Volume 15 Supplement 5
Towards a HPC-oriented parallel implementation of a learning algorithm for bioinformatics applications
© D'Angelo and Rampone; licensee BioMed Central Ltd. 2014
Published: 6 May 2014
The huge quantity of data produced in Biomedical research needs sophisticated algorithmic methodologies for its storage, analysis, and processing. High Performance Computing (HPC) appears as a magic bullet in this challenge. However, several hard to solve parallelization and load balancing problems arise in this context. Here we discuss the HPC-oriented implementation of a general purpose learning algorithm, originally conceived for DNA analysis and recently extended to treat uncertainty on data (U-BRAIN). The U-BRAIN algorithm is a learning algorithm that finds a Boolean formula in disjunctive normal form (DNF), of approximately minimum complexity, that is consistent with a set of data (instances) which may have missing bits. The conjunctive terms of the formula are computed in an iterative way by identifying, from the given data, a family of sets of conditions that must be satisfied by all the positive instances and violated by all the negative ones; such conditions allow the computation of a set of coefficients (relevances) for each attribute (literal), that form a probability distribution, allowing the selection of the term literals. The great versatility that characterizes it, makes U-BRAIN applicable in many of the fields in which there are data to be analyzed. However the memory and the execution time required by the running are of O(n3) and of O(n5) order, respectively, and so, the algorithm is unaffordable for huge data sets.
We find mathematical and programming solutions able to lead us towards the implementation of the algorithm U-BRAIN on parallel computers. First we give a Dynamic Programming model of the U-BRAIN algorithm, then we minimize the representation of the relevances. When the data are of great size we are forced to use the mass memory, and depending on where the data are actually stored, the access times can be quite different. According to the evaluation of algorithmic efficiency based on the Disk Model, in order to reduce the costs of the communications between different memories (RAM, Cache, Mass, Virtual) and to achieve efficient I/O performance, we design a mass storage structure able to access its data with a high degree of temporal and spatial locality. Then we develop a parallel implementation of the algorithm. We model it as a SPMD system together to a Message-Passing Programming Paradigm. Here, we adopt the high-level message-passing systems MPI (Message Passing Interface) in the version for the Java programming language, MPJ. The parallel processing is organized into four stages: partitioning, communication, agglomeration and mapping. The decomposition of the U-BRAIN algorithm determines the necessity of a communication protocol design among the processors involved. Efficient synchronization design is also discussed.
In the context of a collaboration between public and private institutions, the parallel model of U-BRAIN has been implemented and tested on the INTEL XEON E7xxx and E5xxx family of the CRESCO structure of Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), developed in the framework of the European Grid Infrastructure (EGI), a series of efforts to provide access to high-throughput computing resources across Europe using grid computing techniques. The implementation is able to minimize both the memory space and the execution time. The test data used in this study are IPDATA (Irvine Primate splice- junction DATA set), a subset of HS3D (Homo Sapiens Splice Sites Dataset) and a subset of COSMIC (the Catalogue of Somatic Mutations in Cancer). The execution time and the speed-up on IPDATA reach the best values within about 90 processors. Then the parallelization advantage is balanced by the greater cost of non-local communications between the processors. A similar behaviour is evident on HS3D, but at a greater number of processors, so evidencing the direct relationship between data size and parallelization gain. This behaviour is confirmed on COSMIC. Overall, the results obtained show that the parallel version is up to 30 times faster than the serial one.
The huge quantity of data produced in Biomedical research needs sophisticated algorithmic methodologies for its storage, analysis, and processing . Examples of the huge databases available throughout the world are given in the annual Database Issue of Nucleic Acids Research and in the on line Molecular Biology Database Collection [2–4].
Furthermore in many applications one must deal with data that have been collected incompletely [5, 6]. For example in medical studies, measurements on some subjects may be partially lost at certain stages of the treatment ; in DNA analysis, gene- expression microarrays may be incomplete due to insufficient resolution, image corruption, or simply dust or scratches on the slide ; in sensing applications, a subset of sensors may be absent or fail to operate at certain regions .
Incomplete data problems are often solved by filling the missing data with specific values (imputation method). Common algorithms that have been used to complete missing data include: semidefinite programming , the EM algorithm [11, 12], Naïve Bayes classifiers , C4.5 , Gibbs sampling , gradient descent . Since these methods rely on the assumption that data are Missing at Random (MAR)  or they treat the missing data as fixed known data , they suffer of dramatic decrease in accuracy. A full discussion can be found in [17–21].
