Kurtis Cahill
James Badal
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Introduction
Model a Maze as a Markov Chain
Assumptions
First Approach and Example
Second Approach and Example
Experiment
Results
Conclusion
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Problem: To find an efficient approach of solving the
rate of visitation of a cell inside a large maze
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Application: To find the best possible place to intercept
information
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Allows Stochastic principles to be applied to the
problem
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Each maze cell will be model as a state in Markov
Chain
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The Markov Chain will be one recurrent class
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To reduce the complexity of the problem and
simulation, certain assumptions will be applied:
1. Unbiased transition to adjacent cells
2. Random walk can’t be stationary
3. No isolated cells inside the maze
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ri – Steady-state rate of the ith state of the Markov
Chain
pji– Probability of moving from state j to state i on the
next step
The transition matrix for the random walk on this maze
System of Steady State Rate Equations
Row Reduced System of Steady State Rate Equations
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ri – Steady-state rate of the ith state of the Markov
Chain
p – Proportionality constant
ni – Number of connections to the ith cell
Solution to System of Steady State Rate Equations
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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 are measured at the
end of the experiment
The measured result is compared to the calculated
result
Random Walk result of a 2x2 Maze
Random Walk result of a 5x5 Maze
Random Walk result of a 10x10 Maze
Random Walk result of a 20x20 Maze
Random Walk result of a 40x40 Maze
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Modeled the maze as a Markov Chain
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Applied Stochastic principles to the maze
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First Approach is n3 complexity
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Second Approach is n complexity
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Tested the calculated result with the measured result