The plan of solving strategy

Author: Tsinghua University Huanggao

For a specific problem, how to think about the analysis, so that the planning strategy, is very important. The planning process of strategy is a process of divergent thinking. The problem itself is changeable, the strategy of solving the problem is more, the method of plotting strategy is various, according to People's way of thinking, we discuss the following several strategies of thinking.
1, degrade the idea : from the analysis of the special and simple state of the problem to sum up the essence of the problem or law, so as to get the general solution of the problem, that is, we often say "ofspecialty" or called "try to generalize." This kind of thought is very effective in the process of plotting the strategy through the special seeking general, obtains the abstract through the actual, especially when the data given in the topic is relatively large or very abstract and difficult to start, often this kind of thought to plan the strategy.

Question one : (stair problem) a person boarded a level 10 step, you can climb the level 1, you can also step into the 2 level. Q: There are several ways of logging.
The number of steps in this problem is 10, although not very large, but it is difficult to start. We use degraded thinking to analyze this problem, first of all, look at a few simple cases:
A, only 1 steps, there are only 1 methods: Step 1.
B, when there are 2 steps, there are 2 types of logging: Step 2, Level 1 and Level 1 (two steps).
C, there are 3 steps, the logging method has 3 kinds: 1 level + 1 level + 1 level, 1 level + 2 level, 2 level + 1 level.
Here "Level 1 + 2" means the first level 1 re-boarding 2 levels, "2 level + 1 level" means that the first 2 level of re-boarding 1, obviously the two methods are different.
D, when there are 4 steps, there are 5 types of logging:
1 level + 1 level + 1 level + 1 level,
1 level + 1 level + 2 level,
1 level + 2 level + 1 level,
2 level + 1 level + 1 level,
2 level + 2 level.
E, when there are 5 steps, there are 8 kinds of logging methods:
Level 1 + 1 level + 1 level + 1 level + 1 level,
1 level + 1 level + 1 level + 2 level,
1 level + 1 level + 2 level + 1 level,
1 level + 2 level + 1 level + 1 level,
1 level + 2 level + 2 level;
2 level + 1 level +1 Level + 1 level,
2 level + 1 level + 2 level,
2 level + 2 level + 1 level.
The order of the number of steps in increments of stairs is arranged as follows:
1,2,3,5,8 ...
It is easy to think of this as part of the Fibonacci sequence, which finds the law of the problem, and also plots strategies for solving the problem: Mathematical models (rules). Easy to get the following data is: 13,21,34,55,89 ..., so the solution to the problem: 10 steps There are 89 kinds of Deng.
Of course, the correctness of the law found must be proved in detail, not in detail here. The application of this example is to illustrate the superiority of considering the problem from a simple point of view, while pointing out that a keen insight into the ability to have data is also very important for planning strategies.

2, the idea of upgrading: This kind of thought is the opposite of the first thought, it is the abstraction of some too specific problems, the aim is to ignore some of the secondary factors, so as to better grasp the main factors therein, by solving the general problems to solve the specific problems. Sometimes this happens: because the data given in the problem is too specific, it tends to form a mindset that has been analyzed for specific data, ignoring what is obvious in the problem. Using this idea is beneficial to break through the thinking pattern and to find the strategy of solving the problem faster, the key is to analyze and infer the abstract problem.

We abstract the "stair problem", with n-level steps, the rules of the steps are unchanged. There are two scenarios for the last step of the N-level step:
(1) by step n-1 steps (Level 1) to the nth level;
(2) A step (Level 2) is reached at the nth level by the stage n-2.
Obviously, the above two kinds of methods are different, n-level steps of the total Deng method is the sum of these two kinds of methods. We use f (x) to denote the number of =f of the X-step, then there are: F (n) n-1 +f (n-2) and F (1) =1;f (2) = 2.
This quickly gets the mathematical model of the problem and plots the strategy of solving the problem: Mathematical model (regular) strategy.
From the analysis of this example, it can be seen that the abstract generalization of the problem, the search for the general solution of the problem is very simple to solve some problems. Appropriate use of the idea of upgrading can often "one arrow in", quickly describe the nature of the problem, so as to get the strategy of solving problems.

3, the idea of separation : The whole problem is divided into several related sub-problems or a number of consecutive problem-solving steps, and then one by one to try to solve, the final synthesis of the various parts of the solution can get the whole problem. As a kind of thinking method, it is very effective to solve the problem when it is used properly in the analysis of problems. There are many ways to divide the problem, the most basic principle is to make difficult to describe the problem into easy to describe the problem, the difficult to solve the easy to solve.
Still look at the above "stair problem", from the 10 steps to use different steps, can be divided into the following several situations:
A, a total of 10 steps (all are 1 levels each step): There are 1 kinds of logging method;
B, a total of 9 steps (only a certain step to the Level 2, the rest of each step of the 1): there are c19=9 method;
C, a total of 8 steps (only a certain two steps per step 2): There is a c28=28 seed-boarding method;
D, a total of 7 steps (only a certain three steps per step 2 level): There are c37=35 seed-boarding method;
E, a total of 6 steps (only a certain four steps per step 2 level): There are c46=15 seed-boarding method;
F, a total of 5 steps (all are 2 levels per step): there are c55=1 method;
So there are 1+9+28+35+15+1=89 of the total.

Through the analysis of the process of dividing the problem and the result of calculation, we can get the following two kinds of strategies:
(1) Divide and conquer the strategy, directly imitate the above division process by the computer to realize.
(2) Mathematical model (law) strategy, from which summed up: for the N-step of the boarding method has
Two
As can be seen from the above example: the key points in the idea of a "sub" word, that is, how to divide the problem, the Division should pay attention to two points:
1, the division must be consistent with the standard, not repeat, do not miss;
2, the division must be enlightening, that is, the strategy of the problem of planning has some enlightening effect.

4, the idea of change lattice : Through the transformation of certain information, so as to transform the form of the problem, to achieve the implicit, the complexity of the simple, easy to change, the unknown for the purpose of the known, by solving the problem of equivalence to solve the original problems.
Look at the above "step problem", we use the idea of change lattice to transform the problem, consider the position of the person on the steps of the change, such as: When the person is on the 1th step, if the step 1 level can reach the 2nd level step, if the step 2 can reach the 3rd level of stairs. We use a point to represent each step, with the side of the step can be "one-step" relationship, you can get the following diagram:

The problem then translates to: In Figure 6, the total number of paths from point 0 to 10 is obtained. This is a familiar graph theory problem, can be solved by the strategy of poor lifting, of course, this strategy compared to the previous plan of the strategy is not much advantage, but after all, is one of the strategies to solve the problem.
If the application of the lattice thought in the above example has no advantage over other ideas, then recall that when solving the "best airline route Problem", we turned it into "shortest path problem" and "minimum cost flow problem" as an example of the successful application of the variable lattice idea. The key to the application of the idea of "change" is to be ingenious, which is very effective for exploring new problems, especially unknown problems.
The above-mentioned strategies of the specific algorithm on the computer is easy to implement, the specific program is no longer given.