A New Approach of Finding the Steady-State Visit Rates of a Two Dimensional Maze

1. A New Approach of Finding the Steady-State Visit Rates of a Two Dimensional Maze Kurtis Cahill James Badal

2. Outline • Introduction • Model a Maze as a Markov Chain • Assumptions • First Approach and Example • Second Approach and Example • Experiment • Results • Conclusion

3. Introduction • Problem: To find an efficient approach of solving the rate of visitation of a cell inside a large maze • Application: To find the best possible place to intercept information

4. Model a Maze as a Markov Chain • Allows Stochastic principles to be applied to the problem • Each maze cell will be model as a state in Markov Chain • The Markov Chain will be one recurrent class

5. Assumptions • To reduce the complexity of the problem and simulation, certain assumptions will be applied: • Unbiased transition to adjacent cells • Random walk can’t be stationary • No isolated cells inside the maze

6. First Approach • ri – Steady-state rate of the ith state of the Markov Chain • pji– Probability of moving from state j to state ion the next step

7. 3x3 Maze Example The transition matrix for the random walk onthis maze

8. 3x3 Maze Example System of Steady State Rate Equations

9. 3x3 Maze Example Row Reduced System of Steady State Rate Equations

10. Second Approach • ri – Steady-state rate of the ith state of the Markov Chain • p – Proportionality constant • ni– Number of connections to the ith cell

11. 3x3 Maze Example

12. 3x3 Maze Example Solution to System of Steady State Rate Equations

13. Experiment • Random Walker starts at a certain maze location and walks 108 steps • At each step the random walker increments the visit count of the most recently visited cell • The mean and standard deviation aremeasured at the end of the experiment • The measured result is compared to the calculated result

14. Results Random Walk result of a 2x2 Maze

15. Results Random Walk result of a 5x5 Maze

16. Results Random Walk result of a 10x10 Maze

17. Results Random Walk result of a 20x20 Maze

18. Results Random Walk result of a 40x40 Maze

19. Conclusion • Modeled the maze as a Markov Chain • Applied Stochastic principles to the maze • First Approach is n3 complexity • Second Approach is ncomplexity • Tested the calculated result with the measured result