Along with the growth of the data and the need for solutions in the problem of missing data, there is a great necessity of computationally efficient and scalable algorithms able to extract useful information from data sets of very large size [22–28]. This is one of the main challenges in computational biology, since the tools and the methods capable of transforming the heterogeneous available data into biological knowledge  must be implemented efficiently and effectively on the available computer systems.
Recently, in order to deal with incomplete training data, a machine learning algorithm, BRAIN (Batch Relevance-based Artificial INtelligence) , for binary classification rules has been generalized (U-BRAIN) . This algorithm was originally conceived for recognizing splice junctions in human DNA (see also [32, 33]). Splice junctions are points on a DNA sequence at which "superfluous" DNA is removed during the process of protein synthesis in higher organisms . The general method used in the algorithm is related to the STAR technique of Michalski , to the candidate-elimination method introduced by Mitchell , and to the work of Haussler . The BRAIN algorithm was then extended by using fuzzy sets , in order to infer a DNF formula that is consistent with a given set of data which may have missing bits. The new algorithm (U-BRAIN) has low error rates and keeps the polynomial computational complexity of the original BRAIN algorithm.
Unfortunately the algorithm computational complexity, while polynomial, is unaffordable for large scale data. In fact, the algorithm is based on time-consuming nested cycles that need a lot of memory space to store partial results.
Overview of the U-BRAIN algorithm
The U-BRAIN algorithm  is a learning algorithm that finds a Boolean formula (f) in disjunctive normal form (DNF) , of approximately minimum complexity, that is consistent with a set of data (instances). The conjunctive terms of the formula are computed in an iterative way by identifying, from the given data, a family of sets of conditions that must be satisfied by all the positive instances and violated by all the negative ones; such conditions allow the computation of a set of coefficients (relevances) for each attribute (literal), that form a probability distribution, allowing the selection of the term literals.
(1/2 means an uncertain value)
are called negative .
n the number of variables.
2n the number of literals (n in true and n in negated form)
p the number of positive instances.
q the number of negative instances.
i the index of positive instances, ranging from 1 to p.
j the index of negative instances, ranging from 1 to q.
The k-th literal is present in the Sij set if the elements in the position k, belonging to the i-th positive instance (uik) and to the j-th negative instance (vjk), are different or both equal to 1/2.
Depending on the type of pair (uik, vjk) the literal is taken in true (x k ) or negated form . In the following a generic literal will be signed l k for x k and l n + k for .
for literals in negated form.
The literal lk having maximum relevance value is chosen as the next literal of the term of the function f.
The cycle is then repeated and the term is completed when there are no more elements in the Sij sets or there are no more Sij sets. Finally the term is added to the function f. Then the process starts again after erasing from the given data (1) the positive instances satisfying the term found, and updating the uncertain values and the instances. This last step is very important, since each time a term is produced, the implicit choices over the uncertain components of the negative instances, if any, must be explicated to avoid contradiction with the terms to be generated from now on. Moreover, it is possible that there are some instances that are repeated one or more times, either since the beginning or as a result of the reduction step. The results of this updating phase are checked by a consistency test.
The algorithm ends when there are no more data to treat.
Initialize f = Ø
- 2.While(∃ positive instances)
Initialize term = Ø
Build Sij sets
- 2.5.While(∃ elements in Sij)
Compute the Rij relevances
Compute the Ri relevances
Compute the R relevances
Update Sij sets
Add term to f
Update positive instances
Update negative instances
The algorithm complexity refers to both the amount of memory it requires to run to completion (space complexity) and the amount of time it needs to run to completion (time complexity) . According to the Landau's symbol , in the following a big O notation will be used to describe the upper bound complexity.
In order to build a family of sets Sij and to calculate the Rij(lk) elements, U-BRAIN compares each given positive instance with each negative one.
So the space complexity is in the order of O(pqn) ≈ O(n 3 ) for large n.
Since each element of a Rij vector is an element of a probability distribution, it is represented by a floating-point number, which, depending on the coding, occupies several Bytes in a computer's internal memory. Thus, storing of Rij for large scale data in a computer memory is space consuming.
