# 统计代写|算法设计代写Algorithm Design代考|CS3530

## 统计代写|算法设计代写Algorithm Design代考|Esoteric Functions

The bread-and-butter classes of complexity functions were presented in Section 2.3.1 (page 38). More esoteric functions also make appearances in advanced algorithm analysis. Although we will not see them much in this book, it is still good business to know what they mean and where they come from.

• Inverse Ackermann’s function $f(n)=\alpha(n)$ : This function arises in the detailed analysis of several algorithms, most notably the Union-Find data structure discussed in Section 8.1.3 (page 250). It is sufficient to think of this as geek talk for the slowest growing complexity function. Unlike the constant function $f(n)=1, \alpha(n)$ eventually gets to infinity as $n \rightarrow \infty$, but it certainly takes its time about it. The value of $\alpha(n)$ is smaller than 5 for any value of $n$ that can be written in this physical universe.
• $f(n)=\log \log n$ : The “log log” function is just that-the logarithm of the logarithm of $n$. One natural example of how it might arise is doing a binary search on a sorted array of only $\lg n$ items.
• $f(n)=\log n / \log \log n$ : This function grows a little slower than $\log n$, because it is divided by an even slower growing function. To see where this arises, consider an $n$-leaf rooted tree of degree $d$. For binary trees, that is, when $d=2$, the height $h$ is given
$$n=2^h \rightarrow h=\lg n$$
by taking the logarithm of both sides of the equation. Now consider the height of such a tree when the degree $d=\log n$. Then
$$n=(\log n)^h \rightarrow h=\log n / \log \log n$$
• $f(n)=\log ^2 n:$ This is the product of two $\log$ functions, $(\log n) \times(\log n)$. It might arise if we wanted to count the bits looked at when doing a binary search on $n$ items, each of which was an integer from 1 to (say) $n^2$. Each such integer requires a $\lg \left(n^2\right)=2 \lg n$ bit representation, and we look at $\lg n$ of them, for a total of $2 \lg ^2 n$ bits.

## 统计代写|算法设计代写Algorithm Design代考|Data Structures

Putting the right data structure into a slow program can work the same wonders as transplanting fresh parts into a sick patient. Important classes of abstract data types such as containers, dictionaries, and priority queues have many functionally equivalent data structures that implement them. Changing the data structure does not affect the correctness of the program, since we presumably replace a correct implementation with a different correct implementation. However, the new implementation may realize different trade-offs in the time to execute various operations, so the total performance can improve dramatically. Like a patient in need of a transplant, only one part might need to be replaced in order to fix the problem.

But it is better to be born with a good heart than have to wait for a replacement. The maximum benefit from proper data structures results from designing your program around them in the first place. We assume that the reader has had some previous exposure to elementary data structures and pointer manipulation. Still, data structure courses (CS II) focus more on data abstraction and object orientation than the nitty-gritty of how structures should be represented in memory. This material will be reviewed here to make sure you have it down.

As with most subjects, in data structures it is more important to really understand the basic material than to have exposure to more advanced concepts. This chapter will focus on each of the three fundamental abstract data types (containers, dictionaries, and priority queues) and show how they can be implemented with arrays and lists. Detailed discussion of the trade-offs between more sophisticated implementations is deferred to the relevant catalog entry for each of these data types.

## 统计代写|算法设计代写Algorithm Design代考|Esoteric Functions

• 逆阿克曼函数 $f(n)=\alpha(n)$ : 这个函数出现在对几种算法的详细分析中，最值得注意的是第 8.1.3 节 (第 250 页) 中讨论的 Union-Find 数据结构。将其视为增长最慢的复杂性函数的极客谈话就足够了。 与常量函数不同 $f(n)=1, \alpha(n)$ 最後达到无穷大 $n \rightarrow \infty$ ，但它肯定需要时间。的价值 $\alpha(n)$ 小于 5 的 任何值 $n$ 可以写在这个物理宇宙中。
• $f(n)=\log \log n: ” \log \log$ “函数就是这样一一对数的对数 $n$. 它可能如何出现的一个自然示例是对排序 数组进行二进制搜索 $\lg n$ 项目。
• $f(n)=\log n / \log \log n$ : 这个函数的增长速度比 $\log n$ ，因为它被一个增长更慢的函数分割。要查看出 现这种情况的位置，请考虑 $n$-叶根度树 $d$. 对于二叉树，即当 $d=2$ ，高度 $h$ 给出
$$n=2^h \rightarrow h=\lg n$$
通过对等式两边取对数。现在考虑这样一棵树的高度，当度 $d=\log n$. 然后
$$n=(\log n)^h \rightarrow h=\log n / \log \log n$$
• $f(n)=\log ^2 n$ :这是两个的产物 $\log$ 功能， $(\log n) \times(\log n)$. 如果我们想计算在对 $n$ 项目，每个项目都 是从 1 到 (比如说) 的整数 $n^2$. 每个这样的整数都需要一个 $\lg \left(n^2\right)=2 \lg n$ 位表示，我们看看 $l g n$ 其 中，总共 $2 \lg ^2 n$ 位。

## 统计代写|算法设计代写Algorithm Design代考|Data Structures

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