Rectangle Visibility and Elusive K23

A talk detailing my attempts to find the 3D Rectangle Visibility Diagram for the K23 graph using Genetic Algorithms

Transcript

1. Rectangle Visibility and the Elusive K23
3. Which rectangles can “see” each other? A B D C A B C D K4 !
4. Given a Rectangle Diagram, can we come up with a Visibility Graph for it? A E C B D A B D C E This is easy! For a set of rectangles R: For every pair of rectangles (r1 ,r2 ) in R: see if there is a horizontal or vertical line from r1 to r2 , unobstructed by any other rectangle in R. K5 !
5. Easy! Hard! Even deciding on “is there one?” is NP-Complete!
6. Let’s add another dimension to this
7. What if we drew our rectangles not side-by-side on a plane, but one on top of another in 3D? R B R B
8. R G O B Which pairs of rectangles are visible to each other?
9. One rectangle per plane Side View
10. R G O B R B O G
11. Who Cares? 2D Rectangle Visibility Graphs have applications in circuit board layout 3D R.V.G.’s have applications in visualizing software dependency diagrams There are lots of unanswered questions, isn’t that enough?
13. Hard! Determining if a graph has a 3D Rectangle Visibility Representation is just as hard!
14. Special Graph: Sk! 1 2 3 k 4 5 … 2 3 4 5 … k! 1
15. Special Graph: Cn! 1 2 3 n 1 2 3 … n-1! n! In fact, any planar graph has a representation!
16. Special Graph: Km,n! 1 2 3 m 1 n 2 … … 1 2 3 … m! 1 2 … n!
17. What is the largest number of rectangles you can arrange and still have them all be visible to each other?
18. In other words…. What is the largest n such that Kn has a 3D Rectangle Visibility Representation? K4 ? R G O B K5 ? ✔ ✔ K6 ? ✔ …?
19. First, how high can’t we go? No complete graph Kn has a 3D Rectangle Visibility Diagram representation when n ≥ 56! - “A Visibility Representation for Graphs in Three Dimensions”, Bose, et. al. To K55 maybe! And no further! This upper bound is not sharp Doesn’t mean K55 does have one
20. Upper bound of 55 shown via proof by contradiction (trust me) Lower bounds can be shown via construction If we can construct a diagram for some n, we have one for n-1 (just remove the top)
21. K22 !
22. ✔ 22 and down 56 and up ✗ K23 K24 K25 K26 K27 K28 K29 K30 K31 K33 KXX KXX KXX KXX KXX KXX KXX KXX KXX KXX KXX KXX KXX KXX KXX KXX KXX KXX K XX KXX K54 K55
23. How will we find a 3D Rectangle Visibility Representation for K23 ? We can’t just algorithmically convert complete graphs to rectangle diagrams (it’s NP-Complete) What if we could somehow enumerate every possible arrangement of 23 rectangles? Then we could search through them, checking if any of them is fully connected. So we have an enormous search space that we want to search quickly…
24. This looks like a job for Metaheuristics!
25. A crash course in metaheuristics
26. Metaheuristics can find solutions to problems that deterministic optimization algorithms can’t (efficiently). 1.  Examine candidate solution(s) 2.  Evaluate how well solution(s) solve the problem 3.  Modify solution(s) 4.  Repeat Works for problems whose sub-optimal solutions can be compared!
27. Psst… this graph is a perfect use case for a 3D rectangle visibility diagram
28. Genetic Algorithms 1.  Generate P random solutions to your problem (initialization) 2.  Score each solution on well it solves the problem (the fitness) 3.  Take the best N solutions, combine them (crossover) 4.  Slightly modify the combined solutions to get P more (mutation) 5.  Go to 2. Science!
