Frog Jump Problem

Jul 28, 2024

Notes on Frog Jump Problem in Dynamic Programming

Introduction

  • Welcome to Lecture 3 of Dynamic Programming Playlist
  • Important to watch previous lectures for better understanding

Problem Statement

  • A frog starts on the first step of an n-stair staircase
  • The objective is to reach the nth stair with minimal energy
  • Energy lost in jumping from stair i to stair j is calculated as |height[i] - height[j]|
  • The frog can jump to either the next step (i+1) or the step after that (i+2)

Example

  • Given heights: 10, 20, 30, 10
  • Indexing as 0, 1, 2, 3 (0-based indexing): frog needs to go from 0 to 3
  • Calculation paths:
    • Jump 0 to 1 (energy = 10), then 1 to 2 (energy = 10), then 2 to 3 (energy = 20)
      • Total energy = 10 + 10 + 20 = 40
    • Better path: Jump 0 to 1 (energy = 10), jump 1 to 3 (energy = 20)
      • Total energy = 10 + 20 = 30
    • Minimum energy found: 20

Recursion Approach

  • Problem can be approached with recursion: exploring all possible paths
  • Care must be taken as a greedy approach may not yield correct results
  • Example of where greedy fails: greedy chooses the local minimum but ignores future costs

Steps to Write Recursion

  1. Express problem in terms of index: want to reach index n-1
  2. Base case: f(0) = 0 (zero energy needed to stay at the starting point)
  3. Recursive calls: derive the energy needed for jumping from the current index to i-1 and i-2
  4. Calculate minimal energy used

Recursion Functionality

  • Recursive function f(index) for minimal energy:
    • Holds both energy calculations for jumps to i-1 and i-2
  • Memoization can be applied due to overlapping subproblems

Application of Memoization

  • Store results for subproblems in a dp array
  • Time complexity: O(n) due to handling each subproblem once
  • Space complexity: O(n) if using a dp array

Tabulation & Space Optimization

  1. Tabulation (Bottom-Up Approach): rather than recursion, fill the dp array iteratively
  2. Space optimization: find a way to store only necessary parameters (previous two values) rather than entire dp array
  3. Similar step processing: as jump solutions are computed, only store the last two values

Homework Follow-Up Problem

  • Extend the problem to allow K jumps instead of just 1 or 2 jumps
  • Challenge yourself to implement this solution and share in the comments for discussion

Conclusion

  • Understanding all these techniques will aid in solving dynamic programming problems effectively
  • Encourage to comment "understood" if the lecture was helpful!
  • Reminder to check the SD sheet in the description for resources

Final Thoughts

  • Engage with content and share the channel with peers
  • Integration of learning through thorough practice and problem-solving key concepts.