|
Bin Packing Problem
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, splitting a network prefix into multiple subnets, and technology mapping in 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 algorithm provides a fast but often non-optimal solution, involving placing each item into the first bin in which it will fit. It requires ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Optimization Problem
In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ..., an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: * An optimization problem with discrete variables is known as a '' discrete optimization'', in which an object such as an integer, permutation or graph must be found from a countable set. * A problem with continuous variables is known as a '' continuous optimization'', in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems. Search space In the context of an optim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Online Algorithm
In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from the beginning and is required to output an answer which solves the problem at hand. In operations research Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods to improve management and ..., the area in which online algorithms are developed is called online optimization. As an example, consider the sorting algorithms selection sort and insertion sort: selection sort repeatedly selects the minimum element from the unsorted remainder and places it at the front, which requires access to the entire input; it is thus a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Refined 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] [Amazon] |
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 (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...: * 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. The ''load'' of a bin is defined as the sum of sizes of existing items in the bin before placing the new item. ** If such a bin is found, the new item is placed i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Next-fit Bin Packing
Next-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 next-fit algorithm uses the following heuristic: * It keeps a ''current bin'', which is initially empty. * When an item arrives, it checks whether the item fits into the current bin. ** If it fits, it is placed inside it. ** Otherwise, the current bin is closed, a new bin is opened and the coming item is placed inside this new bin. Next-Fit is a bounded space algorithm - it requires only one partially-filled bin to be open at any time. The algorithm was studied by David S. Johnson in his doctoral thesis in 1973. Run time The running time of NextFit can be bounded by \mathcal(n), where n is the number of items in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
David S
David (; , "beloved one") was a king of ancient Israel and Judah and the Kings of Israel and Judah, third king of the Kingdom of Israel (united monarchy), United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Canaanite and Aramaic inscriptions, Aramaic-inscribed stone erected by a king of Aram-Damascus in the late 9th/early 8th centuries BCE to commemorate a victory over two enemy kings, contains the phrase (), which is translated as "Davidic line, House of David" by most scholars. The Mesha Stele, erected by King Mesha of Moab in the 9th century BCE, may also refer to the "House of David", although this is disputed. According to Jewish works such as the ''Seder Olam Rabbah'', ''Seder Olam Zutta'', and ''Sefer ha-Qabbalah'' (all written over a thousand years later), David ascended the throne as the king of Judah in 885 BCE. Apart from this, all that is known of David comes from biblical literature, Historicity of the Bible, the historicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
