Divide and Conquer

This paper presents an experimental comparison of a number of different algorithms for computing the Deluanay triangulation. The algorithms examined are: Dwyer's divide and conquer algorithm, Fortune's sweepline algorithm, several versions of the incremental algorithm (including one by Ohya, Iri, and Murota, a new bucketing-based algorithm described in this paper, and Devillers's version of a Delaunay-tree based algorithm that appears in LEDA), an algorithm that incrementally adds a correct Delaunay triangle adjacent to a current triangle in a manner similar to gift wrapping algorithms for convex hulls, and Barber's convex hull based algorithm.

If it is true that good problems produce good science, then it will be worthwhile to identify good problems, and even more worthwhile to discover the attributes that make them good problems. This discovery process is necessarily empirical, so we examine several challenge problems, beginning with Turing's famous test, and more than a dozen attributes that challenge problems might have. We are led to a contrast between research strategies-the successful "divide and conquer" strategy and the promising but largely untested "developmental" strategy-and we conclude that good challenge problems encourage the latter strategy.

Page 1. Deterministic Algorithms for 3-D Diameter and some 2-D Lower Envelopes * Edgar A. Ramos t Abstract We present a deterministic algorithm for computing the diameter of a set of n points in R3; its run-ning time O(n log n) is worst-case optimal. ...

Formal verification is easy to use and provides significant increases in productivity and quality when used on RTL designs, which fit formal verification tool capacity. However, formal verification can be challenging when used on designs that exceed the capacity of the tools. This paper describes various techniques that were used to overcome these challenges during the verification of a real-life complex interrupt-controller using Cadence's Incisive Formal Verifier. These techniques include divideand-conquer to scale down the design, smart modeling of constraints, lightweight simulation, and use of advanced tool features. The collection of these techniques enabled the full verification of the interrupt-controller and saved many man-months of effort. The paper presents our results and concludes with pointers to areas of further improvement, currently under investigation.

The efficiency of the core Galois field arithmetic improves the performance of elliptic curve based public key cryptosystem implementation. This paper describes the design and implementation of a reconfigurable Galois field multiplier, which is implemented using field programmable gate arrays (FPGAs). The multiplier of Galois field based on Karatsuba's divide and conquer algorithm allows for reasonable speedup of the top-level public key algorithms. Binary Karatsuba multiplier is more efficient if it is truncated at n-bit multiplicand level and use an efficient classic multiplier algorithm. In these work three levels to truncate Binary Karatsuba algorithm (4 bits, 8 bits and 16 bits) are chosen showing that 8 bits is the best level for minimum number of slices and time delay to truncate Binary Karatsuba algorithm which is designed on an Xilinx VirtexE XCV2600 FPGA device. The VHDL hardware models are building using Xilinx ISE foundation software. This work is able to compute GF(2191) multiplication in 45.889 ns. experimental results of comparing block and stream ciphers when used to secure VoIP in terms of end-to-end delay and subjective quality of perceived voice.

When we search from a huge amount of documents, we often specify several keywords and use conjunctive queries to narrow the result of the search. Though the searched documents contain all keywords, positions of the keywords are usually not considered. As the result, the search result contains some meaningless documents. It is therefore eective to rank documents according to proximity of keywords in the documents. This ranking is regarded as a kind of text data mining. In this paper, we propose two algorithms for nding documents in which all given keywords appear in neighboring places. One is based on plane-sweep algorithm and the other is based on divide-and-conquer approach. Both algorithms run in O(n log n) time where n is the number of occurrences of given keywords. We run the plane-sweep algorithm on a large collection of html les and verify its eectiveness.

Parallel algorithms for solving geometric problems on two array processor models-the mesh-connected computer (MCC) and a two-dimensional systolic array-are presented. We illustrate a recursive divide-and-conquer paradigm for MCC algorithms by presenting a time-optimal solution for the problem of finding the nearest neighbors of a set of planar points represented by their Cartesian coordinates. The algorithm executes on a ~/n • x/n MCC, and requires an optimal O(x/n) time. An algorithm for constructing the convex hull of a set of planar points and an update algorithm for the disk placement problem on an nZ/3x n 2/3 twodimensional systolic array are presented. Both these algorithms require O(n 2/3) time steps. The advantage of the systolic solutions lies in their suitability for direct hardware implementation.

