artificial intelligence in game design
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Artificial Intelligence in Game Design. Lecture 11: Path Planning Algorithms. Path Planning Algorithms. Dijkstra’s shortest path algorithm Guaranteed to find shortest path Not fast ( O ( n 2 ) ) A* path algorithm Requires estimation heuristic Much faster - PowerPoint PPT PresentationTRANSCRIPT
Artificial Intelligence in Game Design
Lecture 11: Path Planning Algorithms
Path Planning Algorithms
• Dijkstra’s shortest path algorithm– Guaranteed to find shortest path– Not fast (O (n2) )
• A* path algorithm– Requires estimation heuristic– Much faster – Not guaranteed to find shortest path depending on heuristic
Dijkstra Algorithm Example
• Example:
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7
6
Dijkstra Algorithm Example
• Example as a graph:
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1
1
8
6
5
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7
6B
A
E
D
G
C
Goal: find shortest path from A to G
Dijkstra Algorithm Components
• Tree– Contains all explored nodes– Initially start node– Assumption: know shortest path from
start to all nodes in rest of tree
• Fringe– All nodes adjacent to some tree node– Assumption : know shortest path from
start to all fringe nodes using only nodes in tree
• Unknown– All nodes not in tree or fringe
Dijkstra Algorithm Components
• Data structure for each node stores:– State of that node (tree, fringe, or unknown)– Shortest path (so far) from start to node– Total cost of that path– Initially:
• Fringe nodes have just edge from start• Unknown nodes have no path
State Best Path Path Cost
A
B
C
D
E
G
Dijkstra Algorithm
At each cycle:• Add fringe node n with shortest path to tree• Recheck all nodes f in fringe to see if now shorter path
using n– If current best path to f > path to n + edge from n to f
new path to f = path to n + edge from n to f
• Add unknown nodes u adjacent to n to fringe– u’s path = path to n + edge from n to u
• Done when goal node in tree– At that point, have shortest path to goal from start
Dijkstra Algorithm Example
Initially:• Start node A in tree• Adjacent nodes B, C, and E in fringe• Paths to nodes = edge from start
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6B
A
E
D
G
C
Dijkstra Algorithm Example
State Best Path Path Cost
A Tree A
B Fringe A B 6
C Fringe A C 8
D Unknown
E Fringe A E 2
G Unknown
Dijkstra Algorithm Example
Next step:• Add E to tree (shortest path of B, C, and E)• Check whether creates shorter path to B (no)• Check whether adjacent to unknown nodes (no)
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6B
A
E
D
G
C
Dijkstra Algorithm Example
State Best Path Path Cost
A Tree A
B Fringe A B 6
C Fringe A C 8
D Unknown
E Tree A E 2
G Unknown
Check whether A E B shorter
Cost of A E B = 7, so no change
Dijkstra Algorithm Example
Next step:• Add B to tree (shortest path of B and C)• Check whether creates shorter path to B (yes)• Check whether adjacent to unknown nodes (yes)
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1
1
8
6
5
3
7
6B
A
E
D
G
C
Dijkstra Algorithm Example
State Best Path Path Cost
A Tree A
B Tree A B 6
C Fringe A C A B C 8 7
D Fringe A B D 12
E Tree A E 2
G Fringe A B G 13
Check whether A B C shorterCost of A B C = 7, so use it as path
Paths to new nodes = path to B + edge from B
Dijkstra Algorithm Example
Next step:• Add C to tree (shortest path of C, D, and G)• Check whether creates shorter path to G (yes)
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6B
A
E
D
G
C
Dijkstra Algorithm Example
State Best Path Path Cost
A Tree A
B Tree A B 6
C Tree A B C 7
D Fringe A B D 12
E Tree A E 2
G Fringe A B G A B C G 13 10
Check whether A C G shorterCost of A B C G = 10, so use it as path
Dijkstra Algorithm Example
Next step:• Add goal node G to tree (shortest path of D and G)• Algorithm finished
– No possibility of shorter path through D
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1
1
8
6
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6B
A
E
D
G
C
Dijkstra Algorithm Example
State Best Path Path Cost
A Tree A
B Tree A B 6
C Tree A B C 7
D Fringe A B D 12
E Tree A E 2
G Tree A B C G 10
Final path and path cost
Dijkstra Algorithm Analysis
• Worst case: may need to example all n nodes• For each step, must check all m nodes in fringe
– Worst case: all n nodes in fringe– Unlikely, since most levels far less interconnected
• Number of comparisons: O (nm) – O (n2) in worst case
The A* Algorithm
• Similar in structure to Dijkstra– Tree, fringe, and unknown nodes
– Store “best path so far” fro each node explored
