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

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

Introduction  Effective pathfinding is the key to a good game  …unless your studio wants to be the laughing stalk of gamers worldwide:    But it is difficult to do.  Efficient search strategy/algorithms  A* is the “star of the search”

Case Study—The 8-Puzzle

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.

Case Study: The 8-Puzzle  Rules of the game:  Move free space {left, up, down, right} within boundaries of puzzle LeftUp DownRight

Search Terminology

 Search Tree  Node LeftUp DownRight

Search Terminology LeftUp DownRight “Initial State”

Search Terminology  Successor Function Left Up Down Right “Successors”

Search Terminology  Branching Factor (b) Left Up Down Right b=4

Search Terminology  Step Cost (…)

Search Terminology  Path Cost g(n) (…) g(n) = 1

Search Terminology  Path Cost g(n) (…) g(n) = 2

Search Terminology  Goal State (…)

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

Search Strategies

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

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

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

Uninformed Search Strategies  Breadth-First in Action

Uninformed Search Strategy  Overall Result:

Uninformed Search Strategies  Depth First  Repeatedly traverse one path to completion

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

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

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

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

Heuristics  Admissibility  “Relaxed” Problems  Simplify constraints  Reduces complexity  Example: 8-Puzzle ? Initial StateGoal State

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

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

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

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)

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

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

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|

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

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

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

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

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

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

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

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

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

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

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

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”

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

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

Thank you!  Thanks for listening!