This paper extends the algorithms which were developed in Part I to cases in which there is no a ne schedule, i.e. to problems whose parallel complexity is polynomial but not linear. The natural generalization is to multidimensional schedules with lexicographic ordering as temporal succession. Multidimensional a ne schedules, are, in a sense, equivalent to polynomial schedules, and are much easier to handle automatically. Furthermore, there is a strong connection between multidimensional schedules and loop nests, which allows one to prove that a static control program always has a multidimensional schedule. Roughly, a larger dimension indicates less parallelism. In the algorithm which is presented here, this dimension is computed dynamically, and is just su cient for scheduling the source program. The algorithm lends itself to a \divide and conquer" strategy. The paper gives some experimental evidence for the applicability, performances and limitations of the algorithm.

a) User sketches (b) Automatic segmentation (c) Polycube map (d) GPU subdivision displacement Figure 1: The proposed method allows the users to easily edit and control the polycube map by sketching the features/constraints on the 3D model and the polycube. The generated polycube map is conformal and with low area distortion, thus facilitates some graphics applications such as GPU-based displacement mapping.

The goal of approximating metric spaces by more simple metric spaces has led to the notion ofgraph spanners [PU89, PS891 and to low-distortion embeddings in low-dimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for ...

This paper extends the algorithms which were developed in Part I to cases in which there is no a ne schedule, i.e. to problems whose parallel complexity is polynomial but not linear. The natural generalization is to multidimensional schedules with lexicographic ordering as temporal succession. Multidimensional a ne schedules, are, in a sense, equivalent to polynomial schedules, and are much easier to handle automatically. Furthermore, there is a strong connection between multidimensional schedules and loop nests, which allows one to prove that a static control program always has a multidimensional schedule. Roughly, a larger dimension indicates less parallelism. In the algorithm which is presented here, this dimension is computed dynamically, and is just su cient for scheduling the source program. The algorithm lends itself to a \divide and conquer" strategy. The paper gives some experimental evidence for the applicability, performances and limitations of the algorithm.

The LCS problem is to determine the longest common subsequence (LCS) of two strings. A new linear-space algorithm to solve the LCS problem is presented. The only other algorithm with linear-space complexity is by Hirschberg and has runtime complexity O(mn). Our algorithm, based on the divide and conquer technique, has runtime complexity O(n(m-p)), where p is the length of the LCS.

We present a new algorithm, called marching cubes, that creates triangle models of constant density surfaces from 3D medical data. Using a divide-and-conquer approach to generate inter-slice connectivity, we create a case table that defines triangle topology. The algorithm processes the 3D medical data in scan-line order and calculates triangle vertices using linear interpolation. We find the gradient of the original data, normalize it, and use it as a basis for shading the models. The detail in images produced from the generated surface models is the result of maintaining the inter-slice connectivity, surface data, and gradient information present in the original 3D data. Results from computed tomography (CT), magnetic resonance (MR), and single-photon emission computed tomography (SPECT) illustrate the quality and functionality of marching cubes. We also discuss improvements that decrease processing time and add solid modeling capabilities.

Hill Cipher is one of the symmetric key cryptographic algorithms that has several advantages in data encryption. To avoid key invertible matrices, key matrices are generated using Newton's binomial coefficients. Hill Cipher is one of the symmetric key cryptographic algorithms. The Hill Cipher algorithm uses a matrix measuring m x m as a key for encryption and decryption. The basic matrix theory used in Hill Cipher includes multiplication between matrices and inverses the matrix. Hill Cipher uses mathematical calculations called Linear Algebra and specifically requires users to understand the basis of the matrix. During this time the matrix multiplication used to encrypt and decrypt this method is only done by Devide and Conquer. This is less effective in minimizing the processing time. A good algorithm must not only be correct but also efficient in its use. In addition, an algorithm must also have a low complexity. Algorithms like this must be able to optimize its performance, especially in data processing which is getting more and more in number. An algorithm with low complexity is an algorithm that is able to minimize the use of time and memory space even though more and more data is used. Therefore the author applies the Strassen method in optimizing the matrix multiplication process to encrypt and decrypt it. However, this algorithm will only apply to use of keys and plaintexts of the same order size, which is nxn where n = 2 a , (a> 0). Then the encryption and decryption process will be done by matrix multiplication using the Strassen method. The final results show that by using the Strassen algorithm the encryption and decryption process in the Hill Cipher method can be optimized. Where every time the data to be processed increases the complexity of the time required will not be too high an increase. And this shown that Strassen more effective than the Devide and Conquer method.

