How to implement dynamic programming algorithm in Python

To implement dynamic programming algorithm in Python, you can follow these steps:

1. Defining the state of the problem is crucial, as it can be represented by one or more variables. The choice of state significantly affects the efficiency and correctness of the algorithm.
2. Initialization: Based on the definition of the problem, initialize an array or matrix to represent the initial state. Initializing the state is fundamental to the dynamic programming algorithm.
3. State transition equation: Determine the relationship between states based on the problem definition. Calculate each element in the state array or matrix based on the transition relationship.
4. Return the result: Determine the final outcome based on the definition of the problem. Calculate and return the solution based on the elements in the state array or matrix.

Let’s take the Fibonacci sequence as an example to demonstrate how to implement the dynamic programming algorithm.

def fibonacci(n):
    if n <= 0:
        return 0
    if n == 1:
        return 1
    # 初始化状态数组
    dp = [0] * (n + 1)
    dp[0] = 0
    dp[1] = 1
    # 状态转移方程
    for i in range(2, n + 1):
        dp[i] = dp[i - 1] + dp[i - 2]
    # 返回结果
    return dp[n]

# 测试
print(fibonacci(10))  # 输出：55

In the above code, we define the state of the Fibonacci sequence as dp[i], representing the value of the i-th Fibonacci number. Then, following the definition of the Fibonacci sequence, we initialize the first two elements of the state array dp. Next, we calculate and update each element of the state array based on the state transition equation dp[i] = dp[i – 1] + dp[i – 2]. Finally, we return the last element of the state array as the solution to the problem.