Example 1: Using the Java language and a data set having
p = 2000, q = 3000, n = 560
the Rij and Sij dimensions are about 430 GByte and 200 GByte respectively for the first iteration.
From the time point of view, since the external cycle (2.) is iterated at most p times, the internal cycle (2.5.) is iterated at most n times (the maximum length of a term), the inner relevance computation (2.5.1.-2.5.3.) and the Sij update (2.5.6.) are both of O(pqn), and all the other operations are minorities of these, the overall algorithm time complexity is O(p 2 qn 2 ) ≈ O(n 5 ) for large n.
We find mathematical and programming solutions able to effectively implement the algorithm U-BRAIN on parallel computers. First we give a Dynamic Programming model  of the U-BRAIN algorithm; then we minimize the representation of the relevances; finally, in order to reduce the communication costs between different memories and, then, to achieve efficient I/O performance, a mass storage structure is designed to access its data with a high degree of temporal and spatial locality . Then a parallel implementation of the algorithm is developed by a Single Program Multiple Data (SPMD) technique together to a Message-Passing Programming paradigm.
Dynamic programming model of the U-BRAIN algorithm
In the U-BRAIN algorithm the Sij sets are built in the external cycle (2.4.), and then, for each resulting Sij set, the Rij relevance vectors are calculated (2.5.1.-2.5.3.). Thus, two memory areas are required, one for the Sij sets and one for the relevance vectors. Moreover, for each choice of a literal, a Sij updating step is done (2.5.6.), consisting in a reduction (erasing) of some Sij sets. Starting from here, the inner cycle (2.5.) is repeated and the new Rij relevance vectors, based on the new Sij sets, are calculated, again. This last step is repeated until there are no more Sij sets corresponding to the production of the term.
It is worth to note that for each inner iteration, the Sij sets are not modified but only erased, and so the recalculation of the Rij relevances on the survivor sets it is not necessary because the new Rij relevances are equal to the ones calculated on the first cycle.
Initialize f = Ø
- 2.While(∃ positive instances)
Initialize term = Ø
Compute the Rij relevances
- 2.5.While(∃ elements in Rij)
Compute the Ri relevances
Compute the R relevances
Add term to f
Update positive instances
Update negative instances
These changes involve a dramatic reduction in both space of memory, by avoiding the Sij computation, and execution time, by avoiding the Rij computation in the inner cycle.
Minimizing the Rij representation
Since the Rij relevances are valued for each of the 2n literals as floating-point numbers, a lot of memory space is required to represent them during the algorithm inner cycle execution. Then, a reduced representation form for the Rij's is desirable.
In this way it is possible to view the Rij relevances as function of the four valued coefficients βij, each one representable by 2 bits only. This implies a significant reduction of the required memory space.
Example 2: Compared to the previous Example 1, for
p = 2000, q = 3000, n = 560
the Rij dimension is decreased from about 430 GByte to 1,7 GByte.
β ij data structure and storage system
Since the Rij computation relies on βij's, we define a data structure to hold them. The βij's related to the i-th positive instance form the i-th βi set. We represent this set as an array (inner array). Each element of the inner array contains a βij vectors of 2n dimension whose elements are the 2 bits representation of a βij value. Then we define an array containing each βi, named outer array.
The elements of the mass storage structure are in sequence. Each element contains three different typology of data: Data Size representing the dimension in bytes of the data to be stored, a Delete Flag that indicates whether the item has been deleted and, finally, the Data section. Each Data section contains a βi set, according with the βij data structure reported in Figure 1.
Replacing Rij by βij
Initialize f = Ø
- 2.While(∃ positive instances)
Initialize term = Ø
Compute the β ij values
- 2.5.While(∃ elements in β ij )
Compute the R i relevances
Compute the R relevances
Update β ij
Add term to f
Update positive instances
Update negative instances
Parallel programming model
Here we model a parallel implementation of the U-BRAIN algorithm.
There are three common strategies for creating parallel applications . The first two, implicit parallelism, are based on the automatic parallelization of a sequential program, and on the use of parallel libraries that encapsulate some of the parallel code commonly used. The third one, explicit parallelism, involves the writing of the parallel application from the beginning. It was observed that the use of explicit parallelism, when properly applied, obtains a better efficiency than parallel language or compilers that use implicit parallelism . This is the strategy we adopt here.