We review the three well-known fast algorithms for the solution of Yule-Walker (YW) equations: the Levinson, Euclidean, and Berlekamp-Massey algorithms, and show the relation between each of them and the Pad6 approximation problem. This connection has already been noticed for some of them, but we intend here to offer a synthetic view of these fast algorithms. We classify the algorithms solving YW equations with reference to three criteria, namely: 1) the path they follow in the Pad6 table; 2) the organization of the computation: we distinguish between one-pass and two-pass algorithms; and 3) the auxiliary variables used: some algorithms use the backward predictor of same degree as intermediate variable for computing the forward predictor (or vice versa), while others use two predictors of the same type but of successive degrees. This classification shows that the set of known classical algorithms is not complete, and we propose the missing variants. With these variants of the Berlekamp-Massey and Euclid algorithms, we are able to obtain both forward and backward predictors without additional cost. Furthermore, we give a unified representation of the twopass algorithms, in such a way that the application of the divide and conquer strategy becomes straightforward. A general doubling algorithm which represents all the associated doubling algorithms in an exhaustive way is provided. Hui-Min Zhang, was born in Fujian, China, in 1964. She received the B.S. degree in electric automation from the University of Textiles of China, Shanghai, in 1984, the Engineer Specialization diploma in signaux, images, and formes from the Ecole Suptrieure d'ElectricitC, Metz, France, in 1986, and the Ph.D. degree in digital signal processing from the Ecole Nationale SupCrieure des TClCcommunications and the Centre National d'Etudes des Ttlecommunications, Paris, France, in 1989. After a year of postdoctoral research at Electricit6 de France, she is currently working for Matra Ericsson Telecommunications, Massy, France, on several research projects within the Research for Advanced Communication in Europe (RACE).

Dynamic programming approach is an optimization approach that can solve the complex problem by dividing into overlapping sub-problems and solve those sub-problems stage by stage to get the optimal solution of the problem. Even though we have many problems solving approaches , dynamic programming is the approach the best to get the optimal solution of the problem. In this paper, we would like to discuss about the use of dynamic programming approach and how it is different from other problem solving techniques like divide and conquer, greedy method, branch and bound and backtracking. We discussed the dynamic programming approach in detail and given some of the applications of dynamic programming where we can apply this problem solving technique. Finally we discussed about the limitations of dynamic programming approach.

We present a framework for multi-core implementations of divide and conquer algorithms and show its efficiency and ease of use by applying it to the fundamental geo- metric problem of computing the convex hull of a point set. We concentrate on the Quickhull algorithm intro- duced in (2). In general the framework can easily be used for any D&C-algorithm. It

Non-parametric option pricing models, such as artificial neural networks, are often found to outperform their parametric counterparts in empirical option pricing exercises. In this context, non-parametric models are viewed as more flexible and amenable to adaptive learning. However, the main drawback of non-parametric approaches is their lack of stability, which is detrimental to out-of-sample performance. This is the key reason why one may prefer a parsimonious parametric model. This paper proposes a parametric Takagi-Sugeno-Kang (TSK) fuzzy rule-based option pricing model that requires only a small number of rules to describe highly complex non-linear functions. The findings for this data-driven approach indicate that the TSK model presents a robust option pricing tool that is superior to an array of well-known parametric models from the literature. In addition, its predictive performance is consistently no worse than that of a non-parametric feedforward neural network model.

