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PATHFINDING WITH A* Presented by Joseph Siefers February 19 th, 2008

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Outline Introduction to Pathfinding Search Terminology Search Methods Uninformed Search Informed Search A* Search: How it Works A* Optimizations Conclusion

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Introduction Effective pathfinding is the key to a good game …unless your studio wants to be the laughing stalk of gamers worldwide: http://www.youtube.com/watch?v=wsNK-eisKuU http://www.youtube.com/watch?v=wsNK-eisKuU http://www.youtube.com/watch?v=EH0JRFkpnU0&feature=related http://www.youtube.com/watch?v=EH0JRFkpnU0&feature=related But it is difficult to do. Efficient search strategy/algorithms A* is the “star of the search”

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Case Study—The 8-Puzzle

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Case Study: The 8-Puzzle From the set of AI exercises know as “Toy Problems” Why? Easy to define and understand. Sufficiently complex to analyze search methods.

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Case Study: The 8-Puzzle Rules of the game: Move free space {left, up, down, right} within boundaries of puzzle. 123 46 785 123 46 785 13 426 785 LeftUp 123 486 75 123 46 785 DownRight

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Search Terminology

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Search Tree Node 123 46 785 123 46 785 13 426 785 LeftUp 123 486 75 123 46 785 DownRight

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Search Terminology 123 46 785 123 46 785 13 426 785 LeftUp 123 486 75 123 46 785 DownRight “Initial State”

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Search Terminology Successor Function 123 46 785 123 46 785 13 426 785 Left Up 123 486 75 123 46 785 Down Right “Successors”

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Search Terminology Branching Factor (b) 123 46 785 123 46 785 13 426 785 Left Up 123 486 75 123 46 785 Down Right b=4

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Search Terminology Step Cost 123 46 785 123 46 785 13 426 785 123 486 75 123 46 785 +1 23 146 785 123 746 85 13 426 785 (…)

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Search Terminology Path Cost g(n) 123 46 785 123 46 785 13 426 785 123 486 75 123 46 785 +1 23 146 785 123 746 85 13 426 785 (…) g(n) = 1

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Search Terminology Path Cost g(n) 123 46 785 123 46 785 13 426 785 123 486 75 123 46 785 +1 23 146 785 123 746 85 13 426 785 (…) g(n) = 2

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Search Terminology Goal State 318 64 725 (…) 123 456 78 123 456 78

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Search Terminology “The Fringe” Open/Closed lists “Algorithms which ignore their history are doomed to repeat it.” –AIMA pg 82

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Search Strategies

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Uninformed Search Strategies Uninformed Search Blindly search tree “Driving without a map”

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Uninformed Search Strategies Breadth First Search nodes in “hierarchical order” FIFO Image Credit: Wikimedia Commons

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Uninformed Search Strategies Let’s see breadth-first search in action! “Pathdemo” by Bryan Stout

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Uninformed Search Strategies Breadth-First in Action

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Uninformed Search Strategy Overall Result:

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Uninformed Search Strategies Depth First Repeatedly traverse one path to completion

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Unformed Search Strategies Depth-First in Action? —“No Dice”

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Uninformed Search Strategies Problems: Not practical: (estimated with b= 10, a search rate of 10 5 nodes/sec, and 1 KiB/node) Credit: AIMA fig 3.11, pg 74

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Search Strategies Informed Search Traverse search tree “intelligently” “Driving with a map” A heuristic function = “the map”

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Informed Search Strategies Heuristic function h(n) Estimate path cost to goal Key to improving search problems NEVER overestimate— “be optimistic”

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Heuristics Admissibility “Relaxed” Problems Simplify constraints Reduces complexity Example: 8-Puzzle ? Initial StateGoal State 413 25 786 123 456 78

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A* Search: Introduction A * search: Extremely fast Provably optimal (given an admissible heuristic) Flexible and Adaptable

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A* Search: How it Works Estimated Overall Path Cost = How far I think I have to go + How far I have come

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A* Search: How it Works Node evaluation function Heuristic function Current Path Cost

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A* Search: How it works A* Algorithm Pseudocode 1. Let P= the starting point 2. Assign f, g and h values to P. 3. Add P to the Open list. 4. Let B = the best node (lowest f) from the Open list a. If B is the goal node, quit (path found) b. If the Open list is empty, quit (path impossible) A* Algorithm Pseudocode 1. Let P= the starting point 2. Assign f, g and h values to P. 3. Add P to the Open list. 4. Let B = the best node (lowest f) from the Open list a. If B is the goal node, quit (path found) b. If the Open list is empty, quit (path impossible)

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A* Search: How it works 5. Expand B a. Generate f,g,h values for the successors of B b. Add successors to open list, appropriately 6. Repeat from step 4 5. Expand B a. Generate f,g,h values for the successors of B b. Add successors to open list, appropriately 6. Repeat from step 4

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A* in Map Problems Example Problem—“The hurried traveler”

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A* Map Problem Heuristics A* efficiency dependent on choice of heuristic Euclidean Distance Always admissible Manhattan Distance “City Blocks” Admissible for discrete map problems h = |dx-sx| + |dy-sy|

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A* Map Problem Heuristics Manhattan vs. Euclidean distance comparison Image Credit: Wikimedia Commons

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A* Search: Map Problem Testing Interesting parameters in “Pathdemo” Heuristic weight Sample heuristic functions Manhattan Distance Diagonal Shortcut Euclidean Max

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A* Search: Compare and Contrast Manhattan distance: Weight 0

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A* Search: Compare and Contrast Manhattan distance: Weight 1

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A* Search: Compare and Contrast Manhattan distance: Weight 2

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A* Search: Compare and Contrast Manhattan distance: Weight 3

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A* Search: A Look Back… Breadth-First Search A*, Manhattan, W=3 75% LESS NODES

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A* Optimizations Don’t pathfind Common solution Quick straight line check “Flood insurance” Measure flooding Visually on map

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A* Optimizations Visual representation of flooding Image Credit: AIGPW Figure 3.4.1

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A* Optimizations Don’t pathfind Common solution “Flood insurance” Measure flooding and reduce Visually on map Sometimes futile… Limit map designs?

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A* Optimizations Memory Pooling Allocating memory CPU intensive How much memory? Specific to each game Find “perfect balance” Statistical Analysis Current technology makes pooling practical

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A* Optimizations Open/Closed list implementation “Priority queue” Heap O(lg n) operations Even this might not be fast enough Hash Table O(1) operations Problems fine-tuning “Cheap List”

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Optimization Tools Performance metric for search algorithms Intel VTune In-Code Timers Approximate execution time Multithreading Issues Other threads might be busy and slowing Pathfinding Valuable “Standard Path” No tool stands by itself

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Conclusion A* solves difficult pathfinding problems Ubiquitous across gaming industry Shows up in every type of game A* is highly dependent on a good heuristic Breadth-first if bad heuristic A* can be resource-intensive A* is not a “one-size fits all” solution Plan on spending time optimizing Solved problem

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