From a Flynn's taxonomy  point of view, we adopt a MIMD/Master-Slave strategy, and, specifically, a SPMD programming approach [46, 47], together to the Message-Passing Programming Paradigm. In SPMD, multiple autonomous processors simultaneously execute the same program at independent points. That is, a single program is written so that different processes carry out different actions, and this is achieved by simply having the processes branch on the basis of their process rank. The Message-passing paradigm provides routines to initiate and configure the messaging environment, sending and receiving packets of data between processors of a parallel system.
The portability, the network transparency and the heterogeneity are other goals of interest. Currently, one of the most high-level message-passing systems is MPI (Message Passing Interface) defined by the MPI Forum . MPI is a specification, not an implementation; there are multiple implementations of MPI including versions for COW (Cluster Of Workstation) , distributed-memory multiprocessors (MPP) and shared-memory machines (SMP). Here, we adopt a version for the Java programming language, MPJ . Compared with C or Fortran, the advantages of the Java programming language include higher-level programming concepts, improved compile time and runtime checking, and, as a result, faster problem detection and debugging. In the context of "Java for HPC", the performance evaluation of the Java version reveals that it could achieve comparable performance to the original C code and the Java code performs better in the computation stages .
Although MPI offers great vantages, a significant amount of tasks of the parallelization are delegated to the programmer. So, a design methodology that allows the programmer to focus on machine-independent issues is desirable. According with Foster  we adopt a methodology organized into four stages: partitioning, communication, agglomeration and mapping.
The (22) leads to the implicit decomposition of the other tasks of the algorithm; in fact the uncertainty reduction, the repetition deletion and the updating of positives and negatives instances are performed concurrently by each process on their own partial data.
Agglomeration and mapping
In order to move the previous abstract phases toward the concrete implementation on a specific parallel system, we adapt the number of the partial summations in (23) to exactly one per processor. So, assuming to use nProc processors, each processor will treat, on average, positives instances. If p is not multiple of nProc, the division between p and nProc leads to a remainder (p mod nProc) different from zero. The remaining instances are distributed on p mod nProc processors that are charged of an additional task. In this case, our design is already largely complete, since in defining the nProc tasks that will execute on nProc processors, we have also addressed the mapping problem. This method of load balancing  is static because the tasks are assigned to the processors before the process starts and no information is collected about the state in real time of each single processor. Each processor, by acting on different positive instances, has different execution times. Thus it may finish the job and remains idle. In this case no further load balancing is made, so avoiding NP- Complete problems [54, 55]. Nevertheless, the mapping between tasks and processors, used here, follows a semi-dynamic load balancing algorithm able to adapt the load to the number of available processors. Each process, on the basis of both its own identifier number (RANK) and the given number of positive instances (p), loads a fraction of the positive data. Each processor compute the βij values and stores them in its own random access file according to the mass storage structure of Figure 2; in this way a reduction of data storage for each processor is also obtained and no file access synchronization is required.
Results and discussion
The computing resources and the related technical support used for this work have been provided by CRESCO/ENEAGRID High Performance Computing infrastructure and its staff.
The test data used in this study are IPDATA (Irvine Primate splice-junction data set) , a subset of HS3D (Homo Sapiens Splice Sites Dataset) [57, 58] and a subset of COSMIC (the Catalogue of Somatic Mutations in Cancer) .
IPDATA is a data set of human splice sites, and it consists of 767 donor splice sites, 765 acceptor splice sites, and 1654 false splice sites. According to previous usage  we consider 464 positive instances and 1536 negative instances each one coded by 240 bits.
HS3D is a data set of Homo Sapiens Exon, Intron and Splice sites extracted from GenBank Rel.123. It includes 2796 + 2880 donor and acceptor sites, as windows of 140 nucleotides (560 bits) around a splice site, and 271,937+332,296 windows of false splice sites, selected by searching canonical GT-AG pairs in not splicing positions. In this study we adopt a subset of 2974 donor sites and 161 false ones. COSMIC curates comprehensive information on somatic mutations in human cancer. Release v48 (July 2010) describes over 136,000 coding mutations in almost 542,000 tumour samples. Here we focus on the tumor suppressor p16 . 60 positive instances are generated from missense and nonsense mutations, while 62 negative instances are generated from synonymous mutations . Each instance is of 1884 bits.
