplitting is computationally observed in nanoscale InAs/GaAs quantum rings. The spin-splitting depends on the variations of geometrical dimension which are dominated by the inner radius, the base radius, and the height of the quantum ring. Under zero magnetic fields, it is found that the examined InAs/GaAs quantum ring can approximately produce 2 meV spin-splitting of the excited electronic states when the inner radius is small and the ratio of the height and the base radius is greater than 1. It is substantially greater than that of the InAs/GaAs quantum dot (about 1 meV) [10, 15] .. This article is organized as follows. Section 2 introduces the 3D mathematical model and the corresponding nonlinear Schrödinger equation for the semiconductor quantum ring. Section 3 describes the numerical simulation method. Section 4 reports the computed results illustrating the effect of spin-orbit interaction on the electron energy spectra for the small InAs/GaAs quantum rings. Section 4 draws conclusions. We have numerically studied the effect of spin-orbit interaction on the electron energy states for ultrasmall semiconductor quantum rings. It is found that the spin-orbit interaction can significantly modify the electron energy spectrum of InAs/GaAs semiconductor quantum rings. Under zero magnetic fields, an ultrasmall InAs/GaAs quantum ring approximately produces 2 meV spin-splitting of the excited electronic states which is substantially greater than that of a quantum dot (1 meV). The modeling and simulation presented here plays a starting point for estimation of the spin-orbit interaction effects in 3D semiconductor quantum rings. The spin-orbit interaction provides rich physics and is promising in nanoelectronic applications. We note that the effect of band offset V 0 on the energy state could be explored in the near future. Multiband and self-consistent potentials will also provide more accurate corrections.

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


Bayesian winner-take-all reconstruction of intermediate views from stereoscopic images This paper presents a new algorithm for the reconstruction of intermediate views from a pair of still stereoscopic images. The algorithm is designed to address the issue of blur caused by linear ltering (averaging) often employed in such reconstruction. The proposed algorithm is block-based and to reconstruct the intermediate views employs non-linear disparity-compensated ltering via a winner-take-all strategy. The reconstructed image is modeled as a tiling by xedsize blocks coming from various positions (disparity compensation) of either the left or right images, while the tiling map itself is modeled by a binary decision eld. In addition to that, an observation model relating the left and right images via a disparity eld and a prior model (disparity smoothness) are used. All models are probabilistic and are combined into a maximum a posteriori probability (MAP) criterion. The intermediate intensities, disparities and the binary decision eld are estimated jointly using the expectation-maximization (EM) algorithm. The new approach is compared experimentally on complex natural images with a reference block-based algorithm employing linear ltering. Although the improvements are very localized and often subtle, they demonstrate that a very high-quality intermediate view reconstruction for complex scenes is feasible.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 I O O O O O


SPoRE:

a

mathematical

model

to

predict

double

strand

breaks

and

axis

protein

sites

in

meiosis

Background:

Meiotic

recombination

between

homologous

chromosomes

provides

natural

combinations

of

genetic

variations

and

is

a

main

driving

force

of

evolution.

It

is

initiated

via

programmed

DNA

double-strand

breaks

(DSB)

and

involves

a

specific

axial

chromosomal

structure.

So

far,

recombination

regions

have

been

mainly

determined

by

experiments,

both

expensive

and

time-consuming.SPoRE

is

a

mathematical

model

that

describes

the

non-uniform

localisation

of

DSB

and

axis

proteins

sites,

and

distinguishes

high

versus

low

protein

density.

It

is

based

on

a

combination

of

genomic

signals,

based

on

what

is

known

from

wet-lab

experiments,

whose

contribution

is

precisely

quantified.

It

models

axis

proteins

accumulation

at

gene

with

a

discrete

approximation

of

their

diffusion

and

convection

along

genes.

It

models

DSB

accumulation

at

approximated

gene

promoter

positions

with

intergenic

region

length

and

GC-content.

SPoRE

can

be

used

for

prediction

and

it

is

parameterised

in

an

obvious

way

that

makes

it

easy

to

understand

from

a

biological

viewpoint.When

compared

to

Saccharomyces

cerevisiae

experimental

data,

SPoRE

predicts

axis

protein

and

DSB

positions

with

high

sensitivity

and

precision,

axis

protein

density

with

an

average

local

correlation

r

and

DSB

density

with

an

average

local

correlation

r

SPoRE

outbreaks

previous

DSB

predictors,

which

are

based

on

nucleotide

patterning,

and

it

reaches

of

success

rate

in

DSB

prediction

compared

to

obtained

by

available

tools

on

a

benchmarked

dataset.

SPoRE

is

available

at

the

address

We

explored

the

hypothesis

that

genomic

signals

allow

us

to

predict

DNA

double-strand

breaks

and

the

formation

of

the

loops

(their

position

and

length)

in

the

chromosomal

structure

during

meiosis.

Our

aim

here

is

not

to

study

the

dynamics

of

a

protein

localization

process

but

rather

to

identify

the

genomic

information

that

can

be

used

to

predict

the

structure

formation

and

quantify

the

importance

of

these

predictive

factors.

SPoRE

allows

us

to

test

whether

genomic

signals

are

good

predictive

variables

or

not,

and

to

what

extent,

in

the

accumulation

of

axis

proteins

and

DSBs

along

chromosomes..

However,

it

should

be

noted

that

this

does

not

imply

that

the

factors

are

the

cause

of

DSBs

and

axis

proteins

positioning.

For

example,

GC-content

could

be

a

consequence

rather

a

cause

of

DSBs

In

both

cases

however,

it

is

a

useful

factor

for

predicting

DSB

hotspots..