In this study, a cluster-computing environment is employed as a computational platform. In order to increase the efficiency of the system, a dynamic task scheduling algorithm is proposed, which balances the load among the nodes of the cluster. The technique is dynamic, nonpreemptive, adaptive, and it uses a mixed centralised and decentralised policies. Based on the divide and conquer principle, the algorithm models the cluster as hyper-grids and then balances the load among them. Recursively, the hyper-grids of dimension k are divided into grids of dimensions k − 1, until the dimension is 1. Then, all the nodes of the cluster are almost equally loaded. The optimum dimension of the hyper-grid is chosen in order to achieve the best performance. The simulation results show the effective use of the algorithm. In addition, we determined the critical points (lower bounds) in which the algorithm can to be triggered. 2 System Model & Hyper-Grids A Computational Cluster (CC) is a collection of independent processing-nodes interconnected by a network. We as

Learning a large number of simple local concepts is both faster and easier than learning a single global concept. Inspired by this principle of divide and conquer, a number of modular learning approaches have been proposed by the computational intelligence community. In modular learning, the classification/regression/clustering problem is first decomposed into a number of simpler subproblems, a module is learned for each of these subproblems, and finally their results are integrated by a suitable combining method. Mixtures of experts and clustering are two of the techniques that are describable in this paradigm. In this paper we present a broad framework for Generalized Associative Modular Learning Systems (GAMLS). Modularity is introduced through soft association of each training pattern with every module. The coupled problems of learning the module parameters and learning associations are solved iteratively using deterministic annealing. Starting at a high temperature with only one module, GAMLS framework automatically evolves the required number of modules through a systematic growing and pruning technique. Each phase begins by splitting every module in the previous phase into two, updating these new modules and then pruning and merging any redundant modules. A phase transition is induced by temperature decay. A number of existing modular learning problems, both unsupervised (clustering, mixture model density, mixture of principal components) and supervised (mixture of experts, radial basis function networks), can be effectively tackled in GAMLS. Case studies for clustering and regression using mixture of experts are provided for a number of datasets showing the efficacy of the GAMLS framework in evolving the right number of modules, inducing interpretable localizations among modules and robustness of the solution obtained. More importantly, this framework provides a unifying view for understanding and characterizing modular learning methods.

Description: This seminar course introduces students to the theory and practice of grand strategy, understood as a great design or idea that orders and guides what a state does in interactions with the other actors, whether states or non-states in the international system. Grand strategy is tailored to the state, which means that different kinds of states will pursue various grand strategies. Accordingly, this course surveys the types of grand strategy available to states. First, it expounds on the main tenets of each grand strategy; and then it provides a historical illustration of the grand strategy in action. This exercise prepares students for the tasks of assessing and recommending grand strategy for states in present world politics settings. The states discussed in this course are the great powers. This does not mean that states that are not great powers cannot have grand strategies. But the great powers are the most important and impactful states. Therefore, it stands to reason that any examination of grand strategy archetypes has to begin with the study of the grand strategies they employ. In this respect, the course will cover a large spectrum of great power grand strategic behavior over the last four centuries, including the most significant strategists and most influential states. There are three kinds of great powers: rising powers, status quo powers, and declining powers. Rising powers seek to gain a seat or to improve considerably their position at the great power table. Status quo powers seek to conserve and moderately improve their current position among great powers. Declining powers aim at a minimum to prevent and limit further losses, while at a maximum to recover what they have lost. Hence, rising, status quo, and declining great powers follow different grand strategies. PAGE 15

History indicates that the security community commonly takes a divide-and-conquer approach to battling malware threats: identify the essential and inalienable components of an attack, then develop detection and prevention techniques that directly target one or more of the essential components. This abstraction is evident in much of the literature for buffer overflow attacks including, for instance, stack protection and NOP sled detection. It comes as no surprise then that we approach shellcode detection and prevention in a similar fashion. However, the common belief that components of polymorphic shellcode (e.g., the decoder) cannot reliably be hidden suggests a more implicit and broader assumption that continues to drive contemporary research: namely, that valid and complete representations of shellcode are fundamentally different in structure than benign payloads. While the first tenet of this assumption is philosophically undeniable (i.e., a string of bytes is either shellcode or it is not), truth of the latter claim is less obvious if there exist encoding techniques capable of producing shellcode with features nearly indistinguishable from non-executable content. In this paper, we challenge the assumption that shellcode must conform to superficial and discernible representations. Specifically, we demonstrate a technique for automatically producing English Shellcode, transforming arbitrary shellcode into a representation that is superficially similar to English prose. The shellcode is completely self-containedi.e., it does not require an external loader and executes as valid IA32 code)-and can typically be generated in under an hour on commodity hardware. Our primary objective in this paper is to promote discussion and stimulate new ideas for thinking ahead about preventive measures for tackling evolutions in code-injection attacks.