The dimension of each dataset used is estimated as p×q×n. In this way the size of HS3D sub-dataset is greater than IPDATA which is greater than p16 COSMIC.
where nProc is the number of processors, TS is the execution time of the sequential program and TnProc is the execution time of the parallel one with nProc processors. Ideal speed-up is obtained when S(nProc) = nProc, while for one processor S(1) = 1.
An estimate of this parameter has been taken into account during all the design and the testing of the parallelization process. The use of several data sets has been useful in order to show the effect of the granularity on the performance varying the problem size. The granularity is a qualitative measure of the ratio of computation to communication . Two graphs, reporting the execution time and the speed-up of the parallel version, respectively, are shown for each data set.
As evidenced in Figure 10 and 11, the execution times and the speed-up on IPDATA reach the best values within about 90 processors. Then the parallelization advantage is balanced by the greater cost of non-local communications between the processors.
Overall, the results obtained on the data sets used show that the parallel version is up to 30 times faster than the serial. Moreover, increasing the problem size, at constant number of processors, the speed-up averagely increases.
High-throughput technologies are producing an increasing amount of experimental and clinical data. In such a scenario, large-scale databases and bioinformatics methods are key tools for organizing and exploring biological and biomedical data with the aim to discover new knowledge in biology and medicine.
High-performance computing may play an important role in many phases of life sciences research, from raw data management and processing, to data analysis and integration, till data exploration and visualization. In particular, at the raw data layer, Grid infrastructures may offer the huge data storage needed to store experimental and biomedical data, while parallel computing can be used for basic pre-processing and for more advanced analysis. In such a scenario, parallel architectures coupled with specific programming models may overcome the limits posed by conventional computers to the mining and exploration of large amounts of data.
Here we investigated the problems arising from the HPC implementation of a general purpose learning algorithm able to treat uncertainty on data (U-BRAIN). The U- BRAIN algorithm can be used in many fields of the biology in order to extract the laws that govern the biological process, in the form of mathematical formulas. The U- BRAIN parallel implementation aims to override the computational limits that make the algorithm unaffordable for huge data sets. We found mathematical and programming solutions able to effectively implement the algorithm U-BRAIN on parallel grid computers. The implementation is able to minimize both the memory space and the execution time, while maintaining the results of the sequential version. The solutions adopted in this paper, e.g. dynamic programming, data representation minimization, efficient use of memory, mass storage unit structure with a high degree of temporal and spatial locality, SPMD parallel implementation and Message-Passing Programming Paradigm, are tailored for the U-BRAIN algorithm, but they can be used for many others HPC-oriented parallel implementations.
As evidenced in the experiments, the execution times and the speed-up reach the best values within a data dependent number of processors. Then the parallelization advantage is balanced by the greater cost of non-local communications between the processors. This evidences the direct relationship between data size and parallelization gain.
The obtained results, though not excellent in terms of performance, encourage the algorithm application on larger data sets. By applying the U-BRAIN algorithm on the full HS3D data set (p = 2796, q = 271937, n = 560), and using a single processor, 0,17 × 108 seconds (197 days) are needed to reach the result, while, by assuming a linear relation among the dimension and the execution time, the time reduces to 0,57 × 106 seconds (about 6 days) in a parallel configuration with speed-up = 30.
Open problems rest in adopting a dynamic load balancing algorithm, capable of migrating the load among the processors. Load balancing problem is a most critical point in parallel computing design . A more thorough assessment of how the synchronization and communication costs affect the total performance varying the problem size is another issue that we refer to future works. In order to increase the performance it would be useful implement U-BRAIN algorithm by using a hybrid MPI/OpenMP programming on clusters of multi-core with shared-memory nodes .
This work was supported by the collaboration agreement between Futuridea Association, ENEA, University of Sannio and E4 Computer Engineering S.p.A. The authors wish to thank the CRESCO/ENEAGRID High Performance Computing staff and in particular Ing. Silvio Migliori, Ing. Antonio Perozziello, and Ing. Guido Guarnieri. CRESCO/ENEAGRID High Performance Computing infrastructure is funded by ENEA, the Italian National Agency for New Technologies, Energy and Sustainable Economic Development and by national and European research programs.
The publication costs for this article were funded by DST University of Sannio.
