computer notes - Data Structures - 31
- 1. Class No.31 Data Structures http://ecomputernotes.com
- 2. Timing with Optimization Theorem: A sequence of m union and find operations, n of which are find operations, can be performed on a disjoint-set forest with union by rank (weight or height) and path compression in worst case time proportional to ( m ( n )) ( n )is the inverse Ackermann’s function which grows extremely slowly. For all practical puposes, ( n ) 4. Union-find is essentially proportional to m for a sequence of m operations, linear in m . http://ecomputernotes.com
- 3. Image Segmentation Inclusion criteria for pixels use pixel intensity, threshold of intensity, threshold for difference in intensity of neighbors, texture (ie. a pattern of pixel intensities) http://ecomputernotes.com
- 4. Image Segmentation 0 1 2 3 4 0 0 0 0 4 4 1 2 0 4 4 0 2 4 2 2 4 4 3 4 4 0 4 4 4 0 2 2 4 0 http://ecomputernotes.com
- 5. Image Segmentation 0 1 2 3 4 0 0 0 0 4 4 1 2 0 4 4 0 2 4 2 2 4 4 3 4 4 0 4 4 4 0 2 2 4 0 0 1 2 3 4 0 0 0 0 1 1 1 0 0 1 1 0 2 1 0 0 1 1 3 1 1 0 1 1 4 0 0 0 1 0 Threshold=4 http://ecomputernotes.com
- 6. Image Segmentation 0 1 2 3 4 0 0 0 0 4 4 1 2 0 4 4 0 2 4 2 2 4 4 3 4 4 0 4 4 4 0 2 2 4 0 0 1 2 3 4 0 0 0 0 1 1 1 1 0 1 1 0 2 1 1 1 1 1 3 1 1 0 1 1 4 0 1 1 1 0 Threshold=2 http://ecomputernotes.com
- 8. Maze Generation A random maze generator can use union-find. Consider a 5x5 maze: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 http://ecomputernotes.com
- 9. Maze Generator Initially, 25 cells, each isolated by walls from the others. This corresponds to an equivalence relation -- two cells are equivalent if they can be reached from each other (walls been removed so there is a path from one to the other). http://ecomputernotes.com
- 10. Maze Generator To start, choose an entrance and an exit. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 http://ecomputernotes.com
- 11. Maze Generator Randomly remove walls until the entrance and exit cells are in the same set. Removing a wall is the same as doing a union operation. Do not remove a randomly chosen wall if the cells it separates are already in the same set. http://ecomputernotes.com
- 12. MakeMaze MakeMaze(int size) { entrance = 0; exit = size-1; while (find(entrance) != find(exit)) { cell1 = randomly chosen cell cell2 = randomly chosen adjacent cell if (find(cell1) != find(cell2) { knock down wall between cells union(cell1, cell2) } } http://ecomputernotes.com
- 13. Maze Generator Cell 11, right wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 http://ecomputernotes.com
- 14. Maze Generator Cell 11, right wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S_11 = { 11,12} http://ecomputernotes.com
- 15. Maze Generator Cell 6, bottom wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S_11 = { 11,12} http://ecomputernotes.com
- 16. Maze Generator Cell 6, bottom wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S_11 = { 11,12, 6} http://ecomputernotes.com
- 17. Maze Generator Cell 8, top wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S_11 = { 11,12, 6} http://ecomputernotes.com
- 18. Maze Generator Cell 8, top wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S_11 = { 11,12, 6} S_8 = { 8,3} http://ecomputernotes.com
- 19. Maze Generator Cell 14, top wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S_11 = { 11,12, 6} S_8 = { 8,3} http://ecomputernotes.com
- 20. Maze Generator Cell 14, top wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S_11 = { 11,12, 6} S_8 = { 8,3} S_14 = { 14,9} http://ecomputernotes.com
- 21. Maze Generator Cell 0, bottom wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S_11 = { 11,12, 6} S_8 = { 8,3} S_14 = { 14,9} http://ecomputernotes.com
- 22. Maze Generator Cell 0, bottom wall chosen randomly 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S_11 = { 11,12, 6} S_8 = { 8,3} S_14 = { 14,9} S_0 = { 0,5} http://ecomputernotes.com