Your observation about dynamic programming (DP) problems in competitive programming is insightful. Indeed, many DP problems can be approached systematically by recognizing certain patterns and applying well-established templates. Let’s break down your thoughts and expand on them:

1. Decision Tree and Pruning:

2. When solving DP problems, you can think of it as exploring a decision tree where each node represents a state or subproblem. Pruning (剪枝) is crucial to eliminate redundant calculations and reduce the search space, which is a common technique in both DFS and DP to improve efficiency.

3. Abstraction into Mathematical Sets:

4. Abstracting a problem into mathematical sets and formulating the state transition function is a critical step in solving DP problems. This involves identifying the states, defining the transition between states, and determining the base cases.

5. Templates and Common Patterns:

6. Many DP problems can indeed be categorized into common types, such as the nine types of knapsack problems you mentioned. Recognizing these patterns can significantly simplify the process of solving new problems. Some common DP templates include:

   • Knapsack Problems: This includes variations like 0/1 knapsack, unbounded knapsack, and fractional knapsack.
   • Longest Common Subsequence (LCS): Problems related to finding subsequences or substrings.
   • Matrix Chain Multiplication: Problems involving optimal parenthesization.
   • Fibonacci Sequence: Simple DP problems with overlapping subproblems.
   • Subset Sum: Problems related to finding subsets with a given sum.
   • Coin Change: Problems involving combinations of coins to make a certain amount.
   • Edit Distance: Problems involving operations to convert one string into another.
   • Pathfinding in Grids: Problems involving finding paths in 2D grids with obstacles or costs.
   • Partition Problems: Problems involving dividing sets into subsets with certain properties.

7. Formulating the Dynamic Function:

8. The process of formulating the dynamic function involves:

   • State Definition: Define the state variables that represent the subproblems.
   • State Transition: Define how to move from one state to another (recurrence relation).
   • Base Cases: Define the initial conditions for the smallest subproblems.

9. Example: Knapsack Problem:

10. State Definition: Let ( dp[i][w] ) represent the maximum value that can be achieved using the first ( i ) items and a knapsack capacity of ( w ).
11. State Transition:
    [
    dp[i][w] = \max(dp[i-1][w], dp[i-1][w-w_i] + v_i)
    ]
    where ( w_i ) is the weight of the ( i )-th item and ( v_i ) is its value.

12. Base Case: ( dp[0][w] = 0 ) for all ( w ).

13. Practice and Recognition:

14. The more you practice, the better you become at recognizing these patterns and applying the appropriate template. This recognition process is akin to understanding and applying design patterns in software engineering.

In summary, while DP problems can often be approached using templates and common patterns, the key to mastering them lies in practice and the ability to abstract the problem into a mathematical formulation. By doing so, you can systematically apply the appropriate DP techniques and solve the problem efficiently.