DAY 23 Dynamic Programming: Turning Complexity Into Simplicity

Hey Everyone!

Today was all about Dynamic Programming (DP)—a technique that transforms seemingly complex problems into manageable subproblems. DP problems always feel like solving a series of small puzzles that build up to a grand solution. The key is finding the right recurrence relation and sticking to it like a compass guiding you through a dense forest.

---

How the Day Played Out

1. Climbing Stairs (Easy Difficulty)

   • This classic problem was about finding the number of distinct ways to climb n stairs, where you can take either 1 or 2 steps at a time.
   • The Strategy:
     • Used a DP array where dp[i] represents the number of ways to reach the ith step.
     • The recurrence relation dp[i] = dp[i-1] + dp[i-2] simplified everything.
   • The Fun Part:
     • Realizing how similar this problem is to the Fibonacci sequence was satisfying—it’s always amazing to find patterns in problem-solving.

2. Longest Increasing Subsequence (Medium Difficulty)

   • The challenge was to find the length of the longest increasing subsequence in an array.
   • The Strategy:
     • Used a DP array where dp[i] represents the length of the LIS ending at index i.
     • Updated dp[i] by comparing it with all previous indices to check if extending the subsequence was possible.
   • The Fun Part:
     • Seeing the array evolve with each iteration, forming a chain of increasing numbers, felt like solving a dynamic puzzle.

3. Partition Equal Subset Sum (Medium Difficulty)

   • Determine if an array can be partitioned into two subsets with equal sums.
   • The Strategy:
     • Treated this as a 0/1 Knapsack problem, where the goal was to check if a subset with a sum equal to totalSum / 2 exists.
     • Used a DP table to track possible sums using the array elements.
   • The Fun Part:
     • Turning a subset sum problem into a knapsack problem made it feel like finding hidden connections between different concepts.

---

What Made Today Special

1. Finding Patterns:

   • From Fibonacci-like relations in Climbing Stairs to sequence-building in Longest Increasing Subsequence, today was all about recognizing and leveraging patterns.

2. Breaking Problems Into Steps:

   • DP problems emphasize building solutions incrementally. Solving smaller subproblems first felt like laying a foundation for a bigger structure.

3. Connections Across Problems:

   • Partition Equal Subset Sum showed how seemingly unrelated concepts (like knapsack problems) can offer elegant solutions to new challenges.

---

Key Takeaways

• The Recurrence Relation is Everything:

  • A clear recurrence relation simplifies even the most complex problems. It’s the roadmap to the solution.

• Optimize When Possible:

  • Problems like Climbing Stairs can be solved using constant space by tracking only the last two states instead of using a full DP array.

• Think of Problems as Stories:

  • Each DP problem is like a story where each subproblem is a chapter, building toward the final resolution.

---

Reflections

The Partition Equal Subset Sum problem stood out for its depth and its connection to the knapsack problem—it felt like unlocking a door to a treasure trove of possibilities. On the other hand, Longest Increasing Subsequence was a great exercise in managing dependencies and updating states dynamically.

---

What’s Next?

Tomorrow, I’ll step into the world of Graph Problems, including Number of Islands, Course Schedule, and Shortest Path in Binary Matrix. These will test my ability to traverse and manipulate graphs while solving real-world-inspired challenges.

Thanks for joining me on this exciting journey! Let’s keep exploring and learning together.