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First-fit-decreasing Bin Packing
First-fit-decreasing (FFD) is an algorithm for bin packing. Its input is a list of items of different sizes. Its output is a ''packing'' - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem, so we use an approximately-optimal heuristic. Description The FFD algorithm works as follows. * Order the items from largest to smallest. * Open a new empty bin, bin #1. * For each item from largest to smallest, find the ''first'' bin into which the item fits, if any. ** If such a bin is found, put the new item in it. ** Otherwise, open a new empty bin put the new item in it. In short: FFD orders the items by descending size, and then calls first-fit bin packing. An equivalent description of the FFD algorithm is as follows. * Order the items from largest to smallest. * While there are remaining items: ** ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bin Packing
The bin packing problem is an optimization problem, in which items of different sizes must be packed into a finite number of bins or containers, each of a fixed given capacity, in a way that minimizes the number of bins used. The problem has many applications, such as filling up containers, loading trucks with weight capacity constraints, creating file backups in media and technology mapping in Field-programmable gate array, FPGA semiconductor chip design. Computationally, the problem is NP-hard, and the corresponding decision problem - deciding if items can fit into a specified number of bins - is NP-complete. Despite its worst-case hardness, optimal solutions to very large instances of the problem can be produced with sophisticated algorithms. In addition, many approximation algorithms exist. For example, the First-fit bin packing, first fit algorithm provides a fast but often non-optimal solution, involving placing each item into the first bin in which it will fit. It require ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, short-term goal or approximation. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples that employ heuristics include using trial and error, a rule of thumb or an educated guess. Heuristics are the strategies derived from previous experiences with similar problems. These strategies depend on using readily accessible, though loosely applicable, information to control problem solving in human beings, machines and abstract issues. When an individual applies a heuristic in practice, it generally performs as expected. However it can alternatively cre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-fit Bin Packing
First-fit (FF) is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a ''packing'' - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The first-fit algorithm uses the following heuristic: * It keeps a list of open bins, which is initially empty. * When an item arrives, find the ''first'' bin into which the item can fit, if any. ** If such a bin is found, the new item is placed inside it. ** Otherwise, a new bin is opened and the coming item is placed inside it. Approximation ratio Denote by FF(L) the number of bins used by First-Fit, and by OPT(L) the optimal number of bins possible for the list L. The analysis of FF(L) was done in several steps. * The first upper bound of FF(L) \leq 1.7\mathrm+3 for FF was proven by Ullman in 1971. * In 1972, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David S
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". was, according to the Hebrew Bible, the third king of the United Kingdom of Israel. In the Books of Samuel, he is described as a young shepherd and harpist who gains fame by slaying Goliath, a champion of the Philistines, in southern Canaan. David becomes a favourite of Saul, the first king of Israel; he also forges a notably close friendship with Jonathan, a son of Saul. However, under the paranoia that David is seeking to usurp the throne, Saul attempts to kill David, forcing the latter to go into hiding and effectively operate as a fugitive for several years. After Saul and Jonathan are both killed in battle against the Philistines, a 30-year-old David is anointed king over all of Israel and Judah. Following his rise to power, David ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infimum And Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and maxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Best-fit Bin Packing
Best-fit is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a ''packing'' - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The best-fit algorithm uses the following heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...: * It keeps a list of open bins, which is initially empty. * When an item arrives, it finds the bin with the ''maximum load'' into which the item can fit, if any. ** If such a bin is found, the new item is placed inside it. ** Otherwise, a new bin is opened and the coming item is placed inside it. Approximation ratio De ... [...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] |
Identical-machines Scheduling
Identical-machines scheduling is an optimization problem in computer science and operations research. We are given ''n'' jobs ''J''1, ''J''2, ..., ''Jn'' of varying processing times, which need to be scheduled on ''m'' identical machines, such that a certain objective function is optimized, for example, the makespan is minimized. Identical machine scheduling is a special case of uniform machine scheduling, which is itself a special case of optimal job scheduling. In the general case, the processing time of each job may be different on different machines; in the case of identical machine scheduling, the processing time of each job is the same on each machine. Therefore, identical machine scheduling is equivalent to multiway number partitioning. A special case of identical machine scheduling is single-machine scheduling. In the standard three-field notation for optimal job scheduling problems, the identical-machines variant is denoted by P in the first field. For example, " P, , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