– Choose next fringe node to add to tree
• Idea: Choose fringe node n believed to be part of “shortest path”
– Haven’t explored entire graph yet, so don’t know path length– Requires estimate of total path length
• Estimate is a heuristic H(n)
The A* Algorithm
• Estimated path length for path including n = length of path from start to n + estimated distance from n to goal
Known, since have explored graph from start to n
Requires heuristic H(n) to create an estimate
Start
Goal
A* Algorithm Example
100
5075
60
90
7080
YNG
Goal: find shortest path from YNG to COL
CLE
COL
CAN
WHE
PIT
70AKR
30
A* Algorithm Example
State Best Path Path Cost from Start
Heuristic H (n)
Total Estimated Path Cost
YNG
CLE 100
AKR 90
CAN 75
PIT 150
WHE 90
COL 0
Heuristic H (n) = “as crow flies” distance to COL
A* Algorithm Example
State Best Path Path Cost from Start
Heuristic H (n)
Total Estimated Path Cost
YNG Tree YNG
CLE Fringe YNG CLE 90 100 190
AKR Fringe YNG AKR 70 90 160
CAN Unknown 75
PIT Fringe YNG PIT 70 150 220
WHE Fringe YNG WHE 80 90 170
COL Unknown 0
Best path so far
A* Algorithm Example
State Best Path Path Cost from Start
Heuristic H (n)
Total Estimated Path Cost
YNG Tree YNG
CLE Fringe YNG CLE 90 100 190
AKR Tree YNG AKR 70 90 160
CAN Fringe YNG AKR CAN 100 75 175
PIT Fringe YNG PIT 70 150 220
WHE Fringe YNG WHE 80 90 170
COL Unknown 0
Add CAN to fringe
A* Algorithm Example
State Best Path Path Cost from Start
Heuristic H (n)
Total Estimated Path Cost
YNG Tree YNG
CLE Fringe YNG CLE 90 100 190
AKR Tree YNG AKR 70 90 160
CAN Fringe YNG AKR CAN 100 75 175
PIT Fringe YNG PIT 70 150 220
WHE Fringe YNG WHE 80 90 170
COL Unknown 0
Best path so far
A* Algorithm Example
State Best Path Path Cost from Start
Heuristic H (n)
Total Estimated Path Cost
YNG Tree YNG
CLE Fringe YNG CLE 90 100 190
AKR Tree YNG AKR 70 90 160
CAN Fringe YNG AKR CAN 100 75 175
PIT Fringe YNG PIT 70 150 220
WHE Tree YNG WHE 80 90 170
COL Fringe YNG WHE COL 180 0 180
Add COL to fringe
A* Algorithm Example
State Best Path Path Cost from Start
Heuristic H (n)
Total Estimated Path Cost
YNG Tree YNG
CLE Fringe YNG CLE 90 100 190
AKR Tree YNG AKR 70 90 160
CAN Fringe YNG AKR CAN 100 75 175
PIT Fringe YNG PIT 70 150 220
WHE Tree YNG WHE 80 90 170
COL Fringe YNG WHE COL 180 0 180
Best path so far
A* Algorithm Example
State Best Path Path Cost from Start
Heuristic H (n)
Total Estimated Path Cost
YNG Tree YNG
CLE Fringe YNG CLE 90 100 190
AKR Tree YNG AKR 70 90 160
CAN Tree YNG AKR CAN 100 75 175
PIT Fringe YNG PIT 70 150 220
WHE Tree YNG WHE 80 90 170
COL Fringe YNG AKR CAN COL
175 0 175
Like Dijkstra, reevaluate all other nodes in fringe to see if new tree node gives shorter path
A* Algorithm Example
State Best Path Path Cost from Start
Heuristic H (n)
Total Estimated Path Cost
YNG Tree YNG
CLE Fringe YNG CLE 90 100 190
AKR Tree YNG AKR 70 90 160
CAN Tree YNG AKR CAN 100 75 175
PIT Fringe YNG PIT 70 150 220
WHE Tree YNG WHE 80 90 170
COL Tree YNG AKR CAN COL 175 0 175
When goal node added to tree, algorithm doneBest path
A* Heuristics
• A* heuristic admissible if always underestimates distance to goal
H (n) ≤ actual distance to goal
– True for example– Usually true for “as crow flies” estimates
CLE ARK CAN PIT WHE
Estimated 100 90 75 150 90
Actual 115 105 75 150 100
A* Heuristics
A* Heuristics
• Heuristic may not be valid if “short cuts” in level
– Estimated path using C has length 12.5– Actual path has length 6 (using transporter pad)– Path using A has estimated length 7– Goal added to tree before C ever explored
• However, non-optimal behavior may be more plausible– Character may not know about transporter pad!
Start
GoalA
CB
D
A B C D
Estimated 4 3 6.5 4
Actual 4 11 0 4
“Transporter pad” between C and goal
A* Heuristics
• Algorithm can be inefficient if heuristic severely underestimates distance
– PIT and other obviously implausible nodes (CHI, PHI, NY, TOR, etc.) not explored
• If all estimated distances low (1 for example) then all will be explored
– Adjacent nodes (YNG PIT PHIL) explored before paths with closer but more nodes (YNG AKR CAN COL)
CLE ARK CAN WHE PIT CHI PHI TOR NY …
Estimated 100 90 75 90 150 240 400 210 420
A* Heuristics
• Heuristic underestimates can happen in levels with costly terrain– Estimated distance = 4 squares– Actual distance = 2 + 5 + 5 = 9 due
to desert, mountains
• No perfect solution• Potential hierarchical solution:
– Create high-level map with terrain type of large general areas
– Determine which areas direct path to goal would cross
– Factor in terrain costs
Start
GoalEstimated distance = width of forest area * cost of forest +width of mountain area * cost of mountain