Pathfinding Peril SPA 2016

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Pathﬁnding Peril Chris Parsons

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Huh? What is it all good for... Absolutely nothing?

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Games

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Workshop Plan Introduction Graph Theory Pathﬁnding Algorithms The Maze Contest

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Graph Theory A B D C E F G

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What is it all good for... Graphs can model internet networks, data organisation, computation ﬂow, neural networks, the web, wikis, molecular structures, linguistics, social networks, biological habitats, transport networks, etc.

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Shortest route from B to C A B D C E F G

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“Brute force” A B D C E F G

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“Brute force” A B D C E F G

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“Brute force” A B D C E F G

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Weighted graphs A B D E F G 3 3 17 2 2 3 22 1 C

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B - E - C will cost 19 A B D E F G 3 3 17 2 2 3 22 1 C

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...but B-A-G-D-C is only 9 A B D E F G 3 3 17 2 2 3 22 1 C

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Dijkstra's Algorithm Published 1959 Uses an ordered set of open nodes, tracking minimum cost to each

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Shortest route from B to C A B D E F G 3 3 17 2 2 3 22 1 C

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Visit B Node Cost B 0 A (from B) 3 F (from B) 3 E (from B) 17 A B D E F G 3 3 17 2 2 3 22 1 C

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Visit A Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 E (from B) 17 A B D E F G 3 3 17 2 2 3 22 1 C

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Visit F Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 E (from B) 17 (25 is bigger) A B D E F G 3 3 17 2 2 3 22 1 C

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Visit G Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 D (from G) 8 E (from B) 17 A B D E F G 3 3 17 2 2 3 22 1 C

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Visit D Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 D (from G) 8 C (from D) 9 E (from B) 17 A B D E F G 3 3 17 2 2 3 22 1 C

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Visit C - we’re done! Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 D (from G) 8 C (from D) 9 E (from B) 17 A B D E F G 3 3 17 2 2 3 22 1 C

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Updated weights A B D E F G 3 3 17 2 2 3 3 1 C

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Visit B Node Cost B 0 A (from B) 3 F (from B) 3 E (from B) 17 A B D E F G 3 3 17 2 2 3 3 1 C

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Visit A Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 E (from B) 17 A B D E F G 3 3 17 2 2 3 3 1 C

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Visit F Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 E (from F) 6 A B D E F G 3 3 17 2 2 3 3 1 C

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Visit G Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 E (from F) 6 D (from G) 8 A B D E F G 3 3 17 2 2 3 3 1 C

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Visit E Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 E (from F) 6 C (from E) 8 D (from G) 8 A B D E F G 3 3 17 2 2 3 3 1 C

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Visit C - we’re done! Node Cost B 0 A (from B) 3 F (from B) 3 G (from A) 6 E (from F) 6 C (from E) 8 D (from G) 8 A B D E F G 3 3 17 2 2 3 3 1 C

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A* algorithm Published 1968 Variation on Dijkstra, using a heuristic to cut down on nodes to visit

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Distance Heuristic A 23 B 10 D 12 E 13 F 18 G 20 3 3 17 2 2 3 22 1 C 0

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Visit B Node Cost Order B 0 10 (0+10) F 3 21 (3+18) A 3 26 (3+23) E 17 30 (17+13) A 23 B 10 D 12 E 13 F 18 G 20 3 3 17 2 2 3 22 1 C 0

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Visit F Node Cost Order B 0 10 (0+10) F 3 21 (3+18) A 3 26 (3+23) E 17 30 (17+13) A 23 B 10 D 12 E 13 F 18 G 20 3 3 17 2 2 3 22 1 C 0

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Visit A Node Cost Order B 0 10 (0+10) F 3 21 (3+18) A 3 26 (3+23) G 6 26 (6+20) E 17 30 (17+13) A 23 B 10 D 12 E 13 F 18 G 20 3 3 17 2 2 3 22 1 C 0

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Visit G Node Cost Order B 0 10 (0+10) F 3 21 (3+18) A 3 26 (3+23) G 6 26 (6+20) D 8 20 (8+12) E 17 30 (17+13) A 23 B 10 D 12 E 13 F 18 G 20 3 3 17 2 2 3 22 1 C 0

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Visit D Node Cost Order B 0 10 (0+10) F 3 21 (3+18) A 3 26 (3+23) G 6 26 (6+20) D 8 20 (8+12) C 9 9 (9 +0) E 17 30 (17+13) A 23 B 10 D 12 E 13 F 18 G 20 3 3 17 2 2 3 22 1 C 0

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Visit C - we’re done! Node Cost Order B 0 10 (0+10) F 3 21 (3+18) A 3 26 (3+23) G 6 26 (6+20) D 8 20 (8+12) C 9 9 (9 +0) E 17 30 (17+13) A 23 B 10 D 12 E 13 F 18 G 20 3 3 17 2 2 3 22 1 C 0

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Non-admissible Heuristic A 23 B 10 D 12 E 13 F 18 G 20 3 3 17 2 2 3 22 1 C 0

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Different Heuristics Dijkstra no heuristic A* exact distance (admissible) http://en.wikipedia.org/wiki/A*_search_algorithm http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm A* over-egging distance (non-admissible)

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Summary Brute force approach: O(n²) (ouch!) Dijkstra: guarantees best path but still can be slow A*: depending on Heuristic can trade speed for best path Admissible Heuristics (never overestimating cost) is the sweet spot for A*

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Your turn Wiﬁ: BCSGuest Password: xxxx Visit http://10.30.22.125:3000/rules

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Slides + tournament online https://speakerdeck.com/chrismdp/pathﬁnding-peril- spa-2016 github.com/chrismdp/pathﬁnding-peril

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Sol Trader I’m a web programmer moved back into Games http://soltrader.net

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Thanks! Chris Parsons @chrismdp - http://pa.rsons.org Sol Trader @soltdr - http://soltrader.net