29. Example: Infinite Monkey Theorem Goal: tobeornottobe 2613 combinations! (That’s 2,481,152,873,203,736,576!)
30. Example: Infinite Monkey Theorem Goal: tobeornottobe 2613 combinations! (That’s 2,481,152,873,203,736,576!) vacfnlpingsde sojwsmuelwapy lrkyycpicckxs sfadpfpkxujgm knqruazzbibry ssqsyawkhgesd wlmkglwqwdffl vvbpcmgdjkeqt sdchkmlunsxzf bankorwteyzqt 1 1 0 0 0 0 0 1 0 2
31. Example: Infinite Monkey Theorem Goal: tobeornottobe 2613 combinations! (That’s 2,481,152,873,203,736,576!) bankorwteyzqt vacfnlpingsde bankorwtegsde vacfnlpinyzqt vacfilpinyzqt vacfnlpinyzqh vacfnlpjnyzqt bankorwtfgsde vacfnlpinyzqd bankorwtegsde bankorwteosde badkorwtegsde vacfnlpinpzqt bankorwtegrde 0 0 0 3 0 3 3 3 0 3 tobeorsottnve 10 per generation 50 generations ~1 second
32. BoxCar2D Demo
33. Genetic Algorithms Gotchas May not yield optimal result Local maxima
34. Representing the problem genetically
35. How many possible arrangements of 23 rectangles are there? (0,0)! ∞ ∞ Tons! How can we reduce?
36. How many possible arrangements of 23 rectangles are there? 5? 543? There are only 23 relevant unique positions on an axis on each side
37. How many possible arrangements of 23 rectangles are there? 0! 46! 46!
38. How many possible arrangements of 23 rectangles are there? For any fully connected visibility diagram, there must be a region where all rectangles overlap Let’s call this (0,0)
39. How many possible arrangements of 23 rectangles are there? Each rectangle is (E, N, W, S)! (1, 3, 3, 1)! (3, 2, 1, 1)! (1, 1, 2, 3)! This ensures our rectangles always overlap!
40. How many possible arrangements of 23 rectangles are there? Our rectangle diagram’s “gene”: { 23: ( 2, 20, 14, 7) 22: (15, 1, 9, 1) 21: (13, 12, 17, 11) 20: (10, 18, 13, 10) 19: ( 9, 10, 20, 9) 18: ( 7, 13, 14, 8) 17: ( 5, 20, 16, 7) 16: (19, 2, 11, 5) 15: (17, 3, 10, 4) 14: (18, 6, 6, 3) 13: ( 6, 14, 19, 1) 12: ( 4, 19, 2, 16) 11: ( 3, 15, 3, 15) 10: (14, 4, 1, 20) 9: ( 2, 16, 4, 12) 8: ( 1, 17, 5, 14) 7: (12, 1, 7, 18) 6: (11, 5, 8, 17) 5: ( 8, 7, 9, 19) 4: (20, 8, 12, 2) 3: (15, 9, 18, 6) 2: (16, 11, 15, 13) 1: (22, 22, 22, 22) } 4 * 23 = 92 possible slots 23 possible values per slot = 2392 possible diagrams 190,090,719,111,086,999,142,918,580,674,213, 624,851,393,339,907,036,543,758,193,303,444, 850,234,108,744,968,124,140,618,335,905,659, 363,676,581,401,527,869,136,497,121!
41. Initialization? 23: ( ?, ?, ?, ?) 22: ( ?, ?, ?, ?) 21: ( ?, ?, ?, ?) 20: ( ?, ?, ?, ?) 19: ( ?, ?, ?, ?) 18: ( ?, ?, ?, ?) 17: ( ?, ?, ?, ?) 16: ( ?, ?, ?, ?) 15: ( ?, ?, ?, ?) 14: ( ?, ?, ?, ?) 13: ( ?, ?, ?, ?) 12: ( ?, ?, ?, ?) 11: ( ?, ?, ?, ?) 10: ( ?, ?, ?, ?) 9: ( ?, ?, ?, ?) 8: ( ?, ?, ?, ?) 7: ( ?, ?, ?, ?) 6: ( ?, ?, ?, ?) 5: ( ?, ?, ?, ?) 4: ( ?, ?, ?, ?) 3: ( ?, ?, ?, ?) 2: ( ?, ?, ?, ?) 1: ( ?, ?, ?, ?) Pick a random number from 1 to 23 for each
42. Fitness function? We can quickly determine if any two rectangles can “see” each other (in polynomial time) So we can simply go through every pair of rectangles and check if they can see each other. There are only n(n-1) / 2 possible pairs to check, in other words, the number of edges in Kn! Fitness score = the number of pairs of rectangles that can see each other (the maximum fitness for n rectangles is n(n-1) / 2 )
43. Fitness function? Since every rectangle shares the origin point in the center, two rectangles can see each other if any one of their corners are both free (not blocked by other rectangles) So for any two rectangles A and B, one of the free corner conditions must hold for every rectangle r between A and B: •  FCne (A,B) (northeast free): (Er < min{Ea ,Eb } or Nr < min{Na ,Nb })! •  FCnw (A,B) (northwest free): (Nr < min{Na ,Nb } or Wr < min{Wa ,Wb })! •  FCsw (A,B) (southwest free): (Wr < min{Wa ,Wb } or Sr < min{Sa ,Sb })! •  FCse (A,B) (southeast free): (Sr < min{Sa ,Sb } or Er < min{Ea ,Eb })!
