This document describes an empirical study on pathfinding algorithms. It tests different neighborhood region sizes and heuristic functions using A-Star pathfinding on 6 maps. The results show that neighborhood region strongly affects runtime and path length, while heuristic function has no significant impact. Nodes evaluated models runtime well, while path length is a poor predictor of runtime. The 8-way neighborhood region is recommended to minimize nodes evaluated.

1. Joseph Krall
Master Student, Computer Science
WestVirginia University
April 2010

2.  Introduction
 Neighborhood Regions
 Heuristic Functions
 Algorithms
 Empirical Study
 Analysis
 Results
 Threats toValidity
 Conclusion
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3.  Pathfinding is a subject of high research
interest
 Applications inVideo Games and AI
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4.  Many Problems with PathfindingToday
 http://www.ai-blog.net/archives/000152.html
 In this Project…
 An Empirical Study
 UsingA-Star
 But first a look at Pathfinding Methods…
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5.  Introduction
 Neighborhood Regions
 Heuristic Functions
 Algorithms
 Empirical Study
 Analysis
 Results
 Threats toValidity
 Conclusion
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6.  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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7.  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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8.  Euclidean Distance
 Distance “as the crow flies”
 Not alwaysTrue Distance
 Diagonal Distance
 Combines Manhattan and Euclidean
 AlwaysTrue Distance
.
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9.  Introduction
 Neighborhood Regions
 Heuristic Functions
 Algorithms
 Empirical Study
 Analysis
 Results
 Threats toValidity
 Conclusion
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10.  Three Commonly Known Algorithms…
 Dijkstra’s Algorithm
 Expand outward in all directions until goal found
 GuaranteedOptimal Path, but slow
 Best-First Search
 Expand in direction of goal, until goal is found
 Not GuaranteedOptimal Path, but fast
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11.  A-Star
 Hybrid of first two
 Expand in direction of goal node
 GuaranteedOptimal Path, and is also fast
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12.  Introduction
 Neighborhood Regions
 Heuristic Functions
 Algorithms
 Empirical Study
 Analysis
 Results
 Threats toValidity
 Conclusion
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13.  An experiment using A-Star
 3x3x6 Factorial Design
▪ 3 Neighborhood Regions
▪ 3 Distance Functions
▪ 6 Different Maps
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14.  Factors
 Neighborhood Region
 Heuristic Function
 Map
 DependentVariables
 Nodes Evaluated
 Path Length
 Runtime
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15.  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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16.  Introduction
 Neighborhood Regions
 Heuristic Functions
 Algorithms
 Empirical Study
 Analysis
 Results
 Threats toValidity
 Conclusion
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17.  Tests Used
 TwoWay 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 ExcelTrendlines
▪ For Goals #7 and #8
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18.  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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19.  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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20.  Goodness of Fit Charts
 Path Length vs Runtime
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21.  Goodness of Fit Charts
 Nodes Evaluated vs. Runtime
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22.  Introduction
 Neighborhood Regions
 Heuristic Functions
 Algorithms
 Empirical Study
 Analysis
 Results
 Threats toValidity
 Conclusion
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23.  Array-based A-Star
 Time spent looping through arrays
 ExternalValidity
 Tested using Personal Computer
 Not the best runtimes
 Runtimes scaled higher than usual
 May still be generalizable
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24.  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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