Dynamic Planning Algorithm

Dynamic PlanningAlgorithmIt is usually used to solve a problem with a certain optimum nature.. In the field of mathematics and computer science, dynamic planning is used to solve the problems that can be decomposed into duplicates (overlapping subproblems,Think about recursive factorial.And has the optimal substructure (optimal substructure,Think about the shortest path algorithm). Dynamic Planning takes less time than normal algorithms. In such problems, there may be many feasible solutions. Each solution corresponds to a value. We hope to find a solution with the optimal value. The dynamic planning algorithm is similar to the divide and conquer method.Basic IdeasIt is also to break down the problem to be solved into several subproblems, first solve the subproblems, and then obtain the original solution from the solutions of these subproblems. Different from the Division and control method, it is suitable for Dynamic Planning and solving problems. After decomposition, subproblems are often not independent of each other. If you use the divide and conquer method to solve this type of problem, the number of subproblems obtained by the decomposition is too large, and some subproblems are repeatedly calculated many times.If we can save the answers to the resolved sub-questions and find the answers we have obtained when necessary, we can avoid a large number of repeated computations and save time.. WeYou can use a table to record the answers to all resolved subquestions. Whether or not this subproblem is used in the future, as long as it has been computed, it will fill in the results in the table.. This is the basic idea of dynamic planning. The specific dynamic planning algorithms are diverse, but they have the same table filling format.

Dynamic Planning is widely used, such as the maximum subsequence sum, the optimal binary query tree, And the Fibonacci sequence.

• Example:

1. Maximum subsequence and
Here we repost two good articlesArticle:

Maximum subsequences and Problems

Dynamic Programming Method ------- maximum continuous subsequence and

2. Fibonacci Series

Find the nth number in the Fibonacci sequence and implement it directly based on its mathematical definition:

Function fib (N)
If n = 0
Return 0
Else if n = 1
Return 1
Return fib (n-1) + fib (n-2)

If we call fib (5), a call tree is generated for repeated calculation of the same value multiple times:

FIB (5)
FIB (4) + fib (3)
(FIB (3) + fib (2) + (FIB (2) + fib (1 ))
(FIB (2) + fib (1) + (FIB (1) + fib (0) + (FIB (1) + fib (0 )) + fib (1 ))
(FIB (1) + fib (0) + fib (1) + (FIB (1) + fib (0) + (FIB (1) + fib (0) + fib (1 ))

In particular, FIB (2) is calculated three times. In larger cases, more fib values are repeatedly calculated, which will consume exponential time.

Now, suppose we have a simple map object.MCreates a ing for each computed fib and its return values, modifies the above function fib, and uses and constantly updates it.M. The new function will only requireO(N), Rather than the exponential time:

VaR M: = map (0 → 1, 1 → 1)
Function fib (N)
If map M does not contain key N
M [N]: = fib (n-1) + fib (n-2)
Return M [N]

This technology is called cache, which stores the calculated values.Top-downMethod: divide the problem into several subproblems, and then calculate and store the value.

InBottom-upIn this method, we first calculate a smaller fib, and then calculate a larger fib based on it. This method also takes only linear (O(N) Because it containsN-One cycle. However, this method only requires constants (O(1) space, on the contrary,Top-downThe O (N) To store mappings.

Function fib (N)
VaR previusfib: = 0, currentfib: = 1
If n = 0
Return 0
Else if n = 1
Return 1
Repeat n-1 times
VaR newfib: = previusfib + currentfib
Previusfib: = currentfib
Currentfib: = newfib
Return currentfib

In both examples, we only calculate fib (2) once, and then use it to calculate fib (3) and FIB (4), instead of re-computing each time.

3. Zero currency change

See: currency change

To be continued...