Fun with Mazes
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The document discusses different algorithms for generating mazes, including: 1. Recursive backtracking, which performs a random walk while avoiding previously visited cells, backtracking when no moves remain. 2. Dijkstra's algorithm, which measures the shortest distance between a starting point and all other cells to find the shortest path. 3. Aldous-Broder random walk, which randomly visits neighbor cells to generate an unbiased maze, though it can take a long time to complete.
Fun with Mazes
- 1. Fun with Mazes Hendrik Neumann
- 2. 2 https://pragprog.com/titles/jbmaze/ Standing on the shoulders of giants (one in particular)
- 3. 3 What is a Maze?
- 7. 7 For each cell in the grid, randomly carve either north or east. • strong diagonal texture tending toward the north- east corner of the grid. • corridors run the length of the northern row and the eastern column. Binary Tree
- 8. 8 „every cell can reach every other cell by exactly one path” logical and mathematical purity perfect but yet flawed Perfect maze
- 9. 9 opposite of a perfect maze „characterized by few (if any) dead ends, and passages forming loops” multiple different paths Braid maze
- 10. 10 Easy by hand for a 4x4 maze ;-) Computer should do the work: Dijkstra Solve a maze
- 11. 11 Measures the shortest distance between some starting point (which we specify), and every other cell in the maze. • shortest path from a chosen endpoint to our starting point Dijstra‘s Algorithm 0
- 12. 12 Measures the shortest distance between some starting point (which we specify), and every other cell in the maze. • shortest path from a chosen endpoint to our starting point Dijstra‘s Algorithm 0 1
- 13. 13 Measures the shortest distance between some starting point (which we specify), and every other cell in the maze. • shortest path from a chosen endpoint to our starting point Dijstra‘s Algorithm 0 1 2 2
- 14. 14 Measures the shortest distance between some starting point (which we specify), and every other cell in the maze. • shortest path from a chosen endpoint to our starting point Dijstra‘s Algorithm 0 1 2 3 4 3 2 3 8 5 4 3 6 7 6 5 4 5 6 7 6 7 8 7 8
- 15. 15 Measures the shortest distance between some starting point (which we specify), and every other cell in the maze. • shortest path from a chosen endpoint to our starting point • walk backward from end to start. Dijstra‘s Algorithm 0 1 2 3 4 3 2 3 8 5 4 3 6 7 6 5 4 5 6 7 6 7 8 7 8
- 16. 16 Finding the longest path in a maze • find the longest path between two cells • might not be the longest path of the maze! – but just part of it • run Dijstra again in the other direction Dijstra‘s Algorithm
- 17. 17 Finding the longest path in a maze • find the longest path between two cells • might not be the longest path of the maze! – but just part of it • run Dijstra again in the other direction Dijstra‘s Algorithm
- 18. 18 Finding the longest path in a maze • find the longest path between two cells • might not be the longest path of the maze! – but just part of it • run Dijstra again in the other direction Dijstra‘s Algorithm
- 19. 19 Finding the longest path in a maze • find the longest path between two cells • might not be the longest path of the maze! – but just part of it • run Dijstra again in the other direction Dijstra‘s Algorithm
- 20. 20 Color a maze
- 21. 21 coloring emphazises the maze‘s texture Color a maze Binary Tree Sidewinder Recursive Backtracker
- 22. 22 Texture in a maze is a result of the bias of the algorithm. Other signs might be long passages or the amount of dead ends. Not automatically a bad thing! Example: 2x2 grid perfect maze four possibilities: • Binary Tree: A, B 50% • Sidewinder: A, B, C 75% Bias Random walk for unbiased algorithm
- 23. 23 Aldous-Broder Wilson‘s Unbiased mazes Random walk for unbiased algorithm
- 24. 24 Starting at an arbitrary location in the grid, move randomly from cell to cell. If moving to a previously unvisited cell, carve a passage to it. End when all cells have been visited. Aldous-Broder
- 25. 25 1. Pick a cell at random 2. Pick a random neighbour: east – not visited yet, link cells together 3. Pick next random neighbour.. Aldous-Broder
- 26. 26 Aldous-Broder 4. Pick a random neighbour: west – not visited yet: link cells together 5. Pick a random neighbour: north – already visited: don‘t link cells 6. Finish the maze by randomly visiting neighbours until all are visited
- 27. 27 Starts quickly but can take a very long time to finish. Mazes are guaranteed perfectly random. Unbiased – no preference to any particular texture or feature. Aldous-Broder
- 28. 28 Random Walks… • ..can take a long time (Aldous-Broder) • ..can need a lot of memory (Wilson‘s) Bias not automatically a bad thing! Let’s look a algorithms that add constraints to random walks: • Hunt-and-Kill • Recursive Backtracker Adding Constraints to Random Walks
- 29. 29 Starting at an arbitrary location, perform a random walk, avoiding previously visited cells. When no more moves are possible, backtrack to the most recently visited cell and resume the random walk from there. The algorithm ends when it tries to backtrack from the cell where it started. Recursive Backtrack
- 30. 30 • Start at A4. Put current cell on the stack • Choose an unvisited neighbor randomly - Carve a path and push it on the stack. A3 = current cell Recursive Backtracker stack A4 stack A3 A4 stack
- 31. 31 • Continue with the process.. Put the cells on the stack • We‘re stuck - B4 has no unvisited neighbors Recursive Backtracker stack C4 C3 … A3 A4 stack B4 C4 C3 … A3 A4
- 32. 32 • Pop the dead end from the stack C4 is again our current cell • C4 has an unvisited neighbor D4 Recursive Backtracker stack B4 C4 C3 … A3 A4 stack D4 C4 C3 … A3 A4
- 33. 33 • Continue until every cell has been visited • Final cell is always a dead end backtrack to the beginning. Recursive Backtracker stack A2 A1 B1 … A3 A4 stack A2 A1 B1 … A3 A4
- 34. 34 • Stack is empty. Algorithm is finished. Recursive Backtracker stack deapth-first search
- 35. 35 • long, twisty passages with relatively few dead ends • fast - as it visits every cell only twice • needs considerably memory to keep track of visited cells Recursive Backtrack
- 36. 36 ABAP • colored mazes in html view • more 7.4/7.5 magic Github Javascript mazes… and UI • D3.js • openUI5 Future with Mazes