Dynamic Programming

Dynamic programming is a powerful technique that can dramatically lower the computational costs of many complex algorithms, though it comes with some trade-offs. This chapter introduces dynamic programming and includes a review of greedy algorithms.

We will begin by revisiting the principles of divide-and-conquer strategies to contrast them with dynamic programming. Dynamic programming stands out in algorithm design due to its ability to solve problems that involve overlapping subproblems and optimal substructure. By storing the results of these subproblems, dynamic programming prevents redundant calculations, leading to significant efficiency gains.

Through various examples, we will explore how dynamic programming can be applied to solve classical problems such as the knapsack problem, the longest common subsequence, and the Traveling Salesman Problem. Each example will illustrate the step-by-step approach to breaking down a problem, defining the state space...