We present a de novo hierarchical simulation framework for first-principles based predictive simulations of materials and their validation on high-end parallel supercomputers and geographically distributed clusters. In this framework, highend chemically reactive and non-reactive molecular dynamics (MD) simulations explore a wide solution space to discover microscopic mechanisms that govern macroscopic material properties, into which highly accurate quantum mechanical (QM) simulations are embedded to validate the discovered mechanisms and quantify the uncertainty of the solution. The framework includes an embedded divide-andconquer (EDC) algorithmic framework for the design of linear-scaling simulation algorithms with minimal bandwidth complexity and tight error control. The EDC framework also enables adaptive hierarchical simulation with automated model transitioning assisted by graph-based event tracking. A tunable hierarchical cellular decomposition parallelization framework then maps the O(N) EDC algorithms onto petaflops computers, while achieving performance tunability through a hierarchy of parameterized cell data/ computation structures, as well as its implementation using hybrid grid remote procedure call + message passing + threads programming. High-end computing platforms such as IBM BlueGene/L, SGI Altix 3000 and the NSF TeraGrid provide an excellent test grounds for the framework. On these platforms, we have achieved unprecedented scales of quantum-mechanically accurate and well validated, chemi-cally reactive atomistic simulations-1.06 billion-atom fast reactive force-field MD and 11.8 million-atom (1.04 trillion grid points) quantum-mechanical MD in the framework of the EDC density functional theory on adaptive multigridsin addition to 134 billion-atom non-reactive space-time multiresolution MD, with the parallel efficiency as high as 0.998 on 65,536 dual-processor BlueGene/L nodes. We have also achieved an automated execution of hierarchical QM/MD simulation on a grid consisting of 6 supercomputer centers in the US and Japan (in total of 150,000 processor hours), in which the number of processors change dynamically on demand and resources are allocated and migrated dynamically in response to faults. Furthermore, performance portability has been demonstrated on a wide range of platforms such as BlueGene/L, Altix 3000, and AMD Opteron-based Linux clusters. Downloaded from − − −4 of the bulk density. (c) Energy conservation in EDC-DFT based MD simulation of liquid Rb at 1400 K. The domain and buffer sizes are 16.424 and 8.212 a.u., respectively.

Procedures to divide a cake among n people with n-1 cuts (the minimum number) are analyzed and compared. For 2 persons, cut-and-choose, while envy-free and efficient, limits the cutter to exactly 50% if he or she is ignorant of the chooser's preferences, whereas the chooser can generally obtain more. By comparison, a new 2person surplus procedure (SP'), which induces the players to be truthful in order to maximize their minimum allocations, leads to a proportionally equitable division of the surplus-the part that remains after each player receives 50%-by giving each person a certain proportion of the surplus as he or she values it.

Drawing on Merritt's divide-and-conquer sorting taxonomy [1], we model comparison-based sorting as an abstract class with a template method to perform the sort by relegating the splitting and joining of arrays to its concrete subclasses. Comparison on objects is carried out via an abstract ordering strategy. This reduces code complexity and simplifies the analyses of the various concrete sorting algorithms. Performance measurements and visualizations can be added without modifying any code by utilizing the decorator design pattern. This object-oriented design not only provides the student a concrete way of unifying seemingly disparate sorting algorithms but also help him/her differentiate them at the proper level of abstraction.

This paper investigates the classification of multiclass motor imagery for electroencephalogram (EEG)-based Brain-Computer Interface (BCI) using the Filter Bank Common Spatial Pattern (FBCSP) algorithm. The FBCSP algorithm classifies EEG measurements from features constructed using subject-specific temporal-spatial filters. However, the FBCSP algorithm is limited to binary-class motor imagery. Hence, this paper proposes 3 approaches of multi-class extension to the FBCSP algorithm: One-versus-Rest, Pair-Wise and Divide-and-Conquer. These approaches decompose the multi-class problem into several binary-class problems. The study is conducted on the BCI Competition IV dataset IIa, which comprises single-trial EEG data from 9 subjects performing 4-class motor imagery of left-hand, right-hand, foot and tongue actions. The results showed that the multi-class FBCSP algorithm could extract features that matched neurophysiological knowledge, and yielded the best performance on the evaluation data compared to other international submissions.