This article has been published as part of BMC Bioinformatics Volume 15 Supplement 5, 2014: Italian Society of Bioinformatics (BITS): Annual Meeting 2013. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/15/S5
- Kodama Y, Shumway M, Leinonen R: The sequence read archive: explosive growth of sequencing data. Nucleic Acids Research. 2012, 40: D54-D56. 10.1093/nar/gkr854.PubMed CentralView ArticlePubMedGoogle Scholar
- Baxevanis AD: The Molecolar Biology Database Collection: 2003 update. Nucleic Acids Research. 2003, 31 (1):
- Galperin MY, Fernández-Suárez Xosé M: The 2012 Nucleic Acids Research Database Issue and the online Molecular Biology Database Collection. Nucleic Acids Research. 2012, 40 (Database):
- Fernández-Suárez Xosé M, Galperin MY: The 2013 Nucleic Acids Research Database Issue and the online Molecular Biology Database Collection. Nucleic Acids Research. 2013, 41 (Database):
- Rubin D: Multiple Imputation for Nonresponse in Surveys. 1987, John Wiley & Sons, IncView ArticleGoogle Scholar
- Dick U, Haider P, Scheffer T: Learning from Incomplete Data with Infinite Imputations. Proceedings of the 25th International Conference on Machine Learning. 2008, Helsinki, Finland, 232-239.Google Scholar
- Ibrahim JG: Incomplete data in generalized linear models. Journal of the American Statistical Association. 1990, 85: 765-769. 10.1080/01621459.1990.10474938.View ArticleGoogle Scholar
- Wang X, Li A, Jiang Z, Feng H: Missing value estimation for DNA microarray gene expression data by support vector regression imputation and orthogonal coding scheme. BMC Bioinformatics. 2006, 7: 32-10.1186/1471-2105-7-32.PubMed CentralView ArticlePubMedGoogle Scholar
- Williams D, Carin L: Analytical kernel matrix completion with incomplete multi-view data. Proceedings of the International Conference on Machine Learning (ICML) Workshop on Learning with Multiple Views. 2005, 80-86.Google Scholar
- Graepel T: Kernel matrix completion by semidefinite programming. Proceedings of the International Conference on Artificial Neural Networks. 2002, 2415: 694-699.Google Scholar
- Dempster AP, Laird NM, Rubin DB: Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society Series B (Methodological). 1977, 39 (1): 1-38.Google Scholar
- Tsuda K, Akaho S, Asai K, Williams C: The em algorithm for kernel matrix completion with auxiliary data. Journal of Machine Learning Research. 2003, 4: 67-81.Google Scholar
- Duda RO, Hart PE: Pattern Classification and Scene Analysis. 1973, New York: WileyGoogle Scholar
- Quinlan JR: Programs for Machine Learning. 1993, San Francisco: Morgan Kaufmann PublishersGoogle Scholar
- Geman S, Geman D: Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Transactions on Pattern Analysis and Machine Intelligence. 1984, 6: 721-741.View ArticlePubMedGoogle Scholar
- Ahmad F, Isa NAM., Osman MK, Hussain Z: Performance comparison of gradient descent and Genetic Algorithm based Artificial Neural Networks training. Proceedings of the 10th International Conference on Intelligent Systems Design and Applications (ISDA). 2010, 604-609.Google Scholar
- Rubin DB, Little RJA: Statistical Analysis with Missing Data. 2002, New York: Wiley Interscience, 2Google Scholar
- Rässler S: The Impact of Multiple Imputation for DACSEIS. Technical Report DACSEIS Research Paper Series 5. 2004, Univ. of Erlangen-Numberg, Numberg, GermanyGoogle Scholar
- Schafer JL, Graham JW: Missing Data: Our View of the State of the Art. Psychological Methods. 2002, 7 (2): 147-177.View ArticlePubMedGoogle Scholar
- Chen J, Xue X, Tian F, Huang H: An algorithm for Classifying Incomplete Data With Selective Bayes Classifiers. Proceedings of the IEEE International Conference on Computational Intelligence and Security Workshops. 2007, 445-448.Google Scholar
