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.