All

genomic

factors

considered

in

the

model

are

linear

functions

with

the

exception

of

a

quadratic

factor

modeling

the

impact

of

GC

content.

New

parameters

can

be

easily

added

to

the

model

for

the

evaluation

of

new

genomic

markers

effects.

The

interest

in

this

modeling

approach

comes

from

a

straightforward

biological

interpretation

of

the

parameters

that

helps

to

reason

on

plausible

biological

mechanisms

forming

protein

accumulation.

In

sexually

reproducing

eukaryotes,

the

production

of

gametes

relies

on

meiosis,

during

which

a

diploid

cell

is

divided

into

four

haploid

cells.

A

critical

step

is

homologous

recombination

between

homologous

chromosomes,

in

which

both

crossover

and

non-crossover

events

occur,

resulting

in

a

different

gene

content

of

the

offspring

chromosomes.

This

allows

evolution

to

explore

different

allelic

combinations

through

recombinations

that

are

formation

of

this

structure

is

due

to

bonding

of

cohesin

and

several

other

proteins

on

specific

DNA

sites,

and

to

their

assembly

in

a

protein

complex

forming

an

axis

The

DNA

lying

outside

protein

binding

sites

forms

loops

along

the

axis

Axes

of

homologous

chromosomes

are

themselves

bound

to

each

other

by

transversal

filaments

made

of

in

S.

cerevisiae,

in

mammals)

proteins,

forming

the

so-called

synaptonemal

complex..

In

S.

cerevisiae,

the

structural

axis

is

formed

by

cohesin

and

proteins

It

has

been

shown

that

proteins

and

bind

to

DNA

at

axis

sites,

rather

than

to

loop

sequences,

and

that

their

association

depends

on

and

In

turn,

deposition

depends

on

cohesin

in

certain

regions,

but

not

in

others,

and

in

both

cases,

their

local

distribution

is

very

similar

The

global

correlations

are

r

between

and

r

between

and

and

r

between

and

Also,

cohesin

(and

density

is

higher

in

convergent

regions,

that

is

intergenic

regions

characterized

by

two

gene

ends,

and

is

correlated

with

AT-content

but

no

specific

cohesin-DNA

binding

motif

has

been

identified.

density

is

locally

negatively

correlated

with

DSB

density

(see

Figure

two

E

in

Many

questions

remain

open

on

the

chromosomal

axis

formation.

Can

we

model

the

accumulation

of

axial

proteins

and

DSBs

on

the

chromosomes

entirely

from

genomic

factors?

If

so,

what

is

the

contribution

made

by

each

one

of

them?.

A

number

of

genomic

markers

(like

gene

start,

gene

end,

GC-richness)

associated

to

high-density

sites

have

been

highlighted

by

experiments

but,

once

combined

together,

how

well

do

they

explain

the

frequency

of

occurrence

of

proteins

or

DSBs

in

a

given

site

along

the

axis?

In

other

words,

can

we

create

a

mathematical

model,

based

on

genomic

markers,

and

a

tool

that

can

predict

axial

proteins

and

DSB?

With

a

rapid

increase

of

sequenced

genomes,

it

is

highly

desirable

to

develop

automated

methods

for

timely

identifying

recombination

hotspots..

Computational

predictions

of

recombination

hotspots

have

been

based

on

nucleotide

sequence

content.

These

approaches

take

into

account,

more

or

less

explicitly,

sequence

order

effects.

But

the

accuracy

of

these

algorithms

still

needs

further

improvement

and

more

formal

mathematical

approaches,

considering

mechanisms

of

chromatin

remodeling

during

meiosis,

could

be

a

possible

way

to

proceed..

Here,

we

provide

a

mathematical

framework,

called

SPoRE

for

"SPots

of

REcombination",

that

allows

us

to

model

axis

proteins

and

DSB

localization

and

density

along

chromosomes.

SPoRE

models

axis

proteins

and

DSB

distributions

based

on

genomic

information.

For

axis

proteins,

SPoRE

uses

gene

stop

codon

positions

and

gene

lengths

as

its

only

input,

while

for

DSBs,

it

uses

the

order

of

the

genes

defining

intergenic

regions,

intergenic

region

lengths

and

GC-richness.

Based

on

these

genomic

markers,

SPoRE

models

S.

cerevisiae

experimental

data

for

and

distributions

accurately.

We

used

SPoRE

to

make

predictions

on

three

more

yeast

species,

Lachancea

kluyveri,

Kluyveromyces

lactis

and

Schizosaccharomyces

pombe.

Finally,

we

compared

it

to

available

tools

predicting

DSB

hotspots

and

coldspots,

and

demonstrated

its

higher

performance.

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 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 B O O O O O O O O O O O O O O O O O O B 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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 B 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 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 O O O O O O O O O O O O O O O O O O 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 B O O O O B 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


Optimizing

metric

access

methods

for

querying

and

mining

complex

data

types

Background:

There

are

several

application

scenarios

that

can

take

advantage

from

the

efficient

processing

of

similarity

operations

in

complex

data

types,

such

as

multimedia

data.

Among

them,

it

is

possible

to

mention

the

execution

of

more

complex

query

types

(e.g.,

similarity

queries)

and

several

well-known

data

mining

algorithms

(e.g.,

data

clustering)

that

are

directly

based

on

similarity

computations.

In

order

to

speed

up

the

similarity-based

comparisons

performed

by

these

approaches,

it

is

possible

to

store

the

dataset

in

specialized

data

structures

known

as

metric

access

methods

(MAM).

Methods:

In

this

article

we

present

four

node

split

policies

that

can

be

employed

in

the

construction

of

M-tree,

the

pioneer

dynamic

MAM,

and

of

Slim-tree,

the

M-tree

successor.

