State Transition in Dynamic Programming: The Process from Problem to State Transition Equation

Dynamic programming (DP) solves problems by breaking them down and storing intermediate results to avoid redundant calculations. It is applicable to scenarios with overlapping subproblems and optimal substructure. The core of DP lies in "state transition," which refers to the derivation relationship between states in different stages. Taking the staircase climbing problem as an example: define `dp[i]` as the number of ways to climb to the `i`-th step. The transition equation is `dp[i] = dp[i-1] + dp[i-2]`, with initial conditions `dp[0] = 1` (one way to be at the 0th step) and `dp[1] = 1` (one way to climb to the 1st step). In another extended example, the coin change problem, `dp[i]` represents the minimum number of coins needed to make `i` yuan. The transition equation is `dp[i] = min(dp[i-coin] + 1)` (where `coin` is a usable denomination), with initial conditions `dp[0] = 0` and the rest set to infinity. Beginners should master the steps of "defining the state → finding the transition relationship → writing the equation" and practice to become familiar with the state transition thinking. Essentially, dynamic programming is a "space-for-time" tradeoff, where the state transition equation serves as the bridge connecting intermediate results.