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A* Path Finding

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The document provides an overview of the A* pathfinding algorithm. It begins by defining the pathfinding problem and describing applications like game development. It then introduces the A* algorithm, which finds the shortest path between nodes in a graph by considering both the distance traveled so far (like in Dijkstra's algorithm) and the estimated distance to the goal (like in best-first search). The document provides an example of A* in action and discusses how heuristics can guide the search while ensuring an optimal solution. It also compares the running time of A* to Dijkstra's algorithm.

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A* PATH FINDING ALGORITHM Presented by Daniel Natapov
PROBLEM DEFINITION ,[object Object],[object Object]
APPLICATION ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
GRID = GRAPH ,[object Object],[object Object],[object Object]
EXAMPLE – GET FROM S TO T S T
EXAMPLE – EDGE WEIGHTS ,[object Object],[object Object],[object Object]
WWDD? – WHAT WOULD DIJKSTRA’S DO? Found it! (finally) S S T
WWDD? ,[object Object],[object Object],[object Object],[object Object]
BEST-FIRST-SEARCH ,[object Object],[object Object],[object Object],[object Object]
BEST-FIRST-SEARCH EXAMPLE S S T
HOW TO BREAK IT – OBSTACLES! S Found it! (could’ve taken a better route) S T
BEST-FIRST-SEARCH ,[object Object],[object Object],[object Object],[object Object]
A* ALGORITHM ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
WHAT’S ALL THIS TALK ABOUT ESTIMATES? ,[object Object],[object Object],[object Object],[object Object],[object Object]
MANHATTAN DISTANCE ,[object Object],[object Object]
A* EXAMPLE S T
A* EXAMPLE g = 1 h = 6 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7
A* EXAMPLE g = 1 h = 6 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7
A* EXAMPLE g = 2 h = 3 f = 5 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7 g = 2 h = 5 f = 7
A* EXAMPLE g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7 g = 2 h = 5 f = 7
A* EXAMPLE g = 3 h = 6 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 3 h = 6 f = 9
A* EXAMPLE g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 3 h = 6 f = 9
A* EXAMPLE g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9
A* EXAMPLE g=2 h=7 f=9 g=2 h=7 f=9 g=2 h=7 f=9 g=3 h=8 f=11 g=3 h=8 f=11 g=3 h=8 f=11 g=3 h=8 f=11 g = 3 h = 6 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9
A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9
A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9 g = 5 h = 4 f = 9
A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 6 h = 3 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9 g = 5 h = 4 f = 9 g = 6 h = 3 f = 9
A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 g = 7 h = 2 f = 9 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 6 h = 3 f = 9 g = 7 h = 2 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9 g = 5 h = 4 f = 9 g = 6 h = 3 f = 9
A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 g = 7 h = 2 f = 9 g = 8 h = 1 f = 9 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 6 h = 3 f = 9 g = 7 h = 2 f = 9 g = 8 h = 1 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9 g = 5 h = 4 f = 9 g = 6 h = 3 f = 9
 
MORE ABOUT HEURISTICS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
MORE ABOUT HEURISTICS CONT’D ,[object Object],[object Object],n1 goal h(n1) n2 c(n1->n2) h(n2)
FIDDLING WITH THE HEURISTIC ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
FIDDLING WITH THE HEURISTIC 2 ,[object Object],[object Object],[object Object],[object Object],[object Object]
SPEED-ACCURACY SEE-SAW g(n) h(n) Speed Accuracy
FORMAL DEFINITION ,[object Object],[object Object],[object Object],[object Object]
FORMAL DEFINITION CONT’D ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
PROOF THAT THERE IS NO BETTER PATH ,[object Object],[object Object],[object Object],[object Object],[object Object],COMPLICATED SLIDE. PAY ATTENTION!
CONFUSED? f(n) ≤ f(any) = g(any) + h(any) ≤ g(any) + actual(any) Our actual cost Any other found estimate Any other ‘actual’
WHAT IT ALL MEANS ,[object Object],[object Object],[object Object],[object Object]
RUNNING TIME ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
RUNNING TIME – DIJKSTRA’S S T Area of circle is O(L 2 )
RUNNING TIME – A* S T Area of half ellipse: O(L∙H)
RUNNING TIME – A* S T In total: O((L/n)∙H∙n) = O(L∙H)
ALL DONE! ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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A* Path Finding

  • 1. A* PATH FINDING ALGORITHM Presented by Daniel Natapov
  • 2.
  • 3.
  • 4.
  • 5. EXAMPLE – GET FROM S TO T S T
  • 6.
  • 7. WWDD? – WHAT WOULD DIJKSTRA’S DO? Found it! (finally) S S T
  • 8.
  • 9.
  • 11. HOW TO BREAK IT – OBSTACLES! S Found it! (could’ve taken a better route) S T
  • 12.
  • 13.
  • 14.
  • 15.
  • 17. A* EXAMPLE g = 1 h = 6 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7
  • 18. A* EXAMPLE g = 1 h = 6 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7
  • 19. A* EXAMPLE g = 2 h = 3 f = 5 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7 g = 2 h = 5 f = 7
  • 20. A* EXAMPLE g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7 g = 2 h = 5 f = 7
  • 21. A* EXAMPLE g = 3 h = 6 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 3 h = 6 f = 9
  • 22. A* EXAMPLE g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 3 h = 6 f = 9
  • 23. A* EXAMPLE g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9
  • 24. A* EXAMPLE g=2 h=7 f=9 g=2 h=7 f=9 g=2 h=7 f=9 g=3 h=8 f=11 g=3 h=8 f=11 g=3 h=8 f=11 g=3 h=8 f=11 g = 3 h = 6 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9
  • 25. A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9
  • 26. A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9 g = 5 h = 4 f = 9
  • 27. A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 6 h = 3 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9 g = 5 h = 4 f = 9 g = 6 h = 3 f = 9
  • 28. A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 g = 7 h = 2 f = 9 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 6 h = 3 f = 9 g = 7 h = 2 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9 g = 5 h = 4 f = 9 g = 6 h = 3 f = 9
  • 29. A* EXAMPLE g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 S g = 1 h = 4 f = 5 g = 7 h = 2 f = 9 g = 8 h = 1 f = 9 T g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 1 h = 6 f = 7 g = 2 h = 5 f = 7 g = 6 h = 3 f = 9 g = 7 h = 2 f = 9 g = 8 h = 1 f = 9 g = 3 h = 8 f = 11 g = 2 h = 7 f = 9 g = 3 h = 6 f = 9 g = 4 h = 5 f = 9 g = 5 h = 4 f = 9 g = 6 h = 3 f = 9
  • 30.  
  • 31.
  • 32.
  • 33.
  • 34.
  • 35. SPEED-ACCURACY SEE-SAW g(n) h(n) Speed Accuracy
  • 36.
  • 37.
  • 38.
  • 39. CONFUSED? f(n) ≤ f(any) = g(any) + h(any) ≤ g(any) + actual(any) Our actual cost Any other found estimate Any other ‘actual’
  • 40.
  • 41.
  • 42. RUNNING TIME – DIJKSTRA’S S T Area of circle is O(L 2 )
  • 43. RUNNING TIME – A* S T Area of half ellipse: O(L∙H)
  • 44. RUNNING TIME – A* S T In total: O((L/n)∙H∙n) = O(L∙H)
  • 45.