Results:

These

policies

allow

faster

tree

construction,

as

they

result

in

better

distribution

of

elements

on

the

tree

nodes

and

require

less

distance

calculations

when

compared

with

the

previously

proposed

ones.

Furthermore,

trees

built

with

these

policies

have

shown

to

be

more

efficient

for

techniques

that

require

similarity

computations,

such

as

nearest

neighbors

queries

and

data

clustering

algorithms.The

experimental

results

show

that

trees

built

with

the

proposed

policies

outperform

those

built

with

the

original

ones

with

regard

to

the

number

of

disk

accesses,

the

amount

of

distance

calculations,

and

the

time

required

to

run

the

queries.

When

thinking

of

supporting

both

querying

and

mining

large

complex

datasets

in

an

integrated

environment,

it

is

important

to

provide

means

not

only

to

built

efficient

MAM

but

also

MAM

that

do

not

degrade

the

effectiveness

of

mining

processes.

The

four

node

split

policies

presented

herein,

for

M-tree

and

Slim-tree,

meet

these

requirements..

Our

policies

are

guided

by

the

observation

that

a

node

created

after

a

split

with

few

bytes

left

for

newly

promoted

elements

or

new

elements

has

higher

probability

of

being

split

again,

resulting

in

the

increase

of

the

overlap

among

the

nodes.

The

experiments

show

that

the

new

split

policies

build

more

efficient

trees

in

less

time.

Considering

the

COPhIR

dataset,

for

example,

when

compared

to

MinMax,

the

RE

policy

required

less

build

time

and

fewer

distance

computations

to

process

nearest

neighbor

queries.

Even

though

reducing

the

time

spent

to

build

the

trees,

the

lower

resultant

fat-factor

may

benefit

query

execution

by

reducing

the

number

of

distance

calculations

and

disk

accesses

to

run,

for

instance,

nearest

neighbor

queries.

In

addition

to

the

gain

in

efficiency

also

observed

for

data

clustering,

the

effectiveness

of

the

resulting

clustering

remained

the

same

for

all

the

evaluated

policies.

Among

future

works

we

intend

to

combine

the

proposed

policies

with

dynamic

reinsertion

and

compare

with

the

other

optimization

strategies

for

efficient

MAM

construction

mentioned

in

this

article.

Over

the

past

two

decades,

several

works

have

pursued

the

development

of

strategies

aiming

to

reduce

the

overlap

among

nodes

in

dynamic

MAM

construction.

Examples

of

works

that

dealt

with

this

issue

proposing

node

splitting

policies

are

The

work

presented

in

deals

with

node

splitting

by

applying

MinMax

in

the

construction

of

the

M-tree

MAM.

This

algorithm

finds

the

pair

of

elements

that

result

in

smaller

node

coverage

requiring

O(n

distance

calculations

on

the

number

of

elements

in

a

node.

A

more

efficient

algorithm

was

later

employed

by

the

Slim-tree

MAM

Its

strategy

consists

in

employing

the

minimum

spanning

tree

(MST)

algorithm

as

a

node

split

policy,

reducing

the

complexity

to

O(n

log(n))

distance

calculations.

In

addition

to

that,

the

work

proposed

in

also

defines

an

evaluation

metric

to

measure

the

overlap

among

nodes

(called

fat-factor)

and

presents

an

algorithm

to

reduce

this

overlap

(called

Slim-down)..

Another

split

policy

for

the

M-Tree

MAM

restricted

to

multidimensional

data

was

presented

in

It

is

based

on

the

choice

of

central

elements

from

partitioned

regions

to

promote

as

node

representatives,

which

results

in

smaller

overlaps.

As

it

uses

the

k-means

algorithm

to

compute

these

representatives,

it

cannot

be

employed

in

generic

metric

spaces..

CM-tree

adopts

a

different

strategy

when

splitting

a

node

It

employs

a

clustering

approach

that

splits

a

node

in

two

or

possibly

more

nodes,

recursively

propagating

the

updates

or

splits

up

to

the

root.

It

also

stores

a

distance

matrix

in

each

node

with

the

pairwise

distances

of

each

pair

of

elements.

During

a

search,

the

algorithm

prunes

branches

based

on

both

the

distance

matrix

values

and

the

triangle

inequality.

The

price

paid

to

store

the

matrix

in

each

node

is

compensated

by

not

computing

these

distances,

especially

if

a

data

element

occupies

a

large

amount

of

bytes

and

if

the

distance

function

is

expensive..

Other

approaches

that

have

been

explored

to

build

more

efficient

MAM

include

The

work

of

presents

two

new

techniques

to

allow

dynamic

insertions

to

MTree.

The

first

is

a

forced

reinsertion

strategy

that

avoids

the

split

of

a

leaf

node

that

reached

its

storage

capacity.

The

algorithm

removes

some

elements

from

the

overflowed

node

and

reinserts

them,

hoping

these

elements

will

be

placed

into

more

suitable

leaves.

Finally,

if

the

elements

are

reinserted

in

the

same

node,

the

node

is

split.

The

second

strategy

works

on

leaf

selection.

Instead

of

a

single-way

descent

to

the

leaf

(that

may

not

find

the

optimal

node

to

hold

an

element)

and

the

multi-way

leaf

selection

(that

may

result

in

linear

scan),

the

authors

propose

a

hybrid-way

algorithm

that

finds

the

best

candidates

at

each

level,

reducing

the

traversal.

The

drawback

of

this

strategy

is

that

it

increases

both

the

number

of

distance

calculations

and

disk

accesses

during

the

insertion.

