[PPT] - Pathfinding with Trees Samuel Bader <s.bader@unibas.ch> DMI, PowerPoint Presentation

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Pathfinding with Trees

Samuel Bader

<s.bader@unibas.ch>

DMI, University of Basel

13. Sep. 2016

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Pathfinding

We are given a map a start point a goal point

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Pathfinding

We are given a map a start point a goal point

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Pathfinding

We are given a map a start point a goal point

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Dijkstra's Algorithm

A simple solution: Look at all points neighbouring the start point Continue until the goal is found Follow the arrows back to the start

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Dijkstra's Algorithm

A simple solution: Look at all points neighbouring the start point Continue until the goal is found Follow the arrows back to the start

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Dijkstra's Algorithm

A simple solution: Look at all points neighbouring the start point Continue until the goal is found Follow the arrows back to the start

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Dijkstra's Algorithm -> Tree Cache

Save the search tree for a single root When searching: traverse the tree from start to root traverse the tree from goal to root invert the second part concatenate parts at the root

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Dijkstra's Algorithm -> Tree Cache

Save the search tree for a single root When searching: traverse the tree from start to root traverse the tree from goal to root invert the second part concatenate parts at the root

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Dijkstra's Algorithm -> Tree Cache

Save the search tree for a single root When searching: traverse the tree from start to root traverse the tree from goal to root invert the second part concatenate parts at the root

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Dijkstra's Algorithm -> Tree Cache

Save the search tree for a single root When searching: traverse the tree from start to root traverse the tree from goal to root invert the second part concatenate parts at the root

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Dijkstra's Algorithm -> Tree Cache

Save the search tree for a single root When searching: traverse the tree from start to root traverse the tree from goal to root invert the second part concatenate parts at the root

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Tree Cache: Example 2

Paths can have redundant parts The redundant part can be easily removed by looking for the lowest common ancestor (LCA) of the two nodes

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Tree Cache: Example 2

Paths can have redundant parts The redundant part can be easily removed by looking for the lowest common ancestor (LCA) of the two nodes

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LCA: Naive implementation

A B C D F G H E

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LCA: Naive implementation

A B C D F G H E

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LCA: Naive implementation

A B C D F G H E

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LCA: Naive implementation

A B C D F G H E

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LCA: Naive implementation

A B C D F G H E

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LCA: Faster implementation

A B C D F G H E A depth-first traversal and corresponding levels: A B A C D F D G D H D C E C A 1 1 2 3 2 3 2 3 2 1 2 1

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LCA: Faster implementation

A B C D F G H E A depth-first traversal and corresponding levels: A B A C D F D G D H D C E C A 1 1 2 3 2 3 2 3 2 1 2 1

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LCA: Faster implementation

A B C D F G H E A depth-first traversal and corresponding levels: A B A C D F D G D H D C E C A 1 1 2 3 2 3 2 3 2 1 2 1

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Range minimum query implementation

A B A C D F D G D H D C E C A 1 1 2 3 2 3 2 3 2 1 2 1 The range minimum query can be solved with a few lookups by pre-calculating the minimum for all sub-ranges of length : A A A C D D D D D D C C C A 2 A A A C D D D D C C C A 4 A A A C C C C A 8

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Range minimum query implementation

A B A C D F D G D H D C E C A 1 1 2 3 2 3 2 3 2 1 2 1 The range minimum query can be solved with a few lookups by pre-calculating the minimum for all sub-ranges of length 2i: A A A C D D D D D D C C C A 2 A A A C D D D D C C C A 4 A A A C C C C A 8

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Range minimum query implementation

A B A C D F D G D H D C E C A 1 1 2 3 2 3 2 3 2 1 2 1 The range minimum query can be solved with a few lookups by pre-calculating the minimum for all sub-ranges of length 2i: A A A C D D D D D D C C C A 2 A A A C D D D D C C C A 4 A A A C C C C A 8

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Tree Cache: Example 3

Paths can be bad without redundant parts

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Tree Cache improvement: More than one tree

Generate more than one tree, store distance as well as parent When searching: calculate path distance for each tree choose tree with shortest path construct path

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Tree Cache improvement: More than one tree

Generate more than one tree, store distance as well as parent When searching: calculate path distance for each tree choose tree with shortest path construct path

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Tree Cache: Example 4

Tree Cache generated path far too long without redundancy

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Using Multiple Trees

Using multiple trees generates better path with redundancy

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Both improvements combined

Generate more than one tree, store distance as well as parent, calculate LCA information for each tree When searching: look up LCA and calculate path distance for each tree choose tree with shortest path construct path

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Both improvements combined

Generate more than one tree, store distance as well as parent, calculate LCA information for each tree When searching: look up LCA and calculate path distance for each tree choose tree with shortest path construct path

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Experiments

Maps from the Gridbased Path Planning Competition Most maps 512x512 tiles, the largest 768x768

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Results

map size:768x768 time per tree memory per tree Trees 230 ms 10 MB Trees + LCA 400 ms 110 MB

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Results

map size:768x768 time per tree memory per tree Trees 230 ms 10 MB Trees + LCA 400 ms 110 MB

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Results: Root placement

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Conclusions

Tree Cache finds potentially bad paths very fast Path quality can be improved significantly Combination of improvements big amounts of memory Good results using only a few trees

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Conclusions

Tree Cache finds potentially bad paths very fast Path quality can be improved significantly Combination of improvements big amounts of memory Good results using only a few trees

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Conclusions

Tree Cache finds potentially bad paths very fast Path quality can be improved significantly Combination of improvements big amounts of memory Good results using only a few trees

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Conclusions

Tree Cache finds potentially bad paths very fast Path quality can be improved significantly Combination of improvements big amounts of memory Good results using only a few trees

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Future Work

advanced root placement strategies compress tree information adapt to directed graphs

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Future Work

advanced root placement strategies compress tree information adapt to directed graphs

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Future Work

advanced root placement strategies compress tree information adapt to directed graphs

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