動態(tài)規(guī)劃（DP）通過將復(fù)雜問題分解為更簡單的子問題并存儲其結(jié)果以避免重復(fù)計算，來優(yōu)化求解過程。主要方法有兩種：1. 自頂向下（記憶化）：遞歸分解問題，使用緩存存儲中間結(jié)果；2. 自底向上（表格化）：從基礎(chǔ)情況開始迭代構(gòu)建解決方案。適用于需要最大/最小值、最優(yōu)解或存在重疊子問題的場景，如斐波那契數(shù)列、背包問題等。在Python中，可通過裝飾器或數(shù)組實現(xiàn)，并應(yīng)注意識別遞推關(guān)系、定義基準情況及優(yōu)化空間復(fù)雜度。

Dynamic programming (DP) is a method for solving complex problems by breaking them down into simpler subproblems. It's especially useful when those subproblems overlap — meaning the same subproblem shows up multiple times in the bigger problem.

Unlike divide-and-conquer methods (like recursion), which solve subproblems independently, DP stores the results of subproblems so they don’t need to be recomputed every time they come up. This technique is called memoization or caching.

In Python, you can implement dynamic programming using two main approaches: top-down with memoization and bottom-up with tabulation.

1. Understand the Two Main Approaches

There are two primary ways to implement dynamic programming:

• Top-down (Memoization):
  You start from the original problem and break it down recursively. Along the way, you store computed results to avoid redundant calculations.
  In Python, this is often done using dictionaries or the lru_cache decorator from the functools module.

• Bottom-up (Tabulation):
  You start from the base cases and build up solutions for larger and larger subproblems until you reach the original problem.
  This is usually implemented using loops and an array (or list) to store intermediate values.

Both approaches aim to reduce computation time by avoiding repeated work, but each has its own use case depending on the problem structure.

2. Recognize Problems That Benefit From DP

Some classic signs that a problem might benefit from dynamic programming:

• The problem asks for:

  • Maximum or minimum value
  • Number of ways to do something
  • Optimal solution under certain constraints

• Subproblems overlap (e.g., computing Fibonacci(n) requires both Fibonacci(n-1) and Fibonacci(n-2))

Common examples include:

• Fibonacci sequence
• Knapsack problem
• Longest common subsequence (LCS)
• Coin change problem
• Edit distance

If you find yourself writing recursive code that gets slower as input size increases, DP might help speed things up.

3. Implementing DP in Python – A Simple Example

Let’s take the classic Fibonacci example to show how to apply both techniques.

Top-down with memoization:

from functools import lru_cache

@lru_cache(maxsize=None)
def fib_memo(n):
    if n <= 1:
        return n
    return fib_memo(n - 1)   fib_memo(n - 2)

This uses Python’s built-in cache decorator to remember previously computed values.

Bottom-up with tabulation:

def fib_tab(n):
    if n <= 1:
        return n
    dp = [0] * (n   1)
    dp[1] = 1
    for i in range(2, n   1):
        dp[i] = dp[i - 1]   dp[i - 2]
    return dp[n]

Here we build the solution iteratively, storing each result in a list.

You’ll notice the second version avoids recursion depth issues and may be more memory-efficient depending on how you manage storage.

4. Tips for Using DP Effectively

When applying dynamic programming:

• Start by identifying the recurrence relation — how the current state relates to previous states.
• Define your base cases clearly.
• Think about space optimization — many DP problems can reduce memory usage by only keeping track of necessary previous steps.
• Use lru_cache or cache decorators for quick memoization in Python 3.9 .
• Practice on common patterns like 1D or 2D DP tables.

Also, not all DP problems require full arrays — sometimes just a few variables are enough to hold what you need.

It takes practice to get comfortable spotting where DP applies and choosing between top-down and bottom-up. But once you get the hang of it, it becomes a powerful tool for optimizing performance.基本上就這些。

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