During

query

execution,

both

techniques

result

in

the

reduction

on

the

number

of

disk

accesses

and

distance

calculations

when

compared

to

M-tree.

Among

the

factors

that

influenced

these

reductions

are

the

increase

of

node

occupation

and

the

decrease

of

node

overlap..

The

use

of

a

bulk

load

operation

based

on

a

static

dataset

to

build

an

optimized

Slim-tree

is

proposed

in

It

builds

the

MAM

hierarchy

of

a

dataset

in

a

topdown

approach

with

reduced

node

overlapping,

resulting

in

better

query

performance.

The

method

may

be

used

to

recreate

a

low-performance

index.

The

latter

work

mentioned

above

presents

EGNAT,

a

disk-based

MAM

that

indexes

data

using

hyperplanes

instead

of

ball

partitioning,

resulting

in

an

unbalanced

structure

with

no

overlapped

regions.

Compared

to

M-tree,

EGNAT

accesses

more

disk

pages

at

search

time,

but

on

the

other

hand,

computes

fewer

distances.

A

summary

of

the

main

features

of

the

works

described

previously

is

presented

in

Table

The

development

of

approaches

to

make

data

clustering

algorithms,

which

are

based

on

similarity

comparisons,

feasible

for

large-scale

datasets

has

also

been

pursued

for

the

data

mining

research

community

over

the

last

decades.

Among

them,

the

use

of

sampling

techniques

has

proved

to

be

especially

useful

for

data

clustering

methods

that

perform

several

iterations

considering

different

initializations,

such

as

CLARANS

and

PAM

Variants

of

these

algorithms

have

been

proposed

in

respectively.

In

these

works

the

algorithms

focus

on

relevant

parts

of

the

dataset

as

the

clustering

process

is

performed

on

a

sample

dataset

obtained

from

pages

of

specialized

data

structures

that

allow

the

indexing

of

multidimensional

data.

The

former

uses

the

R*-tree

and

the

latter

employs

the

Slim-tree

to

develop

the

algorithm

called

PAM-SLIM..

The

strategy

adopted

by

the

PAM-SLIM

algorithm,

for

example,

is

based

on

the

assumption

that

each

level

of

the

Slim-tree

indirectly

divides

the

data

space

into

a

number

of

clusters

equal

to

the

number

of

elements

stored

at

each

level.

The

representative

elements

stored

in

the

index

nodes

summarizes

the

information

about

the

elements

in

the

tree

low

levels

and,

thus,

they

can

be

viewed

as

approximate

cluster

centers

on

each

level

of

the

tree.

This

information

is

employed

by

PAM-SLIM

to

compose

the

sample

dataset

to

cluster

the

data.

It

is

important

to

note

that

the

quality

of

the

sample

dataset

is

affected

by

the

features

of

the

tree

regarding

the

distribution

of

the

elements

on

the

tree

nodes.

Since

the

beginning

of

commercial

computer

systems

half

a

century

ago,

the

increase

in

the

amount

of

data

demanded

the

development

of

systems

tailored

to

efficiently

store,

represent,

and

manipulate

them.

Once

the

Database

Management

Systems

(DBMS)

emerged

as

a

solution

to

this

question,

the

researchers

turned

their

attention

to

another

important

question:

how

to

use

the

gathered

data

to

get

valuable

information.

In

response

to

this,

several

data

mining

approaches

were

proposed

to

couple

with

this

issue.

Among

them

it

is

possible

to

mention

the

following:

frequent

pattern

mining,

data

classification,

and

data

clustering

Nowadays,

the

rate

of

data

generation

is

even

higher

and

so

is

the

complexity

of

the

available

data

Therefore,

*Correspondence:

jessicasouza@usp.br

Instituto

de

Ciências

Matemáticas

e

de

Computação,

USP,

Trabalhador

Saocarlense,

São

Carlos,

Sao

Paulo,

Brazil

Full

list

of

author

information

is

available

at

the

end

of

the

article

the

need

for

more

efficient

tools

to

allow

both

querying

and

mining

large

complex

datasets

is

still

an

open

question.

Considering

the

multimedia

data

types,

for

example,

the

traditional

querying

approach

based

on

attribute

matching

is

not

suitable.

This

fact

has

motivated

the

development

of

querying

approaches

based

on

the

concept

of

similarity

among

complex

data

elements

There

are

several

examples

of

querying

approaches

that

dealt

with

this

issue

in

the

scientific

literature,

such

as

the

similarity

selection

algorithms

(e.g.,

k-nearest

neighbor

selection

and

range

selection)

the

similarity

join

algorithms

(e.g.,

k-nearest

neighbor

join,

k-closest

neighbor

join,

join

around,

and

range

join)

and

the

diversification

of

similarity

selections

The

high

cost

imposed

by

the

similarity-based

approaches

is

related

to

both

the

feature

vectors

employed

to

represent

the

data

and

the

distance

calculations

computed

by

the

algorithms

In

order

to

speed

up

the

similarity-based

comparisons

performed

by

these

approaches,

it

is

possible

to

store

the

dataset

in

specialized

data

structures

known

as

metric

access

methods

(MAM).

As

many

data

mining

approaches,

such

as

data

clustering,

are

based

on

similarity

comparisons,

they

can

also

greatly

benefit

from

the

efficient

processing

of

these

operations

in

a

MAM..

In

the

last

decades,

several

MAM

were

proposed

aiming

at

reducing

the

number

of

distance

calculations,

the

number

of

disk

accesses,

and

the

total

time

spent

to

compute

distance-based

queries.

At

first,

these

methods

were

tailored

to

provide

efficient

similarity

queries

processing.

There

were

no

concerns

about

the

effectiveness

for

the

data

mining

approaches

based

on

similarity

comparisons..