We propose a stable and efficient divide-and-conquer algorithm for computing the eigendecomposition of a symmetric tri-block-diagonal matrix. The matrix can be derived from discretizing Laplace operator eigenvalue in some two-dimensional graphs. All numerical results show that our algorithm is competitive with other methods, such as e.g QR algorithm, Cuppen's divide-and-conquer algorithm. We also show how to improve our algorithm by Fast Multipole Method.

We prwcnt two new algorithms, ADT and MDT, for solving order-n Toeplik systema of linear equations Tz-b in time O(n log'n) and space O(n). The fastest algorithma previously known, wch as Trench's algorithm, require time Q(n2) and require that all principal submatri~ of T be nonsin&u.

The goal of the PRISM project is the development of infrastructure and algorithms for the parallel solution of eigenvalue problems. We are currently investigating a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). After brie y reviewing SYISDA, we discuss the algorithmic highlights of a distributed-memory implementation of this approach. These include a fast matrix-matrix multiplication algorithm, a new approach to parallel band reduction and tridiagonalization, and a harness for coordinating the divide-and-conquer parallelism in the problem. We also present performance results of these kernels as well as the overall SYISDA implementation on the Intel Touchstone Delta prototype.

When applied to a system, the doctrine of successive refinement is a divide-and-conquer strategy. Complex systems are sucessively divided into pieces that are less complex, until they are simple enough to be conquered. This decomposition results in several structures for describing the product system and the producing system. These structures play important roles in systems engineering and project management. Many of the remaining sections in this chapter are devoted to describing some of these key structures. Structures that describe the product system include, but are not limited to, the requirements tree, system architecture and certain symbolic information such as system drawings, schematics, and data bases. The structures that describe the producing system include the project's work breakdown, schedules, cost accounts and organization.

The computation of the spectral decomposition of a symmetric arrowhead matrix is an important problem in applied mathematics . It is also the kernel of divide and conquer algorithms for computing the Schur decomposition of symmetric tridiagonal matrices and diagonal-plus-semiseparable matrices . The eigenvalues of symmetric arrowhead matrices are the zeros of a secular equation and some iterative algorithms have been proposed for their computation ]. An important issue of these algorithms is the choice of the initial guess. Let α1 ≤ α2 ≤ . . . ≤ αn−1 be the entries of the main diagonal of a symmetric arrowhead matrix of order n. Denoted by λi, i = 1, . . . , n, the corresponding eigenvalues, it is well know that αi ≤ λi+1 ≤ αi+1, i = 1, . . . , n − 2. An algorithm for computing each eigenvalue λi, i = 1, . . . , n, of a symmetric arrowhead matrix with monotonic quadratic convergence, independent of the choice of the initial guess in the interval ]αi−1, αi[ is proposed in this paper. Although the eigenvalues of a symmetric arrowhead matrix can be computed efficiently, a loss of orthogonality can occur in the computed matrix of eigenvectors .In this paper we propose also a simple, stable and efficient way to compute the eigenvectors of arrowhead matrices.

Due to wide application of evapotranspiration (ET) data, a number of indirect methods for estimation of reference ET (ETo) have been developed. Therefore, it becomes impractical for many users to select the best ETo estimation method for the available data and climatic condition. To overcome this problem, a decision support system (DSS ET) was developed which supports 22 ETo estimation methods with varied options for calculation of various intermediate parameters, generalized data input format with copy-paste option from spreadsheet applications, visualize/check input data and results, features to estimate missing data, and user-friendly graphical user interface (GUI) that enhances its applicability as a handy research and teaching tool. Monthly meteorological data for 32 years (1971-2002) were procured from IMD, Pune, for 133 selected stations evenly distributed over five climatic ecosystems of India. Altogether 18 ETo estimation methods, including the globally accepted standard method FAO-56 Penman-Monteith (PM), were found applicable for the data availability conditions used in this study. In absence of lysimeter measured ETo data, the FAO-56 PM ETo values were used to estimate the normal monthly ETo for each station and climatic condition. Finally, ranks were assigned to each applicable ETo method based on weighted standard error of estimate (WSEE) with respect to the FAO-56 PM method for different climatic ecosystems. If, because of poor data availability condition, it becomes unavoidable to use methods other than FAO-56 PM, then these ranks can readily be used to select the best applicable method in the existing data and climatic conditions to get ETo values closer to FAO-56 PM estimates.