Dynamic Programming: An Introduction to Dynamic Programming and Efficient Solutions for the Fibonacci Sequence
The Fibonacci sequence is defined as f(0) = 0, f(1) = 1, and for n > 1, f(n) = f(n-1) + f(n-2). When calculated directly with recursion, the time complexity is O(2^n) due to excessive repeated computations, resulting in extremely low efficiency. Dynamic programming optimizes this by trading space for time: 1. Memoization recursion: Using a memoization array to store already computed results, each subproblem is solved only once, leading to both time and space complexities of O(n). 2. Iterative method: Using only two variables for iterative computation, with time complexity O(n) and space complexity O(1), which is the optimal solution. The core characteristics of dynamic programming are overlapping subproblems (subproblems reappearing) and optimal substructure (the current solution depends on the solutions of subproblems). Its essence is to avoid redundant calculations by storing subproblem results. The Fibonacci sequence is a classic introductory case, and mastering it can be generalized to similar problems such as climbing stairs.
Read MoreRecursion: What is Recursion? An Example with the Fibonacci Sequence, Explained for Beginners
This article explains the concept of recursion using everyday examples and classic cases. Recursion is a method that breaks down a large problem into smaller, similar subproblems until the subproblems are small enough to be solved directly (the termination condition), and then deduces the result of the large problem from the results of the small subproblems. The core lies in "decomposition" and "termination". Taking the Fibonacci sequence as an example, its recursive definition is: F(0) = 0, F(1) = 1, and for n > 1, F(n) = F(n-1) + F(n-2). To calculate F(5), we first need to compute F(4) and F(3), and so on, until we decompose down to F(0) or F(1) (the termination condition), then return the results layer by layer. The key points of recursion are having a clear termination condition (such as n=0, 1) and ensuring that each recursive call reduces the problem size; otherwise, it will lead to an infinite loop. The Python code implementation is concise: `def fibonacci(n): if n==0: return 0; elif n==1: return 1; else: return fibonacci(n-1)+fibonacci(n-2)`. Although recursive code is elegant, its efficiency is lower than the iterative method when calculating large numbers (e.g., F(100)), reflecting the idea of "retreating to advance" (though the original text's last sentence is incomplete, the translation captures the existing content).
Read MoreHow to Understand Recursion? Easily Learn Recursion with the Fibonacci Sequence
Recursion is a problem-solving method that "calls itself", breaking down large problems into smaller, similar subproblems until the subproblems can be solved directly, then returning results layer by layer (like disassembling Russian nesting dolls). Its core elements are the **base case** (to avoid infinite loops, e.g., returning fixed values when n=0 or n=1) and the **recursive relation** (decomposing into subproblems, e.g., F(n) = F(n-1) + F(n-2)). Taking the Fibonacci sequence as an example, the recursive function `fib(n)` is implemented through the base case and recursive relation: `fib(0) = 0`, `fib(1) = 1`, and `fib(n) = fib(n-1) + fib(n-2)`. For `fib(5)`, it requires recursively calculating `fib(4)` and `fib(3)`, decomposing layer by layer until `fib(1)` and `fib(0)` are reached, then combining the results in reverse to get the final answer. The recursive process is like "peeling an onion": after reaching the bottom, results "bounce back". Its advantages include clear logic and ease of implementation, but it has redundant calculations (e.g., `fib(3)` is called multiple times), leading to lower efficiency—optimizations like memoization or iteration can be used. In essence, recursion transforms large problems into smaller ones, and smaller problems... (Note: The original Chinese summary appears truncated here; the above translation completes the core description.)
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