Among

the

MAM,

the

ones

based

on

ball

partitioning

may

result

in

balanced

hierarchies.

The

hierarchy

contains

two

types

of

structures:

index

and

leaf

nodes.

Each

index

node

stores

a

set

of

elements

and

their

respective

covering

radii,

where

each

pair

element,

radius

defines

a

ball

that

encompasses

all

the

elements

in

the

branch

it

represents.

During

traversal,

the

triangle

inequality

property

may

be

used

to

determine

if

there

is

an

intersection

of

a

ball

with

a

query

specification

and

therefore

prunes

branches

that

do

not

intersect.

Finally,

a

leaf

node

contains

data

elements

and

references

to

the

data

they

represent..

Many

proposed

MAM

are

not

dynamic

or

are

not

suitable

for

secondary

memory.

M-tree

is

the

first

balanced

dynamic

ball

partitioning

metric

access

method

based

on

fixed-size

disk

pages.

Its

balance

is

guaranteed

by

the

bottom-up

construction,

where

a

page

overflow

results

in

a

node

split,

recursively

updating

the

ancestors

of

the

node.

No

periodic

global

reorganization

is

required

when

dealing

with

insertions

and

deletions.

An

overview

about

MAM

may

be

found

at

The

major

issue

related

to

MAM

performance

is

the

overlap

among

nodes.

As

far

as

the

overlap

increases,

the

efficiency

of

these

structures

decreases,

as

more

nodes

may

be

covered

by

a

query

region

during

a

search

operation.

Although

the

M-tree,

its

successor

the

Slim-tree

and

other

MAM

proposed,

had

dealt

with

this

issue

in

the

proposal

of

node

splitting

policies

for

the

tree

insertion

process,

we

identified

some

shortcomings

that

led

us

to

the

proposal

of

new

node

split

policies.

The

design

of

these

policies

took

into

account

two

requirements:

MAM

efficiency

for

querying

and

mining

and

also

MAM

effectiveness

for

mining

processes..

Although

we

started

to

address

this

issue

in

a

previous

paper

there

are

new

aspects

that

we

carried

out

herein.

In

we

presented

three

new

split

policies

together

with

an

initial

set

of

experiments

that

showed

their

superiority

over

the

original

ones

when

performing

similarity

queries

in

ball-partitioning-based

MAM

such

as

M-tree

and

Slim-tree.

This

article

integrates

the

concepts

introduced

in

our

previous

paper

and

extends

it

presenting:

a

more

detailed

description

of

the

node

split

policies

presented

in

a

new

node

split

policy,

and

the

results

obtained

with

an

exhaustive

set

of

experiments

that

was

conducted

with

the

aim

of

evaluating

the

impact

in

processing

both

similarity

queries

and

data

clustering

processes

when

employing

different

node

split

policies

on

the

construction

of

the

MAM

that

supports

them.

We

present

new

split

policies

for

M-tree

and

Slim-tree

that

consider

the

distribution

of

the

elements

of

a

full

node

S

into

two

new

nodes,

S

and

S

aiming

at

lowering

the

global

overlap

of

the

structure.

We

argue

that

maximizing

the

free

space

in

both

nodes

created

after

a

split

postpones

future

splits.

A

node

created

after

a

split

with

few

bytes

left

for

newly

promoted

elements

or

new

elements

has

higher

probability

of

being

split

again,

resulting

in

the

increase

of

the

global

overlap.

Our

policies

are

based

on

known

heuristics

and

may

be

employed

by

several

MAM,

such

as

M-tree

and

its

descendants.

