Laboratory work №7 on discrete analysis, 4 semester, Moscow Aviation Institute (MAI).

There is a natural number n. You can perform 3 kinds of operations with it:

1. subtract 1
2. divide by 2
3. divide by 3

Transformation - the sequence of the above-mentioned operations. The cost of each operation is current value of n. The cost of the transformation is summary cost of all operations in the transformation. The task is to find a transformation with the lowest cost, that will transform input number to n.

2 ≤ n ≤ 10^7

In the first string print the lowest cost. In the second one print the transformation itself. Possible operations are: /2, /3, -1. All of them must be divided by space character.

To solve this problem I used a dynamical programming method. The task can be easily divided into separated subtasks, and if we solve them and unite the solutions, we will get the solution of the main task. As a subtask we can take counting minimal cost of each number less than n. Using the solutions we can count minimal cost of n and this will be an answer.

We will need an array with 2 lines and n columns. In the first line we will keep the lowest price for the number, operations in the second one. The minimal cost search algorithm works like this:

1. The number we begin with is 2
2. The lowest cost of the current number i will be the lowest cost of numbers i - 1, i / 2, i / 3. We put the cost to the first line of the array where i is column index.
3. In the second line we put an operation that allowed us to get the lowest cost.
4. Go to the next number (increment i) and do step 2.

The process will repeat until the cost of n will be counted.