Joseph Krall Master Student, Computer Science West Virginia University April 2010

Introduction Neighborhood Regions Heuristic Functions Algorithms Empirical Study Analysis Results Threats to Validity Conclusion

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Pathfinding is a subject of high research interest

Applications in Video Games and AI

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Many Problems with Pathfinding Today

http://www.ai-blog.net/archives/000152.html

In this Project…

An Empirical Study

Using A-Star

But first a look at Pathfinding Methods…

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Introduction Neighborhood Regions Heuristic Functions Algorithms Empirical Study Analysis Results Threats to Validity Conclusion

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A set of accessible nodes surrounding a node

4-Way System

8-Way System

16-Way System

Steps to neighbors have associated costs

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Estimate distance from node to goal

Manhattan Distance

Step only {Up, Down, Left, Right} and count

D(n) = |X1 – X2| + |Y1 – Y2|

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Euclidean Distance

Distance “as the crow flies”

Not always True Distance

Diagonal Distance

Combines Manhattan and Euclidean

Always True Distance

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Introduction Neighborhood Regions Heuristic Functions Algorithms Empirical Study Analysis Results Threats to Validity Conclusion

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Three Commonly Known Algorithms…

Dijkstra’s Algorithm Expand outward in all directions until goal found

Guaranteed Optimal Path, but slow

Best-First Search

Expand in direction of goal, until goal is found

Not Guaranteed Optimal Path, but fast

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A-Star

Hybrid of first two

Expand in direction of goal node

Guaranteed Optimal Path, and is also fast

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Introduction Neighborhood Regions Heuristic Functions Algorithms Empirical Study Analysis Results Threats to Validity Conclusion

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An experiment using A-Star 3x3x6 Factorial Design

▪ 3 Neighborhood Regions

▪ 3 Distance Functions

▪ 6 Different Maps

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Factors Neighborhood Region

Heuristic Function

Map

Dependent Variables

Nodes Evaluated

Path Length

Runtime

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Research Goals 1. Does Neighborhood Region affect Runtime?

2. Does Heuristic Function affect Runtime?

3. Does Neighborhood Region affect Nodes Evaluated?

4. Does Heuristic Function affect Nodes Evaluated?

5. Does Neighborhood Region affect Path Length?

6. Does Heuristic Function to Path Length ?

7. Does Path Length affect Runtime ?

8. Does Nodes Evaluated affect Runtime ?

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Introduction Neighborhood Regions Heuristic Functions Algorithms Empirical Study Analysis Results Threats to Validity Conclusion

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Tests Used

Two Way ANOVA

▪ 99% Confidence for Goals #1 and #2

▪ 75% Confidence for Goals #3 and #4

▪ 95% Confidence for Goals #5 and #6

Goodness of Fit quantified by R-Squared

▪ Using Excel Trendlines

▪ For Goals #7 and #8

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Research Goals

1. Neighborhood Region strongly affects Runtime F = 6.99 | F_Crit = 5.11

2. Heuristic Function has no significance on Runtime F = 0.01 | F_Crit = 5.11

3. Neighborhood Region slightly affects Nodes Evaluated F = 1.596 | F_Crit = 1.143

4. Heuristic Function has no significance on Nodes Evaluated F = 1.596 | F_Crit = 1.143

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Research Goals

5. Neighborhood Region affects Path Length F = 3.432 | F_Crit = 3.204

6. Heuristic Function does not affect Path Length F = ~zero | F_Crit = 3.204

7. Path Length does not model Runtime very well R-Squared = 0.388

8. Nodes Evaluated models Runtime fairly well R-Squared = 0.898 Model: Runtime = 18.89*e0.111[Nodes Evaluated]

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Goodness of Fit Charts Path Length vs Runtime

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Goodness of Fit Charts Nodes Evaluated vs. Runtime

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Introduction Neighborhood Regions Heuristic Functions Algorithms Empirical Study Analysis Results Threats to Validity Conclusion

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Array-based A-Star

Time spent looping through arrays

External Validity

Tested using Personal Computer

Not the best runtimes

Runtimes scaled higher than usual

May still be generalizable

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Want to minimize Nodes Evaluated

Avoid searching Swamps (dead-ends)

Use an appropriate Neighborhood Region

4-Way is best, but impractical

8-Way is the way to go

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