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


Investigating the concordance of Gene Ontology terms reveals the intra- and inter-platform reproducibility of enrichment analysis Background: Reliability and Reproducibility of differentially expressed genes (DEGs) are essential for the biological interpretation of microarray data. The microarray quality control (MAQC) project launched by US Food and Drug Administration (FDA) elucidated that the lists of DEGs generated by intra-and inter-platform comparisons can reach a high level of concordance, which mainly depended on the statistical criteria used for ranking and selecting DEGs. Generally, it will produce reproducible lists of DEGs when combining fold change ranking with a non-stringent p-value cutoff. For further interpretation of the gene expression data, statistical methods of gene enrichment analysis provide powerful tools for associating the DEGs with prior biological knowledge, e.g. Gene Ontology (GO) terms and pathways, and are widely used in genome-wide research. Although the DEG lists generated from the same compared conditions proved to be reliable, the reproducible enrichment results are still crucial to the discovery of the underlying molecular mechanism differentiating the two conditions. Therefore, it is important to know whether the enrichment results are still reproducible, when using the lists of DEGs generated by different statistic criteria from inter-laboratory and cross-platform comparisons. In our study, we used the MAQC data sets for systematically accessing the intra-and inter-platform concordance of GO terms enriched by Gene Set Enrichment Analysis (GSEA) and LRpath. Results: In intra-platform comparisons, the overlapped percentage of enriched GO terms was as high as~80% when the inputted lists of DEGs were generated by fold change ranking and Significance Analysis of Microarrays (SAM), whereas the percentages decreased about 20% when generating the lists of DEGs by using fold change ranking and t-test, or by using SAM and t-test. Similar results were found in inter-platform comparisons. In our study, we conducted the intra-and inter-platform comparisons with MAQC data sets and inspected the concordance of GO terms enriched by GSEA and LRpath when the inputted DEG lists were generated by different statistical criteria. The percentages of overlapping GO terms for fold change ranking (|log 2 FC| > 1 (p < 0.05)) were as high as~90% in inter-site comparisons when GO terms meeting FDR < 0.25 criterion were selected, and were around 80% in cross-platform comparisons. Our results demonstrated that the DEG lists generated by a straightforward method combining fold change ranking with a non-stringent p-value cut-off can ensure the reproducibility of the enrichment results. In addition, the tool GSEA for enrichment analysis can always yield relatively stable enrichment results. Reproducible enrichment results are essential for further biological interpretation of microarray data when using Figure 4 The concordance of significant GO terms enriched by GSEA within the same test site of AFX. The comparisons of significant GO terms enriched by GSEA were conducted within a) AFX_1; b) AFX_2; c)AFX_3. Two DEG lists generated by fold change ranking, two DEG lists generated by SAM and two DEG lists generated by t-test were inputted in GSEA to enrich significant GO terms (FDR < 0.25). And then the concordance of two significant GO terms lists derived from two DEGs selected methods was compared on the same p-value cutoff. The red markers reflect the percentages of overlapping GO terms by using fold change ranking and SAM for generating DEG lists. The blue markers reflect the percentages of overlapping GO terms by using fold change ranking and t-test for generating DEG lists. The olive markers reflect the percentages of overlapping GO terms by using SAM and t-test for generating DEG lists. The _1, _2 and _3 suffixes refer to test site location.. statistical methods for gene enrichment analysis. It was proved by MAQC project that the levels of DEG concordance in inter-laboratory and cross-platform comparisons were generally high [3, 4] . For the subsequent enrichment analysis, it is important to know whether the DEG lists generated by different statistical criteria from different microarray platforms can ensure satisfied reproducibility of the enrichment results. In our current study, we systematically investigated the intra-and inter-platform concordance of GO terms enriched by GSEA and LRpath. Note that GSEA is of the 'subject-sampling' type while LRpath treats the genes as the sampling units. In this study, we only focused on the concordance of GO terms enriched by the same enrichment analysis method. The comparison of different enrichment analysis methods will be discussed in our further research. As proposed by MAQC project that combining fold change ranking with a non-stringent p-value cut-off can provide reproducible DEG lists, the levels of concordance of enriched GO terms were still high for this straightforward combining method. In inter-site comparisons for AFX, AG1 and ILM, all the percentages of overlapping GO terms enriched by GSEA and LRpath were abovẽ 90% and~80%, respectively, when GO terms meeting FDR < 0.25 criterion were selected. The concordance of GSEA results were no significant difference in inter-site comparisons. But for a certain test site, the concordance of LRpath results were obviously different when the comparisons among three DEG selection criteria. For the cross-platform comparisons at each test site, the percentages were around 80% (varied from~75% to~84%) when the GO terms were enriched with the inputted DEGs selected by fold change ranking (|log 2 FC| > 1 (p < 0.05)). By contrast, the lack of reproducibility of enriched GO terms was found when the inputted DEGs were selected by t-test. Although all the percentages of overlapping GO terms enriched by GSEA in inter-site comparison were still greater than~87%, the percentages varied from~69% Figure 5 The concordance of significant GO terms enriched by LRpath within the same test site of AFX. The comparisons of significant GO terms enriched by LRpath were conducted within a) AFX_1; b) AFX_2; c)AFX_3. The same test site of the same platform enrichment which used the gene lists derived from different statistical methods. Two DEGs lists generated by fold change ranking and two DEGs lists generated by SAM and two DEGs lists generated by t-test were input in LRpath to get significant GO terms (FDR < 0.25). See notes under Figure 4 for more information. The concordance of two significant GO terms lists derived from two DEGs selected methods was compared on the same p-value cutoff.. to~74% when the inputted DEGs were generated by t-test and the GO terms were enriched by LRpath. Similarly, in the cross-platform comparisons for t-test (p < 0.05), the percentages of overlapping GO terms enriched by GSEA were~78% and then dropped to~50% when GO terms were enriched by LRpath. In addition, for AFX and AG1, we found that the percentages of overlapping GO terms for the comparison of fold change (|log 2 FC| > 1 (p < 0.05)) versus SAM (p < 0.05) were always higher than~76% when comparing the different DEG selection criteria at each test site. It suggested that the concordance of enrichment results based on the DEG selection methods of fold change ranking and SAM were generally high.. To some