- Williams D, Liao X, Xue Y, Carin L, Krishnapuram B: On Classification with Incomplete Data. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2007, 29 (3): 427-436.View ArticlePubMedGoogle Scholar
- Li D, Zhong C, Li J: An attribute weighted fuzzy c-means algorithm for incomplete data sets. Proceedings of the IEEE International Conference on System Science and Engineering (ICSSE). 2012, 449-453.Google Scholar
- Thangaparvathi B, Anandhavalli D, Mercy Shalinie S: A high speed decision tree classifier algorithm for huge dataset. Proceedings of the IEEE International Conference on Recent Trends in Information Technology (ICRTIT). 2011, 695-700.View ArticleGoogle Scholar
- Menon AK: Large-Scale Support Vector Machines: Algorithms and Theory. Research Exam. 2009, University of California, San Diego, 1-17.Google Scholar
- Guosheng W: A Survey on Training Algorithms for Support Vector Machine Classifiers. Proceedings of the Fourth IEEE International Conference on Networked Computing and Advanced Information Management, NCM '08. 2008, 123-128.Google Scholar
- Lu C, Li X, Pan H: Application of SVM and Fuzzy Set Theory for Classifying with Incomplete Survey Data. Proceedings of the IEEE International Conference on Service Systems and Service Management. 2007, 1-4.Google Scholar
- Chen J, Xue X, Fengzhan T, Huang H: An Algorithm for Classifying Incomplete Data with Selective Bayes Classifiers. Proceedings of the IEEE International Conference on Computational Intelligence and Security Workshops, CISW. 2007, 445-448.Google Scholar
- Amado N, Gama J, Silva F: Parallel Implementation of Decision Tree Learning Algorithms. Progress in Artificial Intelligence Lecture Notes in Computer Science. 2001, 2258: 6-13. 10.1007/3-540-45329-6_4.View ArticleGoogle Scholar
- Larranaga P, Calvo B, Santana R, Bielza C, Galdiano J, Inza I, Lozano JA, Armananzas R, Santafe G, Perez A, Robles V: Machine learning in bioinformatics. Briefings in bioinformatics. 2006, 7 (1): 86-112. 10.1093/bib/bbk007.View ArticlePubMedGoogle Scholar
- Rampone S: Recognition of spline-junctions on DNA sequences by BRAIN learning algorithm. Bioinformatics Journal. 1998, 14 (8): 676-684. 10.1093/bioinformatics/14.8.676.View ArticleGoogle Scholar
- Rampone S, Russo C: A fuzzified BRAIN algorithm for learning DNF from incomplete data. Electronic Journal of Applied Statistical Analysis (EJASA). 2012, 5 (2): 256-270.Google Scholar
- Rampone S: An Error Tolerant Software Equipment For Human DNA Characterization. IEEE Transactions on Nuclear Science. 2004, 51 (5): 2018-2026.View ArticleGoogle Scholar
- Aloisio A, Izzo V, Rampone S: VLSI implementation of greedy-based distributed routing schemes for ad hoc networks. Soft Computing. 2007, 11 (9): 865-872. 10.1007/s00500-006-0138-7.View ArticleGoogle Scholar
- Green MR: Pre-mRNA splicing. Annual Review of Genetics. 1986, 20: 671-708. 10.1146/annurev.ge.20.120186.003323.View ArticlePubMedGoogle Scholar
- Michalski RS: A theory and methodology of inductive learning. Artificial Inteligence. 1983, 20: 111-116. 10.1016/0004-3702(83)90016-4.View ArticleGoogle Scholar
- Mitchell TM: Generalization as search. Artificial Inteligence. 1982, 18: 203-226. 10.1016/0004-3702(82)90040-6.View ArticleGoogle Scholar
- Haussler D: Quantifying inductive bias: AI learning algorithms and Valiant's learning framework. Artificial Inteligence. 1988, 36: 177-222. 10.1016/0004-3702(88)90002-1.View ArticleGoogle Scholar
- Zadeh LA: Fuzzy sets. Information and Control. 1965, 8 (3): 338-353. 10.1016/S0019-9958(65)90241-X.View ArticleGoogle Scholar
- Mendelson E: Introduction to Mathematical Logic. 1997, London: Chapman & HallGoogle Scholar
- Bürgisser P, Clausen M, Shokrollahi MA: Algebraic Complexity Theory. 1997, SpringerView ArticleGoogle Scholar
- Knuth D: Big Omicron and big Omega and big Theta. SIGACT News. 1976, 18-24. Apr.- JuneGoogle Scholar
