Recursion is beautiful. Seashell contains recursive patterns.

Demystifying Dynamic Programming: Beyond the Template

Many resources offer fantastic explanations of Dynamic Programming (DP) for solving algorithmic problems. While I’ve devoured these articles and tackled numerous DP problems, a nagging feeling remained: I wasn’t truly grasping the concept. I’d simply apply a memorized solution template without a deeper understanding.

Recently, LeetCode.com significantly expanded their DP problem set https://leetcode.com/studyplan/dynamic-programming. This meticulously crafted curriculum progressively increases difficulty, exposing learners to diverse applications of DP. Solving these problems (available on my GitHub https://github.com/jolly-fellow/rust_algo_tasks) — implemented in Rust, a new language for me!) was immensely rewarding, and I highly recommend them for practice.

Breaking Free from the Template Trap

This time, here’s what resonated:

1. DP: Not Programming as You Know It

For beginners, the term “Dynamic Programming” can be misleading. Relax, it has nothing to do with traditional programming! The name predates computers, stemming from mathematical optimization https://en.wikipedia.org/wiki/Mathematical_Programming. In essence, DP is a clever brute-force optimization technique.

2. Recursion for Natural Solutions (But Be Mindful)

You’ve likely encountered the advice to leverage recursion for DP problems. However, some, myself included, have an optimization obsession. We know recursion can be memory-intensive and computationally less efficient. This tempts us towards iterative approaches, which can be convoluted and lengthy compared to the elegance of recursion for these problems.

Think of it like weightlifting. A beginner wouldn’t start with a 100kg barbell. They’d begin with manageable weights and gradually increase the challenge. Similarly, when tackling a new DP problem, embrace recursion first. It often leads to more intuitive and natural solutions. Here’s the Fibonacci sequence as an example:

  fn fibonacci(n: u32) -> u32 {
      if n <= 1 {
          n
      } else {
          fibonacci(n - 1) + fibonacci(n - 2)
      }
  }
  

This beautifully mirrors the mathematical formula:

Fn = Fn-1 + Fn-2

While the iterative solution below is faster and uses less memory, it’s less clear for grasping the core concept:

  fn iterative_fibonacci(n: u32) -> u32 {
      if n <= 1 {
          return n;
      }
      let mut a = 0;
      let mut b = 1;
      let mut c: u32;
      for _ in 2..=n {
          c = a + b;
          a = b;
          b = c;
      }
      b
  }

Remember, understand the problem first, then optimize later. Premature optimization can be a pitfall, but if a single iteration solves it cleanly, the urge to optimize is tempting!

3. Mastering the Initial Case and Recursive Formula

Solid DP articles consistently emphasize these steps. Take the Fibonacci sequence:

Initial Cases: fibonacci(0) = 0, fibonacci(1) = 1
Recursive Formula: fibonacci(n) = fibonacci(n — 1) + fibonacci(n — 2)

Don’t rush to code. Write these down, understand them thoroughly. This clarity will prevent errors and lead to a well-structured solution.

Conclusion

By embracing recursion for initial understanding and following a step-by-step approach, we can break free from the template trap and develop a true grasp of Dynamic Programming.