Coin Change: Complete DP Breakdown

Dynamic programming is a method for solving optimization problems by breaking them down into simpler overlapping subproblems. The technique works when a problem exhibits both optimal substructure (the optimal solution contains optimal solutions to subproblems) and overlapping subproblems (the same subproblems are solved multiple times in a naive recursive approach). For this problem, we first define the state that captures all the information needed to make optimal decisions. Then we establish the recurrence relation that expresses the solution to the current state in terms of previously computed states. The base cases provide the starting values from which all other states are computed. We can implement the solution top-down using memoization (recursion plus a cache) or bottom-up using tabulation (filling a DP table iteratively). The bottom-up approach is generally preferred as it avoids recursion overhead and makes space optimization easier. In many DP problems, the current state depends only on the previous row or the previous two values, allowing us to reduce space from O(n^2) to O(n) or even O(1). Always analyze whether the DP table can be compressed. Practice identifying DP patterns: linear DP on sequences, grid DP on matrices, interval DP on subarrays, tree DP on hierarchical structures, and knapsack DP for subset selection problems.