44. Crossover? 23: ( 2, 20, 14, 7) 22: (15, 1, 9, 1) 21: (13, 12, 17, 11) 20: (10, 18, 13, 10) 19: ( 9, 10, 20, 9) 18: ( 7, 13, 14, 8) 17: ( 5, 20, 16, 7) 16: (19, 2, 11, 5) 15: (17, 3, 10, 4) 14: (18, 6, 6, 3) 13: ( 6, 14, 19, 1) 12: ( 4, 19, 2, 16) 11: ( 3, 15, 3, 15) 10: (14, 4, 1, 20) 9: ( 2, 16, 4, 12) 8: ( 1, 17, 5, 14) 7: (12, 1, 7, 18) 6: (11, 5, 8, 17) 5: ( 8, 7, 9, 19) 4: (20, 8, 12, 2) 3: (15, 9, 18, 6) 2: (16, 11, 15, 13) 1: (22, 22, 22, 22) 23: (23, 23, 23, 23) 22: (16, 20, 18, 19) 21: (23, 6, 15, 18) 20: (17, 19, 16, 13) 19: (21, 3, 13, 14) 18: (15, 22, 3, 17) 17: (11, 18, 21, 5) 16: (19, 4, 22, 3) 15: (12, 21, 1, 23) 14: (10, 3, 17, 12) 13: (18, 7, 9, 6) 12: ( 7, 19, 5, 21) 11: ( 6, 22, 7, 15) 10: ( 5, 23, 8, 23) 9: (23, 12, 11, 2) 8: ( 3, 22, 12, 16) 7: ( 1, 23, 14, 15) 6: (16, 15, 19, 10) 5: (21, 20, 15, 5) 4: (22, 20, 6, 13) 3: (13, 1, 7, 20) 2: (18, 1, 22, 17) 1: (23, 23, 23, 23)
45. Mutation? 23: (23, 23, 23, 23) 22: (16, 20, 18, 19) 21: (23, 6, 15, 18) 20: (17, 19, 16, 13) 19: (21, 3, 13, 14) 18: (15, 22, 3, 17) 17: (11, 18, 21, 5) 16: (19, 4, 22, 3) 15: (12, 21, 1, 23) 14: (10, 3, 17, 12) 13: (18, 7, 9, 6) 12: ( 7, 19, 5, 21) 11: ( 6, 22, 7, 15) 10: ( 5, 23, 8, 23) 9: (23, 12, 11, 2) 8: ( 3, 22, 12, 16) 7: ( 1, 23, 14, 15) 6: (16, 15, 19, 10) 5: (21, 20, 15, 5) 4: (22, 20, 6, 13) 3: (13, 1, 7, 20) 2: (18, 1, 22, 17) 1: (23, 23, 23, 23) 6,
46. Library Code: https://github.com/rodhilton/metaheuristics-engine Language: Java License: LGPL v3.0 App Code: https://github.com/rodhilton/rectangle_visibility Language: Groovy License: Apache v2.0 Rectangle Visibility Searcher
47. Worst. Results. Ever.
48. Uniquely assign position values Try alternative metaheuristics Try regularly “packing” rectangles Tool for experimenting by hand with partial result Is K22 the upper bound?
49. Questions ?