Largest Differencing Method
   HOME





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]  


picture info

Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). 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 preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Balanced Number Partitioning
In telecommunications and professional audio, a balanced line or balanced signal pair is an electrical circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths, to ground, and to other circuits. The primary advantage of the balanced line format is good rejection of common-mode noise and interference when fed to a differential device such as a transformer or differential amplifier.G. Ballou, ''Handbook for Sound Engineers'', Fifth Edition, Taylor & Francis, 2015, p. 1267–1268. As prevalent in sound recording and reproduction, balanced lines are referred to as balanced audio. A common form of balanced line is twin-lead, used for radio frequency communications. Also common is twisted pair, used for traditional telephone, professional audio, or for data communications. They are to be contrasted to unbalanced lines, such as coaxial cable, which is designed to have its return conductor connected to ground, or circuits wh ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Joseph Ibn Habib
Joseph ibn Habiba (), 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 Nedarim was also incl ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Nachmanides
Moses ben Nachman ( ''Mōše ben-Nāḥmān'', "Moses son of Nachman"; 1194–1270), commonly known as Nachmanides (; ''Nakhmanídēs''), and also referred to by the acronym Ramban (; ) and by the contemporary nickname Bonastruc ça Porta (; literally "Mazel tov, Mazel Tov near the Gate", see ), was a leading medieval Jewish scholar, Jews of Catalonia, Catalan rabbi, philosophy, philosopher, physician, Kabbalah, kabbalist, and Jewish commentaries on the Bible, 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 Siege of Jerusalem (1099), destruction by the Crusaders in 1099. Name "Nachmanides" () is a Greek language, 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 Cata ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Balanced Partition Problem
In telecommunications and professional audio, a balanced line or balanced signal pair is an electrical circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths, to ground, and to other circuits. The primary advantage of the balanced line format is good rejection of common-mode noise and interference when fed to a differential device such as a transformer or differential amplifier.G. Ballou, ''Handbook for Sound Engineers'', Fifth Edition, Taylor & Francis, 2015, p. 1267–1268. As prevalent in sound recording and reproduction, balanced lines are referred to as balanced audio. A common form of balanced line is twin-lead, used for radio frequency communications. Also common is twisted pair, used for traditional telephone, professional audio, or for data communications. They are to be contrasted to unbalanced lines, such as coaxial cable, which is designed to have its return conductor connected to ground, or circuits wh ...
[...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]  


picture info

Significant Figures
Significant figures, also referred to as significant digits, are specific digits within a number that is written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcome of a measurement (such as length, pressure, volume, or mass), if the number of digits exceeds what the measurement instrument can resolve, only the digits that are determined by the resolution are dependable and therefore considered significant. For instance, if a length measurement yields 114.8 mm, using a ruler with the smallest interval between marks at 1 mm, the first three digits (1, 1, and 4, representing 114 mm) are certain and constitute significant figures. Further, digits that are uncertain yet meaningful are also included in the significant figures. In this example, the last digit (8, contributing 0.8 mm) is likewise considered significant despite its uncertainty. Therefore, this measurement contains ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Richard E
Richard is a male given name. It originates, via Old French, from compound of the words descending from Proto-Germanic language">Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include " Richie", " Dick", " Dickon", " Dickie", " Rich", " Rick", "Rico (name), Rico", " Ricky", and more. Richard is a common English (the name was introduced into England by the Normans), German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Portuguese and Spanish "Ricardo" and the Italian "Riccardo" (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * ...
[...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 Longest-processing-time-first scheduling, LPT scheduling. It processes the inputs by descending order of value, fro ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

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 and objective 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 polytope. A linear programming algorithm finds a point in the polytope where this function has the largest (or smallest) value if such a point exists. Linear programs are problems that can be expressed in standard form as: : \begin & ...
[...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 partition of a set, 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 Heuristic, 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 problem, 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 ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]