Understanding the structural differences in problems that allow for a greedy solution versus those requiring DP.
A fundamental skill in algorithm design is discerning when to use a greedy approach versus when a problem requires the more comprehensive exploration of Dynamic Programming. The core difference lies in the nature of the choices made. A greedy algorithm makes a single, locally optimal choice at each step, without looking back. It hopes this sequence of local optima leads to a global optimum. DP, on the other hand, explores all possible choices at each step and computes solutions for subproblems based on these choices, ultimately selecting the best one. The 0/1 Knapsack problem is a classic illustration of this difference. A greedy approach might be to take the items with the highest value-to-weight ratio first. This can fail. For example, a high-ratio item might take up so much weight that it prevents you from taking two other items that have a slightly lower ratio but a greater combined value. DP solves this correctly by considering both options for each item: either take it or don't take it. The key distinction is that for a greedy strategy to work, the problem must have the 'greedy choice property'—the locally optimal choice must always be part of some globally optimal solution. Proving this property is often the hardest part of designing a greedy algorithm.