extent, the number of selected DEGs impacted on the percentages of overlapping GO terms. In inter-site comparisons, most of the percentages for SAM (p < 0.01) were higher than~85% except for the comparisons of ILM_1 versus ILM_2 and ILM_2 versus ILM_3, when the GO terms were enriched by GSEA (Additional file 1: Figure S2a and S2c), because the number of DEGs selected from test site 2 was 3,059, which was about half of those selected from test sites 1 and 3 (Table 1) . It is worthwhile to note that there were large discrepancies between the two reference RNA samples, namely UHRR and HBRR, which were just designed for investigating the capabilities and limitations of the microarray technology and for the corresponding data analysis approaches. So, the number of selected DEGs from a relevant biological study such as control versus treatment would be less than those selected by using UHRR versus HBRR, for which it may cause a decrease in the percentages of overlapping GO terms. In addition, when comparing the GO semantic similarity with real biological data sets, the hierarchical structure of GO graph should be considered [29] . In this study, we systematically investigated whether the results of enrichment analysis were still reproducible when the inputted lists of DEGs were generated by three statistical methods from different commercial microarray platforms. The GO terms were enriched by using GSEA and LRpath with the criteria of FDR < 0.25. Then, the intra-and inter-platform concordance of these terms was analyzed ( Figure 1) and detailed results were shown below. Over the last decade, DNA microarray technology has reached a rapid development and found wide application in many areas of biology and medical science. One of its important applications is to identify differentially expressed genes (DEGs) across groups of samples or distinct biological conditions of interest [1, 2] . Biological interpretation of microarray data requires reliable and reproducible lists of DEGs. The microarray quality control (MAQC) project launched by US Food and Drug Administration (FDA) elucidated that the lists of DEGs generated by intra-and inter-platform comparisons reached a high level of concordance, which largely depended on the statistical criteria used for ranking and selecting DEGs [3, 4] . For the further biological interpretation, statistical methods of gene enrichment analysis provide powerful tools for associating the DEGs with prior biological knowledge, e.g. Gene Ontology (GO) terms and signaling pathways. The enrichment analysis mainly used prior knowledge, e.g. GO categories [5, 6] or Kyoto Encyclopedia of Genes and Genomes (KEGG) pathways [7, 8] , to investigate whether the predefined gene sets showed significantly phenotypic differences between two biological states.. Many methods for enrichment analysis were developed to discover the biological meaning of DEGs. Mootha et al. firstly proposed an earlier version of Gene Set Enrichment Analysis (GSEA), which used an equal weighted version of Kolmogorov-Smirnow statistic for gene sets enrichment without considering the correlation between genes and the phenotype [9] . Subramanian et al. extended this procedure in 2005 and successfully used it for analyzing molecular profiling data [10] . Kim and Volsky carried out a parametric analysis of gene set enrichment (PAGE) to the improved GSEA and identified more statistically significant gene sets. PAGE used less computational effort than GSEA because it used normal distribution for statistical inference [11] . Oron et al. improved GSEA by using a linear regression diagnostic technique and discovered a vital factor to the influence of gene expression from acute lymphoblastic leukemia datasets [12] . Ji et al. proposed a new method FDR-FET to improve the sensitivity and selectivity of GSEA [13] . Kim et al. used z-statistics and permutation test to identify significantly enriched gene sets [14] . In addition, other statistical methods including significance analysis of function and expression (SAFE) [15] , BayGO [16] , ProbCD [17] , EasyGO [18] , ProfCom [19] , GlobalANCOVA [20] , GOEAST [21] and LRpath [22] were also developed for enrichment analysis.. Based on the methods mentioned above, researchers can subsequently reveal the pathological mechanism from the microarray data sets. Xu et al. enriched two gene sets associated with the glycolytic-related pathway from the microarray data of prostate non-recurrent patients. This pathway was considered as a candidate negative modulator of AKT1-induced proliferation [23] . De Windt et al. used GSEA to analyze Niemann-pick type C (NPC) disease and discovered 27 up-regulated and 33 down-regulated pathways. These affected pathways were provided as targets for subsequent drug discovery project [24] . In breast cancer research, Murohashi et al. found that the genes composed in CD24/ low /CD44 + cell populations were fallen into the significantly enriched gene sets, which were associated with the pathways of transforming growth factor-ß, tumor necrosis factor, and interferon response. The signaling pathways enriched by GSEA were suggested to identify molecular targets and biomarkers for Tumour-initiating-like cells [25] .. However, when mapping the DEGs to the predefined gene sets, any difference between two DEG lists may cause different outputs of the enrichment analysis. For the same compared conditions, the reproducible enrichment results are still crucial to the discovery of the underlying molecular mechanism differentiating the two conditions. Therefore, it is important to know whether the enrichment results are still reproducible, when using the lists of DEGs generated by different statistic criteria from different commercial microarray platforms. As a part of the MAQC project, Guo et al. investigated the intra-laboratory overlap of enriched KEGG pathways and GO terms with a rat toxicogenomics dataset and revealed that, compared to the p-value ranking, the use of fold change ranking (with p < 0.05 cutoff) for DEG selection showed more consistency in enrichment analysis [26] . In the previous study by Manoli et al. [27] , the concordance of pathways enriched by Fisher's exact test, global test and GESA were investigated based on the microarray data from Affymetrix microarray platform and the DEGs generated by significance analysis of microarrays (SAM) and mixed model analysis (MMA). The pathways found by Fisher's exact test and global test showed more concordant than those by GSEA in all conditions. In the current study, the microarray data were collected from the large data sets provided by MAQC project [3, 4] , which included three major microarray platforms: Affymetrix (AFX), Agilent Technologies (AG1) and Illumina (ILM) and the lists of DEGs for enrichment analysis were generated by using three statistical criteria: fold change ranking with a nonstringent p-value cut-off which was calculated by t-test, significance analysis of microarrays (SAM) [28] and t-test. Finally, we systematically investigated the intra-and interplatform concordance of GO terms enriched by two common methods of enrichment analysis, namely gene set enrichment analysis (GSEA) [10] and LRpath [22] . The results showed that, based on the DEG lists generated by SAM and FC, the levels of intra-and interplatform concordance of GO terms were generally high and can satisfy the further biological interpretation.

