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# 数学代写|计算复杂度代写Computational Complexity代考|COS522 Big-Oh notation

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## 数学代写|计算复杂度代写Computational Complexity代考|Big-Oh notation

As mentioned above, we will typically measure the computational efficiency algorithm as the number of a basic operations it performs as a function of its input length. That is, the efficiency of an algorithm can be captured by a function $T$ from the set of natural numbers $\mathbb{N}$ to itself such that $T(n)$ is equal to the maximum number of basic operations that the algorithm performs on inputs of length $n$. However, this function is sometimes be overly dependant on the low-level details of our definition of a basic operation. For example, the addition algorithm will take about three times more operations if it uses addition of single digit binary (i.e., base 2) numbers as a basic operation, as opposed to decimal (i.e., base 10) numbers. To help us ignore these low level details and focus on the big picture, the following well known notation is very useful:

DEFINITION $1.2$ (BIG-OH NOTATION)
If $f, g$ are two functions from $\mathbb{N}$ to $\mathbb{N}$, then we (1) say that $f=O(g)$ if there exists a constant $c$ such that $f(n) \leq c \cdot g(n)$ for every sufficiently large $n$, (2) say that $f=\Omega(g)$ if $g=O(f)$, (3) say that $f=\Theta(g)$ is $f=O(g)$ and $g=O(f),(4)$ say that $f=o(g)$ if for every $\epsilon>0, f(n) \leq \epsilon \cdot g(n)$ for every sufficiently large $n$, and $(5)$ say that $f=\omega(g)$ if $g=o(f)$.

To emphasize the input parameter, we often write $f(n)=O(g(n))$ instead of $f=O(g)$, and use similar notation for $o, \Omega, \omega, \Theta$.

## 数学代写|计算复杂度代写Computational Complexity代考|Modeling computation and eﬃﬃﬃciency

We start with an informal description of computation. Let $f$ be a function that takes a string of bits (i.e., a member of the set ${0,1}^$ ) and outputs, say, either 0 or 1 . Informally speaking, an algorithm for computing $f$ is a set of mechanical rules, such that by following them we can compute $f(x)$ given any input $x \in{0,1}^$. The set of rules being followed is fixed (i.e., the same rules must work for all possible inputs) though each rule in this set may be applied arbitrarily many times. Each rule involves one or more of the following “elementary” operations:

1. Read a bit of the input.Read a bit (or possibly a symbol from a slightly larger alphabet, say a digit in the set ${0, \ldots, 9})$ from the “scratch pad” or working space we allow the algorithm to use.
2. Based on the values read,
3. Write a bit/symbol to the scratch pad.
4. Either stop and output 0 or 1 , or choose a new rule from the set that will be applied next.
5. Finally, the running time is the number of these basic operations performed. Below, we formalize all of these notions.

## 数学代写|计算复杂度代写计算复杂度代考|Big-Oh符号

.

$f, g$ 是两个函数 $\mathbb{N}$ 到 $\mathbb{N}$，那么我们(1)说 $f=O(g)$ 如果存在一个常数 $c$ 如此这般 $f(n) \leq c \cdot g(n)$ 对于每一个足够大的 $n$，(2)说 $f=\Omega(g)$ 如果 $g=O(f)$，(3)说 $f=\Theta(g)$ 是 $f=O(g)$ 和 $g=O(f),(4)$ 说出来 $f=o(g)$ 如果对于每一个 $\epsilon>0, f(n) \leq \epsilon \cdot g(n)$ 对于每一个足够大的 $n$，以及 $(5)$ 说出来 $f=\omega(g)$ 如果 $g=o(f)$.

## 数学代写|计算复杂度代写Computational Complexity代考|建模计算和效率

.计算复杂度代考|

1. 读取输入的一个bit。从我们允许算法使用的“草稿板”或工作空间中读取一点(或者可能是稍大一点的字母表中的一个符号，比如集合${0, \ldots, 9})$中的一个数字)。
2. 根据读取的值，
3. 向刮痧板上写入位/符号。
4. 要么停止并输出0或1，要么从接下来将应用的集合中选择一个新规则。
5. 最后，运行时间是执行这些基本操作的次数。下面，我们将这些概念形式化。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。