DAY 43 Bit Manipulation Meets Multidimensional Thinking

Hello Everyone!

Day 3 of Week 9 brought an exciting mix of Bit Manipulation and Multidimensional Dynamic Programming. The tasks required a blend of creativity and structured thinking, from solving range-based problems with bits to managing triangular grids with DP. It felt like combining precision tools with flexible strategies.

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How the Day Played Out

1. Bitwise AND of Number Ranges (Medium Difficulty)

   • Compute the bitwise AND of all numbers in a given range [m, n].
   • The Strategy:
     • Observed that as the range grows, the trailing bits of numbers in the range become zero due to overlapping.
     • Used bit shifting to identify the common prefix of m and n by right-shifting both numbers until they were equal, then shifted the result back.
   • The Fun Part:
     • Watching the common bits emerge step by step felt like peeling layers off a problem to reveal its core structure.

2. Triangle (Medium Difficulty)

   • Find the minimum path sum from the top to the bottom of a triangular grid.
   • The Strategy:
     • Used bottom-up dynamic programming to calculate the minimum path sum for each level, starting from the second-last row and moving up.
     • Updated each cell with the minimum sum of its possible paths (left or right child) plus its current value.
   • The Fun Part:
     • The process of collapsing the triangle layer by layer into a single solution was satisfying—it felt like solving a layered puzzle.

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What Made Today Special

1. Bitwise Elegance:

   • The solution for Bitwise AND of Number Ranges showcased how simple bit shifts can replace brute force, making the solution both elegant and efficient.

2. Multidimensional Problem-Solving:

   • Triangle required managing dependencies across multiple dimensions, emphasizing the power of structured thinking in DP.

3. Different Perspectives:

   • From low-level bit manipulation to high-level grid-based DP, today’s problems highlighted the versatility required for competitive programming.

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Key Takeaways

• Bitwise AND with Ranges:

  • Problems like Bitwise AND of Number Ranges emphasize how recognizing patterns (like trailing zeros) simplifies operations on ranges.

• Bottom-Up DP for Efficiency:

  • Solving Triangle bottom-up reduced space complexity, demonstrating the efficiency of in-place updates in dynamic programming.

• Layered Problem-Solving:

  • Both problems involved breaking down complex scenarios (ranges and grids) into manageable layers, leading to concise solutions.

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Reflections

The Bitwise AND of Number Ranges problem was a fascinating exercise in recognizing patterns and leveraging bit-level operations for efficiency. Meanwhile, Triangle was a rewarding DP problem that required systematic optimization from the ground up. Together, these tasks reinforced the importance of adaptability and structured thinking.

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What’s Next?

On Monday, I’ll conclude Week 9 with Multidimensional DP Problems, tackling Minimum Path Sum and Unique Paths II. These challenges will further test my ability to manage dependencies and optimize grid-based solutions.

Thank you for following along! Let’s keep learning, solving, and growing together.