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# 计算机代写|算法和结构代写Data Structures and Algorithms代考|The Average Case

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## 计算机代写|算法和结构代写Data Structures and Algorithms代考|The Average Case

Indeed, in a worst-case scenario, Selection Sort is faster than Insertion Sort. However, it is critical that we also take into account the average-case scenario.
Why?

By definition, the cases that occur most frequently are average scenarios. The worst- and best-case scenarios happen only rarely. Let’s look at this simple bell curve:

Best- and worst-case scenarios happen relatively infrequently. In the real world, however, average scenarios are what occur most of the time.

And this makes a lot of sense. Think of a randomly sorted array. What are the odds that the values will occur in perfect ascending or descending order? It’s much more likely that the values will be all over the place.
Let’s examine Insertion Sort in context of all scenarios.
We’ve looked at how Insertion Sort performs in a worst-case scenario-where the array sorted in descending order. In the worst case, we pointed out that in each passthrough, we compare and shift every value that we encounter. (We calculated this to be a total of $\mathrm{N}^2$ comparisons and shifts.)

In the best-case scenario, where the data is already sorted in ascending order, we end up making just one comparison per passthrough, and not a single shift, since each value is already in its correct place.

Where data is randomly sorted, however, we’ll have passthroughs in which we compare and shift all of the data, some of the data, or possibly none of data. If you’ll look at our preceding walkthrough example, you’ll notice that in Passthroughs # 1 and #3, we compare and shift all the data we encounter. In Passthrough #4, we compare and shift some of it, and in Passthrough #2, we make just one comparison and shift no data at all.

While in the worst-case scenario, we compare and shift all the data, and in the best-case scenario, we shift none of the data (and just make one comparison per passthrough), for the average scenario, we can say that in the aggregate, we probably compare and shift about half of the data.

## 计算机代写|算法和结构代写Data Structures and Algorithms代考|A Practical Example

Let’s say that you are writing a JavaScript application, and somewhere in your code you find that you need to get the intersection between two arrays. The intersection is a list of all the values that occur in both of the arrays. For example, if you have the arrays $[3,1,4,2]$ and $[4,5,3,6]$, the intersection would be a third array, $[3,4]$, since both of those values are common to the two arrays.

JavaScript does not come with such a function built in, so we’ll have to create our own. Here’s one possible implementation:
function intersection(first_array, second_array) {
var result $=[] ;$
for (var $i=0 ; i<$ first_array.length; $i++){$
$\quad$ for (var $j=0 ; j<$ second_array.length; $j++){$
$\quad$ if (first_array[i]== second_array[j]) {
$\quad$ result.push(first_array[i]);
$\quad}$
return result;
Here, we are running a simple nested loop. In the outer loop, we iterate over each value in the first array. As we point to each value in the first array, we then run an inner loop that checks each value of the second array to see if it can find a match with the value being pointed to in the first array.

There are two types of steps taking place in this algorithm: comparisons and insertions. We compare every value of the two arrays against each other, and we insert matching values into the array result. The insertions are negligible beside the comparisons, since even in a scenario where the two arrays are identical, we’ll only insert as many values as there are in one of the arrays. The primary steps to analyze, then, are the comparisons.

If the two arrays are of equal size, the number of comparisons performed are $\mathrm{N}^2$. This is because for each element of the first array, we make a comparison of that element to each element of the second array. Thus, if we’d have two arrays each containing five elements, we’d end up making twenty-five comparisons. So this intersection algorithm has an efficiency of $\mathrm{O}\left(\mathrm{N}^2\right)$.
(If the arrays are different sizes-say $\mathrm{N}$ and $\mathrm{M}-$ we’d say that the efficiency of this function is $\mathrm{O}\left(\mathrm{N}^* \mathrm{M}\right)$. To simplify this example, however, let’s assume that both arrays are of equal size.)

## 计算机代写|算法和结构代写Data Structures and Algorithms代考|A Practical Example

JavaScript 没有内置这样的函数，因此我们必须创建自己的函数。这是一个可能的实现:
function intersection(first_array, second_array) {
var result $=$ प] if $\left(\right.$ first $\left._a r r a y[i]==\operatorname{second}_a r r a y[j]\right) \$$\result.push \left(\right. first _a rray \left.[i]\right) ; \$$ 返回结果； 在这里，我们正在运行一个简单的嵌套循环。在外循环中，我们遍历第一个数组中的每个值。当我们指向第一 个数组中的每个值时，我们然后运行一个内部循环来检查第二个数组的每个值，看它是否可以找到与第一个数 组中指向的值匹配的值。 该算法中有两种类型的步祭: 比较和揷入。我们将两个数组的每个值相互比较，并将匹醀值揷入到数组结果 中。除了比较之外，揷入可以忽略不计，因为即使在两个数组相同的情况下，我们也只会揷入与其中一个数组 中一样多的值。那么，分析的主要步㡜就是比较。 如果两个数组大小相等，则执行的比较次数为$\mathrm{N}^2$. 这是因为对于第一个数组的每个元㛃，我们将该元责与第二 比较。所以这个交集算法的效率是$\mathrm{O}\left(\mathrm{N}^2\right)$. (如果数组大小不同 — 比如$\mathrm{N}$和$\mathrm{M}$-我们会说这个函数的效率是$\mathrm{O}\left(\mathrm{N}^* \mathrm{M}\right)\$. 但是，为了简化此示例，我们假 设两个数组的大小相同。)

## 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。