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Largest Differencing Method
In computer science, the largest differencing method is an algorithm for solving the partition problem and the multiway number partitioning. It is also called the Karmarkar–Karp algorithm after its inventors, Narendra Karmarkar and Richard M. Karp. It is often abbreviated as LDM. The algorithm The input to the algorithm is a set ''S'' of numbers, and a parameter ''k''. The required output is a partition of ''S'' into ''k'' subsets, such that the sums in the subsets are as nearly equal as possible. The main steps of the algorithm are: # Order the numbers from large to small. #Replace the largest and second-largest numbers by their difference. #If two or more numbers remain, return to step 1. # Using backtracking, compute the partition. Two-way partitioning For ''k''=2, the main step (2) works as follows. * Take the two largest numbers in ''S'', remove them from ''S'', and insert their difference (this represents a decision to put each of these numbers in a different subset ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Number Partitioning
Balanced number partitioning is a variant of multiway number partitioning in which there are constraints on the number of items allocated to each set. The input to the problem is a set of ''n'' items of different sizes, and two integers ''m'', ''k''. The output is a partition of the items into ''m'' subsets, such that the number of items in each subset is at most ''k''. Subject to this, it is required that the sums of sizes in the ''m'' subsets are as similar as possible. An example application is identical-machines scheduling where each machine has a job-queue that can hold at most ''k'' jobs. The problem has applications also in manufacturing of VLSI chips, and in assigning tools to machines in flexible manufacturing systems. In the standard three-field notation for optimal job scheduling problems, the problem of minimizing the largest sum is sometimes denoted by "P , # ≤ k , ''C''max". The middle field "# ≤ k" denotes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Ibn Habib
Joseph ibn Habiba ( he, יוסף חביבא), also known as Joseph Havivah and Nimmukei Yosef, after the title of his book, was a Spanish Talmudist who flourished in the 14th and 15th centuries. He lived in Barcelona. Nimmukei Yosef Like his predecessor, R. Nissim ben Reuben (RaN), Ibn Ḥabib wrote a commentary on the ''halachot'' of Isaac Alfasi, entitled ''Nimmuḳei Yosef,'' published with the text and the commentary of R. Nissim (Constantinople, 1509). Against the opinion of David Conforte that Ibn Ḥabib wrote commentaries only upon those tractates which R. Nissim had omitted, Azulai''Shem ha-Gedolim'' proved that Ibn Ḥabib's ''Nimmuḳei Yosef'' covered the entire ''halachot'' of Isaac Alfasi, but a part of it had remained unpublished, and that the commentary to the ''halachot'' of Moed Katan and Makkot, attributed to R. Nissim, belongs to Ibn Ḥabib. The latter quotes Asher ben Jehiel, Ritva, his master RaM, and R. Nissim himself. ''Nimmuḳei Yosef'' on Ketubot and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nachmanides
Moses ben Nachman ( he, מֹשֶׁה בֶּן־נָחְמָן ''Mōše ben-Nāḥmān'', "Moses son of Nachman"; 1194–1270), commonly known as Nachmanides (; el, Ναχμανίδης ''Nakhmanídēs''), and also referred to by the acronym Ramban () and by the contemporary nickname Bonastruc ça Porta (literally "Mazel Tov near the Gate", see astruc), was a leading medieval Jewish scholar, Sephardic rabbi, philosopher, physician, kabbalist, and biblical commentator. He was raised, studied, and lived for most of his life in Girona, Catalonia. He is also considered to be an important figure in the re-establishment of the Jewish community in Jerusalem following its destruction by the Crusaders in 1099. Name "Nachmanides" (Ναχμανίδης) is a Greek-influenced formation meaning "son of Nahman". He is also commonly known by the Hebrew acronym (Ra-M-Ba-N, for ''Rabbeinu Mōšeh bēn-Nāḥmān'', "Our Rabbi Moses son of Nahman"). His Catalan name was Bonastruc ça Porta (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Partition Problem
Balanced number partitioning is a variant of multiway number partitioning in which there are constraints on the number of items allocated to each set. The input to the problem is a set of ''n'' items of different sizes, and two integers ''m'', ''k''. The output is a partition of the items into ''m'' subsets, such that the number of items in each subset is at most ''k''. Subject to this, it is required that the sums of sizes in the ''m'' subsets are as similar as possible. An example application is identical-machines scheduling where each machine has a job-queue that can hold at most ''k'' jobs. The problem has applications also in manufacturing of VLSI chips, and in assigning tools to machines in flexible manufacturing systems. In the standard three-field notation for optimal job scheduling problems, the problem of minimizing the largest sum is sometimes denoted by "P , # ≤ k , ''C''max". The middle field "# ≤ k" denotes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anytime Algorithm
In computer science, an anytime algorithm is an algorithm that can return a valid solution to a problem even if it is interrupted before it ends. The algorithm is expected to find better and better solutions the longer it keeps running. Most algorithms run to completion: they provide a single answer after performing some fixed amount of computation. In some cases, however, the user may wish to terminate the algorithm prior to completion. The amount of computation required may be substantial, for example, and computational resources might need to be reallocated. Most algorithms either run to completion or they provide no useful solution information. Anytime algorithms, however, are able to return a partial answer, whose quality depends on the amount of computation they were able to perform. The answer generated by anytime algorithms is an approximation of the correct answer. Names An anytime algorithm may be also called an "interruptible algorithm". They are different from contract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Significant Figures
