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Harmonic Bin Packing
Harmonic bin-packing is a family of online algorithms for 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 ma .... The input to such an algorithm is a list of items of different sizes. The 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 harmonic bin-packing algorithms rely on partitioning the items into categories based on their sizes, following a Harmonic progression. There are several variants of this idea. Harmonic-''k'' The Harmonic-''k'' algorithm partitions the interval of sizes (0,1] harmonically into k-1 pieces I_j := (1/(j+1),1/j] for 1\leq j < k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, 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 an offline algorithm. On the other hand, insertion sort considers one input element per iteration and produces a partial solution without considering future elements. Thus insertion sort is an online algorithm. Note that the final result of an insertion sort ... [...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 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 '' Θ''(''n'' log ''n'') time, where ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Progression (mathematics)
In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequence of the form : \frac,\ \frac,\ \frac,\ \frac, \cdots, where ''a'' is not zero and −''a''/''d'' is not a natural number, or a finite sequence of the form : \frac,\ \frac,\ \frac,\ \frac, \cdots,\ \frac, where ''a'' is not zero, ''k'' is a natural number and −''a''/''d'' is not a natural number or is greater than ''k''. Examples * 1, 1/2, 1/3, 1/4, 1/5, 1/6, sometimes referred to as the ''harmonic sequence'' * 12, 6, 4, 3, \tfrac, 2, … , \tfrac, … * 30, −30, −10, −6, − \tfrac, … , \tfrac * 10, 30, −30, −10, −6, − , … , \tfrac Sums of harmonic progressions Infinite harmonic progressions are not summable (sum to infinity). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
