Interesting algorithm world - algorithm root (1)

Welcome to the world of interesting algorithms.

Last time we started talking to her - a wonderful algorithm and gave a definition. Although some disputes exist, do you still have to resign a guide? Haha, that's good, we continue our algorithm journey.

And in fact, the real scenery begins now! If she (algorithm) is a big tree, then I don't want to lead your eyes to the lush branches, leading to the flowers, Zi Zihong ... Oh, no, let's take your attention Stretching the roots, here is the place where you are born.

One hundred years ago, at 1900 at the International Mathematician meeting held in Paris, Hilbert issued a famous speech entitled "Mathematical Problem", and its main part told 23 mathematical problems, and was unified as Hilbert issues. Among them, the 10th problem is about the algorithm.

The problem is this:

(10) Can I determine whether there is a decix solution in a limited step? Alternatively, a algorithm is designed to test if the polynomial is integrated.

Interestingly, there is no concept of algorithms at the time, so he did not use the term "algorithm", and used this phrase: "The process that can be decided by limited operations." [Note 1]

This speech is significant, it has unveiled the sequence of mathematics development in the 20th century. In particular, this 10th problem is to be related to the computer's interest, although it is proven to be inevitable, many by-products have driven the development of computer science and deeply reveal the root cause of the algorithm.

There is no doubt that we can see math as the soil she rooted. In fact, in ancient mathematics, the shadow of an algorithm has already appeared, and it can even be said to be a good algorithm. For example, the European Ricide Algorithm, that is, the process of seeking two largest number of conventions in the "geometric original" (Euclid's Elements, VII, propositions I and II) (ie rushed down) ), This is a typical. In the present way, we can narrate that the Euculsi Economics E is as follows: E = "Nature X, Y: 1) Repeat the following operation until y = 0. 2) assignment X ← X mod y. 3) The value of X and Y is exchanged. 4) Output x. "

In modern mathematics, the algorithm can be said to be a bright spot in the flowers, often can see her beautiful figure; especially in computer mathematics, finding algorithms can be said to be one of the core issues we face. Looking at an example, I left the question "King's Castle" on the last time, as a simple analysis, it is easy to know that it is a chart question, combined with K-connect map and optimization knowledge. Of course, the problem is more difficult, we simplify, you will understand. For example, it can be considered that the step G is N, we want to construct K-connectivity on the basis of G, now assume that all edge costs (ie, the cost of building roads) is a constant, with the right of edge (the length of the road) Norway, if the cost is consuming, we can make the edges of G as small as possible, but in the case of knowing N and K, the minimum number of K-Connect Diagram G is a problem. The above said Hilbert's speech, but there is no algorithm concept. Hilbert 10 Issues seem to have a hypothesis that he thinks this algorithm is existing, and people's tasks are to find "her." But unfortunately this problem has no algorithm. However, at that time, people could not find this anyway. Only intuitive concepts may write several effective algorithms, but if you do not have a profound roots, you can't prove that there is no algorithm solution. This foundation was established in 1936. Chuqi and Tu Leades issued the paper, defined from the perspective of mathematics. What we mainly introduce is one of the achievements, the basic mathematical model of the computer, the map spirit. Among them, it can be presented that they link the connection between the non-forms and precise definitions of the algorithm, the first time, ie,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

The intuition concept of the algorithm is equal to the graphic algorithm

Before introducing the map forever, we have prepared work.

The development of history is always very interesting. In terms of mathematics, it seems that very elegant thinking is often seen as a monster during his birth, and it is refused to have a thousand miles away. The initial development of abstract algebra is starting from the concept of group, but the concept of group has been re-recognized from the proposal to re-recognize it, and experienced a long process. But don't say it, we will see the knowledge of abstract algebra; as long as we don't cool our "algorithm", I am happy for today's era.

The concept of collection, everyone has, such as the natural number of natural numbers in our elementary school. Speaking of the operation, everyone has intuitive concepts, such as the addition of " ", on N can be performed: 1 1 = 2, 1 2 = 3, 2 3 = 5, ... but honestly, compare Just finding the difference between the differences and regularities, I was in the preliminary time in the high school collections, I still don't realize that the collection can be combined with the operation, and the essence of the operation can reveal from another angle. Instead, we have obtained the "prior" concept throughout and highlighting: The operation is a verb in mathematics, 1 1 is equal to 2. (You said that I am not the laughter, haha, joke, now I can do a good guide :)

Yes, combined with the assembly together, and there is a new thing: . The addition operation " " is actually a map of the collections: 1 and 2 belong to N, 1 1 mapping 2. Undoubtedly, in this sense, our " " operation is closed on N, that is, two natural numbers, the result of the " " arithmetic map is still in N. Of course, you can define an operation such as "△", where 1 △ 1 = 100? No problem, but its nature needs you to study. Set A as a collection, " " is arbitrary as defined on A, if " " is closed on A, it is called an algebraic structure . General, " " can be exchanged to a collection of calculations, and to ensure that each of them is closed on a. Group is a very important algebra structure. But we can't say the knowledge of abstract algebra. If you are interested, you can check out the relevant information, the mathematics book is very good, and more than the computer book is cheap. Broadly speaking, the map spirit can be said to be a complex algebra structure, which is a universal computer model that can do everything that can do actually. Of course, the map spirit also has problems that cannot be solved, but in fact these problems have exceeded the ability of the computer; we are now basically thinking.

Take a look at time, oh is not early, then we will follow this journey next time, observe the roots of the beautiful "algorithm". Thank you for your attention. Thank you.

[Note 1] From the intuition of the mathematician, the finite to the algorithm can be supported. In fact, poorness is a lot of books that emphasize, and there are subtle. But after the next article is written, I will give you a reply, why I have mentioned the algorithm in infinite exercises in "What is Algorithm".

The title of this post: There is a rich man, live in the private villa, he needs his window to have such a function: (1) If it is dark, automatically shuts down; (2) If the weather is allowed, if the weather is allowed, automatically opens; (3) If the weather changes in the case, the weather changes is not allowed, it is automatically turned off. There is no doubt that this requires something called a microcontroller and several electronic products to complete control, but if you call the problem, you can write this algorithm sequence?