- Cormen TH, Leiserson CE, Rivest RL, Stein C: Introduction to Algorithms. 2009, Boston: The MIT Press, 3Google Scholar
- Vitter JS: External Memory Algorithms and Data Structures: Dealing with Massive Data. ACM Computing Surveys. 2001, 33 (2): 209-271. 10.1145/384192.384193. JuneView ArticleGoogle Scholar
- Kasim H, March V, Zhang R, See S: Survey on Parallel Programming Model. NPC Proceedings of the IFIP International Conference on Network and Parallel Computing. 2008, 266-275.Google Scholar
- Flynn MJ: Very high-speed computing systems. Proceedings of the IEEE. 1966, 54 (12): 1901-1909.View ArticleGoogle Scholar
- Auguin M, Larbey F: OPSILA: an advanced SIMD for numerical analysis and signal processing. Microcomputers: developments in industry, business, and education, Ninth EUROMICRO Symposium on Microprocessing and Microprogramming, Madrid, September 13-16. 1983, 311-318.Google Scholar
- Darema F: SPMD model: past, present and future, Recent Advances in Parallel Virtual Machine and Message Passing Interface. Proceedings of the 8th European PVM/MPI Users' Group Meeting. 2001, 2131: 1-1. Santorini/Thera, Greece, September 23-26, Lecture Notes in Computer ScienceGoogle Scholar
- Message Passing Interface Forum. 2013, Online, last access October 4, [http://www.mpi-forum.org/]
- Jing Y, Weichang S, Gongxiao Y: Construct COW Based on MPICH in Linux Environment. Proceedings of the First International Workshop on Education Technology and Computer Sciense. 2009, 895-898.Google Scholar
- MPJ Express. 2013, Online, last access October 4, [http://mpj-express.org/]
- Shafi A, Hussain A, Raza J: A Parallel Implementation of the Finite- Domain Time-Difference Algorithm using MPJ Express. Proceedings of the IEEE International Symposium on Parallel and Distributed Processing. 2008, IPDPS, 1-6.Google Scholar
- Foster I: Designing and Building Parallel Programs. 1996, Addison WesleyGoogle Scholar
- Nian S, Guangmin L: Dynamic Load Balancing Algorithm for MPI Parallel Computing. Proceedings of the IEEE International Conference on New Trends in Information and Service Science. 2009, 95-99.Google Scholar
- Ullman JD: NP-Complete Scheduling Problems. Journal of Computer and System Sciences. 1975, 10: 384-393. 10.1016/S0022-0000(75)80008-0.View ArticleGoogle Scholar
- Sinnen O, Sousa LA, Sandnes FE: Toward a Realistic Task Scheduling Model. IEEE Transactions on Parallel and Distributed Systems. 2006, 17 (3): 263-275.View ArticleGoogle Scholar
- Bache K, Lichman M: UCI Machine Learning Repository. 2013, Irvine, CA: University of California, School of Information and Computer Science, [http://archive.ics.uci.edu/ml]Google Scholar
- Pollastro P, Rampone S: HS3D: Homo Sapiens Splice Site Data Set. Nucleic Acids Research. 2003, Annual Database
- Pollastro P, Rampone S: HS3D, a Dataset of Homo Sapiens Splice Regions, and its Extraction Procedure from a Major Public Database. International Journal of Modern Physics C. 2003, 13 (8): 1105-1117.View ArticleGoogle Scholar
- Forbes SA: COSMIC: mining complete cancer genomes in the Catalogue of Somatic Mutations in Cancer. Nucleic Acids Research. 2011, 39 (suppl 1): D945-D950.PubMed CentralView ArticlePubMedGoogle Scholar
- Liggett WH, Sidransky D: Role of the p16 tumor suppressor gene in cancer. J Clin Oncol. 1998, 16 (3): 1197-206. MarPubMedGoogle Scholar
- Barney B: Introduction to Parallel Computing. Lawrence Livermore National Laboratory. 2013, Online, last access October 4, [https://computing.llnl.gov/tutorials/parallel_comp/]Google Scholar
- Rabenseifner R, Hager G, Jost G: Hybrid MPI/OpenMP Parallel Programming on Clusters of Multi-Core SMP Nodes. Proceedings of the 17th Euromicro International Conference on Parallel Distributed and Network-based Processing. 2009, IEEE Press, 427-436.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.