How do you know if dynamic programming is working for you?

1 Understand the problem

The first step to apply dynamic programming is to understand the problem you are trying to solve. What are the inputs, outputs, constraints, and objectives? How can you formulate the problem as a recurrence relation, that is, a function that expresses the optimal solution in terms of smaller subproblems? What are the base cases, that is, the simplest subproblems that can be solved directly? What are the overlapping subproblems, that is, the subproblems that are repeated and can be reused?

2 Choose a strategy

The next step is to choose a strategy for implementing dynamic programming. There are two main approaches: top-down and bottom-up. The top-down approach starts from the original problem and recursively solves the subproblems until reaching the base cases. It uses memoization, that is, storing the results of previously solved subproblems in a table or a cache, to avoid recomputing them. The bottom-up approach starts from the base cases and iteratively builds up the solution by combining the results of smaller subproblems. It uses tabulation, that is, filling up a table or an array with the optimal values for each subproblem, to store and access the results.

3 Analyze the complexity

The third step is to analyze the complexity of your dynamic programming solution. You need to consider both the time and space complexity, that is, how much time and memory your solution requires as a function of the input size. The time complexity depends on the number of subproblems and the time to solve each subproblem. The space complexity depends on the size of the table or cache you use to store the results. You can use asymptotic notation, such as O(n), to express the upper bound of the complexity. You can also compare your dynamic programming solution with other possible solutions, such as brute force or greedy algorithms, to see if you have achieved a significant improvement.

4 Test and debug

The fourth step is to test and debug your dynamic programming solution. You need to verify that your solution works correctly for different inputs, especially edge cases and boundary conditions. You can use examples, test cases, or unit tests to check your solution. You can also use debugging tools, such as breakpoints, print statements, or visualizers, to trace your solution and identify any errors or bugs. You can also use code quality tools, such as linters, formatters, or refactoring tools, to improve your code readability and style.

5 Optimize and generalize

The final step is to optimize and generalize your dynamic programming solution. You can look for ways to improve your solution by reducing the time or space complexity, such as using better data structures, pruning unnecessary subproblems, or applying advanced techniques, such as bit manipulation or matrix multiplication. You can also look for ways to generalize your solution by applying it to other problems that have similar structures, such as variations, extensions, or combinations of the original problem. You can also learn from other dynamic programming solutions by studying different examples, patterns, or categories of problems that can be solved by dynamic programming.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?