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expressing the result of a measurement (e.g., length, pressure, volume, or mass) has more digits than the number of digits allowed by the measurement resolution, then only as many digits as allowed by the measurement resolution are reliable, and so only these can be significant figures. For example, if a length measurement gives 114.8 mm while the smallest interval between marks on the ruler used in the measurement is 1 mm, then the first three digits (1, 1, and 4, showing 114 mm) are certain and so they are significant figures. Digits which are uncertain but ''reliable'' are also considered significant figures. In this example, the last digit (8, which adds 0.8 mm) is also considered a significant figure even though ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greedy Number Partitioning
In computer science, greedy number partitioning is a class of greedy algorithms for multiway number partitioning. The input to the algorithm is a set ''S'' of numbers, and a parameter ''k''. The required output is a partition of ''S'' into ''k'' subsets, such that the sums in the subsets are as nearly equal as possible. Greedy algorithms process the numbers sequentially, and insert the next number into a bin in which the sum of numbers is currently smallest. Approximate algorithms The simplest greedy partitioning algorithm is called list scheduling. It just processes the inputs in any order they arrive. It always returns a partition in which the largest sum is at most 2-\frac times the optimal (minimum) largest sum. This heuristic can be used as an online algorithm, when the order in which the items arrive cannot be controlled. An improved greedy algorithm is called LPT scheduling. It processes the inputs by descending order of value, from large to small. Since it needs to pre-o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Integer Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where this function has the smallest (or largest) value if such a point exists. Linear programs are problems that can be expressed in canonical form as : \begin & \text && \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multifit Algorithm
The multifit algorithm is an algorithm for multiway number partitioning, originally developed for the problem of identical-machines scheduling. It was developed by Coffman, Garey and Johnson. Its novelty comes from the fact that it uses an algorithm for another famous problem - the bin packing problem - as a subroutine. The algorithm The input to the algorithm is a set ''S'' of numbers, and a parameter ''n''. The required output is a partition of ''S'' into ''n'' subsets, such that the largest subset sum (also called the makespan) is as small as possible. The algorithm uses as a subroutine, an algorithm called '' first-fit-decreasing bin packing'' (FFD). The FFD algorithm takes as input the same set ''S'' of numbers, and a bin-capacity ''c''. It heuristically packs numbers into bins such that the sum of numbers in each bin is at most ''C'', aiming to use as few bins as possible. Multifit runs FFD multiple times, each time with a different capacity ''C'', until it finds some ''C'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Problem
In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset ''S'' of positive integers can be partitioned into two subsets ''S''1 and ''S''2 such that the sum of the numbers in ''S''1 equals the sum of the numbers in ''S''2. Although the partition problem is NP-complete, there is a pseudo-polynomial time dynamic programming solution, and there are heuristics that solve the problem in many instances, either optimally or approximately. For this reason, it has been called "the easiest hard problem". There is an optimization version of the partition problem, which is to partition the multiset ''S'' into two subsets ''S''1, ''S''2 such that the difference between the sum of elements in ''S''1 and the sum of elements in ''S''2 is minimized. The optimization version is NP-hard, but can be solved efficiently in practice. The partition problem is a special case of two related problems: * In the subset sum problem, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Scientist
__NOTOC__ ''American Scientist'' (informally abbreviated ''AmSci'') is an American bimonthly science and technology magazine published since 1913 by Sigma Xi, The Scientific Research Society. In the beginning of 2000s the headquarters was in New Haven, CT. Each issue includes feature articles written by prominent scientists and engineers who review research in fields from molecular biology to computer engineering. Each issue also includes the work of cartoonists, including those of Sidney Harris, Benita Epstein Benita L. Epstein is a prolific gag cartoonist for magazines, greeting cards, websites and newspapers. She was a regular contributor to the comic strip ''Six Chix'', distributed by King Features Syndicate. Before becoming a cartoonist, Epstein ea ..., and Mark Heath. Also included is the ''Scientists' Nightstand'' that reviews a vast range of science-related books and novels. ''American Scientist Online'' () was launched in May 2003. References External links * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