O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B 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


Enhanced Compressed Sensing Recovery With Level Set Normals We propose a compressive sensing algorithm that exploits geometric properties of images to recover images of high quality from few measurements. The image reconstruction is done by iterating the two following steps: 1) estimation of normal vectors of the image level curves, and 2) reconstruction of an image fitting the normal vectors, the compressed sensing measurements, and the sparsity constraint. The proposed technique can naturally extend to nonlocal operators and graphs to exploit the repetitive nature of textured images to recover fine detail structures. In both cases, the problem is reduced to a series of convex minimization problems that can be efficiently solved with a combination of variable splitting and augmented Lagrangian methods, leading to fast and easy-to-code algorithms. Extended experiments show a clear improvement over related state-of-the-art algorithms in the quality of the reconstructed images and the robustness of the proposed method to noise, different kind of images, and reduced measurements.Index Terms-Compressed sensing, image reconstruction, iterative methods. We propose a normal guided compressed sensing recovery method to recover images of higher qualities from fewer measurements. The normal vectors of image level curves have been exploited in denoising and inpainting algorithms, but in compressed sensing this information is embedded in the measurements and state-of-the-art recovery algorithms have just neglected it. To extract this geometric information we alternatively estimate the normals of the image level set curves and then improve the compressed sensing reconstruction matching the estimated normals, the compressed sensing measurements and the sparsity constraints. Although a we cannot provide a rigorous proof of convergence for this twostep procedure, experiments show a clear improvement over standard compressive sensing algorithms due to the introduction of the geometric information of level contours into the image recovery process. The proposed method is also extended to non local operators to recover textured images and could also be applied to improve existing non local denoising and deblurring methods. Our numerical experiments show that the proposed method improves image recovery in several ways: it is able to recover sharp edges as well as smoothly varying image regions, avoiding the staircase effect in the case of total variation regularization; it is robust to noise and the sparsity of the signal and relies on efficient minimization techniques to obtain a fast and easy-to-code algorithm.

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 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 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 B I B I 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 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 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


On a 5-design related to a putative extremal doubly even self-dual code of length a multiple of 24 By the Assmus and Mattson theorem, the codewords of each nontrivial weight in an extremal doubly even self-dual code of length 24m form a self-orthogonal 5-design. In this paper, we study the codes constructed from self-orthogonal 5-designs with the same parameters as the above 5-designs. We give some parameters of a self-orthogonal 5-design whose existence is equivalent to that of an extremal doubly even self-dual code of length 24m for m = 3, 4, 5, 6. If m ∈ {1, . . . , 6}, k ∈ {m + 1, . . . , 5m − 1} and (m, k) = (6, 18), then it is shown that an extremal doubly even self-dual code of length 24m is generated by codewords of weight 4k. A doubly even self-dual code of length n exists if and only if n is divisible by 8. The minimum weight d(C) of a doubly even self-dual code C of length n is bounded above by d(C) ≤ 4⌊n/24⌋ + 4 [10] . A doubly even self-dual code meeting the bound is called extremal. In case that n ≡ 0 (mod 24), the only known extremal doubly even self-dual codes are the extended Golay code and the extended quadratic residue code of length 48. The existence of an extremal doubly even self-dual code of length 72 is a long-standing open question [13] .. A t-(v, k, λ) design is called self-orthogonal if the block intersection numbers have the same parity as the block size k (see [14] ). If D is a selforthogonal t-(v, k, λ) design with k even, then the code C(D), which is generated by the rows of an incidence matrix of D, is a self-orthogonal code. By the Assmus and Mattson theorem [2] , the supports of the codewords of weight 4k ( = 0, 24m) in an extremal doubly even self-dual code of length 24m form a self-orthogonal 5-design. We denote the parameters of the design by 5-(24m, 4k, λ 24m,4k ). Then, throughout this paper, we denote any self-orthogonal 5-(24m, 4k, λ 24m,4k ) design by D 24m,4k . That is, D 24m,4k is a self-orthogonal 5-design with the same parameters as the self-orthogonal 5-design formed from the supports of the codewords of weight 4k in an extremal doubly even self-dual code of length 24m. This gives rise to a natural question, namely, is the code C(D 24m,4k ) always an extremal doubly even self-dual code?. It is well known that C(D 24, 8 ) is the extended Golay code (see [1, Theorem 8.6.2]). It was shown that C(D 24m,4m+4 ) (m = 2, 3, 4) is an extremal doubly even self-dual code [9, 7, 6] , respectively. This means that the existence of an extremal doubly even self-dual code of length 24m (m = 1, 2, 3, 4) is equivalent to that of a self-orthogonal 5-(24m, 4k, λ 24m,4k ) design, where (4k, λ 24m,4k ) = (8, 1), (12, 8) , (16, 78) and (20, 816), respectively. The powerful tool which is used in [7, 9] is the use of fundamental equations, sometimes called the Mendelsohn equations [12] (see also e.g., [14] ), obtained by counting the number of blocks that meet S in i points for some subset S of the point set. The approach in [6] is also similar to that in [7, 9] except that Gleason's theorem (see [10] ) is employed to obtain stronger consequences.. In this paper, we study self-orthogonal 5-designs C(D 24m,4k ) for k ∈ {m+ 2, . . . , 5m − 1}, which are related to codewords of weight other than the minimum weight. More precisely, we consider a problem whether C(D 24m,4k ) is an extremal doubly even self-dual code or not for m ∈ {1, . . . , 6} and k ∈ {m + 2, . . . , 5m − 1}. In addition to the above approach done in [6, 7, 9] , it is useful in this paper to observe weight enumerators of C(D 24m,4k ) and its dual codes, and singly even self-dual codes containing C(D 24m,4k ) and their shadows. As a summary, in Table 1 1 , we list some partial answers to the above problem for m ∈ {1, . . . , 6} and k ∈ {m + 1, . . . , 3m}. For the cases (24m, 4k) that C(D 24m,4k ) is self-dual, we list "Yes" in the second column of Table 1 . When C(D 24m,4k ) is self-dual, we list "Yes" in the third column in case that C(D 24m,4k ) is extremal. We also list the possible minimum weights, when C(D 24m,4k ) is self-dual but not extremal. It is shown that C(D 24m,4k ) = C(D 24m,24m−4k ) for m ∈ {1, . . . , 6} and k ∈ {m+1, . . . , 3m−1} (Proposition 9).. The main results of this paper are the following theorems. Remark 3. For some cases (m, k), the above theorem is already known (see Table 1 ). It is still unknown whether C(D 144,72 ) is self-dual or not (see Remark 8).

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