关于DP算法的讨论

通常可以使用DP算法解决的问题有两个特点：

 

There are two key attributes that a problem must have in order for dynamic programming to be applicable: optimal substructure and overlapping subproblems which are only slightly smaller. When the overlapping problems are, say, half the size of the original problem the strategy is called "divide and conquer" rather than "dynamic programming". This is why mergesort, quicksort, and finding all matches of a regular expression are not classified as dynamic programming problems.

 

一般把DP算法分为两类：

 

• Top-down approach: This is the direct fall-out of the recursive formulation of any problem. If the solution to any problem can be formulated recursively using the solution to its subproblems, and if its subproblems are overlapping, then one can easily memoize or store the solutions to the subproblems in a table. Whenever we attempt to solve a new subproblem, we first check the table to see if it is already solved. If a solution has been recorded, we can use it directly, otherwise we solve the subproblem and add its solution to the table.
• Bottom-up approach: This is the more interesting case. Once we formulate the solution to a problem recursively as in terms of its subproblems, we can try reformulating the problem in a bottom-up fashion: try solving the subproblems first and use their solutions to build-on and arrive at solutions to bigger subproblems. This is also usually done in a tabular form by iteratively generating solutions to bigger and bigger subproblems by using the solutions to small subproblems. For example, if we already know the values of F₄₁ and F₄₀ （Fibonacci series）, we can directly calculate the value of F₄₂.

wikipedia here