A* Path Finding Algorithm

1
A Path Finding Algorithm

• Presented by Daniel Natapov

2
Problem Definition

• Find the shortest (weighted) path from a start
  node to a goal node in a graph (or grid).
• What if we are informed with heuristics? Can we
  look-ahead and direct our search?

3
Application

• Games NPC movement. Needs to be smart and fast.
• Games use grids to describe the environment.
  These slides do too.
• Many other applications
• Network routing
• Image processing
• A.I. Path finding
• ...

4
Grid Graph

• Grid allows movement between adjacent cells in 4
  or 8 possible directions.
• Each direction may have a different cost.

5
Example Get from S to T




S




T



6
Example Edge Weights

• In a game, edge weights depend on various
  factors, ie travel on road vs. grass.
• For simplicity lets say all horizontal and
  vertical costs are the same.
• Also assume no diagonal paths.

7
WWDD? What Would Dijkstras Do?




S




T



S
Found it! (finally)
8
WWDD?

• Dijkstras algorithm guarantees shortest path.
• But searches a lot of unneeded area.
• We know where the destination node is, (just not
  how to get there).
• We can try to direct the search with greedy
  Best-First-Search.

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Best-First-Search

• Similar to Dijkstras, but is informed.
• Has some estimate of how far from the goal each
  vertex is look-ahead.
• This estimate is a heuristic.
• It prioritizes vertices which it believes to be
  closets to the goal, as opposed to vertices
  closest to the start.

10
Best-First-Search example




S




T



S
11
How to break it Obstacles!




S




T



S
Found it! (couldve taken a better route)
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Best-First-Search

• Best-First-Search works faster than Dijkstras.
• But does not guarantee an optimal-path.
• We want some combination of Dijkstras and
  Best-First-Search.
• Enter A!

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A algorithm

• Prioritizes its search based on
• The distance traveled (Dijkstras)
• The distance remaining (Best-First-Search)
• g(n) Distance traveled from the start to a
  cell.
• h(n) Estimated distance from a cell to the
  target.
• Value of a cell is f(n) g(n) h(n).
• The algorithm prioritizes cells whose f(n) is
  lowest.

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Whats all this talk about estimates?

• An estimate of the distance between a cell and a
  target is a heuristic.
• May be able to estimate distance between two
  cells.
• Choosing a good heuristic is important, and can
  be difficult.
• In our simplified case it is easy the Manhattan
  Distance
• h(n) cell.x goal.x cell.y goal.y

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Manhattan Distance

• Good for our case. Actual distance can never be
  less (more on this later).
• Lets go through a complete example.

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A example


S T


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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

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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

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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 2 h 3 f 5
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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

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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
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
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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
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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 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
g2 h7 f9
g3 h8 f11
g2 h7 f9
g3 h8 f11
g2 h7 f9
g3 h8 f11
g3 h8 f11
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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
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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
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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
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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
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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
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(No Transcript)
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More about heuristics

• Depending heuristic, A can be admissible.
• This guarantees an optimal solution, despite
  using an estimate.
• For A to be admissible and guarantee an optimal
  solution we need
• ?n, h(n) h(n)
• h(n) is the actual distance.
• If the heuristic overestimates the actual
  distance, an optimal solution is not guaranteed.

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More about heuristics contd

• For admissibility we also need monotonicity.
• Satisfy triangle inequality h(n1) c(n1 ? n2)
  h(n2)

n1
h(n1)
c(n1-gtn2)
goal
n2
h(n2)
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Fiddling with the heuristic

• Use the heuristic to balance speed vs. accuracy.
• If h(n) 0, then f(n) g(n).
• In other words, A becomes Dijkstras.
• If h(n) gtgt g(n), g(n) can be ignored.
• f(n) h(n). A becomes Best-First-Search.
• In general
• The bigger g(n) is, the more it expands, which
  makes it slower.
• The bigger h(n) is, the more direct the search
  is, but better paths could be missed.

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Fiddling with the heuristic 2

• If h(n) h(n), then A will find the optimal
  solution, and not expand anything unnecessary.
• Straight to the target.
• Only possible with good heuristic and no
  obstacles.
• Can fiddle with the heuristic and set it
  depending on the need.
• Sometimes okay to get an approximate solution at
  the cost of a speed-up.

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Speed-Accuracy see-saw
g(n)
h(n)
Speed
Accuracy
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Formal Definition

• preCond Input a grid/graph G with positive edge
  weights, a source node s, a target node t.
• Also given a admissible heuristic for estimating
  distances.
• postCond Finds a shortest weighted path from s
  to t.
• Loop Invariant So far, the nodes have been
  handled in order of f(n), where f(n) g(n)
  h(n).

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Formal Definition Contd

• Step Handle the found (not handled) node with
  min f(n).
• Store the parent for each cell the cell through
  which the shortest path from s came.
• Exit Stop when t has been found.
• Obtaining the post condition LI Exit Code gt
  PostCond.
• Proving the path we traced back is shortest
• Prove there is a path of this length We have
  one.
• Prove there is no shorter path ...

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Proof that there is no better path
COMPLICATED SLIDE. PAY ATTENTION!

• We know our heuristic is admissible, h(n)
  h(n).
• By LI, our path handles cells in order of the
  minimum of f(n) g(n) h(n).
• All unhandled paths have a larger f(n) than ours.
  
• We found t. So h(n)0. f(n) becomes our actual
  -g(n). In other words, our actual cost is lower
  than the actualestimated of any other found
  node.
• The estimated cost is always less than the actual
  cost. Meaning our actual cost is less than any
  other actual cost.

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Confused?
f(n)
f(any) g(any) h(any)
g(any) actual(any)
Our actual cost
Any other found estimate
Any other actual
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What it all means

• If heuristic is admissible, A returns the
  shortest path.
• It will find it by (likely) expanding and
  searching less cells than Dijkstras.
• But if the condition that h(n) h(n) is
  violated, we can no longer ensure optimality.
• Should it always be optimal?

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Running Time

• Well....
• Dijkstras O(E V log V )
• V number of vertices, E number of edges.
• Obviously it is possible for A to search every
  edge as well, so we have no savings in the worst
  case.
• Lets focus on the nodes instead
• Dijkstras O(V2)

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Running time Dijkstras
L
S
T
Area of circle is O(L2)
43
Running time A
H
L
S
T
Area of half ellipse O(LH)
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Running time A
T
S
H
L/n
In total O((L/n)Hn) O(LH)
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All Done!

• Thank you. Questions?
• The algorithm was first described in 1968 by
  Peter Hart, Nils Nilsson, and Bertram Raphael
• References Resources
• http//theory.stanford.edu/amitp/GameProgramming/
  (Great source)
• http//www.policyalmanac.org/games/aStarTutorial.h
  tm
• http//en.wikipedia.org/wiki/A_search_algorithm
• http//www.cse.yorku.ca/course_archive/2008-09/W/3
  402/slides/Week3.pdf