Growing and Shrinking Polygons for Random Testing of Computational Geometry Algorithms

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1 Growing and Shrinking Polygons for Random Testing of Computational Geometry Algorithms Experience Report Ilya Sergey ICFP 2016 September 20th, 2016

2 Polygons

3 Polygons

4 Polygons

5 Polygons

6 Polygons are Trees

7 Polygons are Trees We can grow them;

8 Polygons are Trees We can grow them; We can also trim them;

9 Polygons are Trees We can grow them; We can also trim them; We can use QuickCheck to test their properties.

13 Would you like to run a scenario project this year?

14 Would you like to run a scenario project this year? Sure, why not.

15 Scenario Project Requirements

16 Scenario Project Requirements For 2nd year undergrads, team work,

17 Scenario Project Requirements For 2nd year undergrads, team work, One week-long,

18 Scenario Project Requirements For 2nd year undergrads, team work, One week-long, Should involve math and programming,

19 Scenario Project Requirements For 2nd year undergrads, team work, One week-long, Should involve math and programming, Challenging for students, but easy to assess,

20 Scenario Project Requirements For 2nd year undergrads, team work, One week-long, Should involve math and programming, Challenging for students, but easy to assess, To be delivered in about a month.

21 Scenario Project Requirements For 2nd year undergrads, team work, One week-long, Should involve math and programming, Challenging for students, but easy to assess, To be delivered in about a month.

26 How many guards do we really need?

27 How many guards do we really need? The answer depends on the shape of the gallery.

28 How many guards do we really need? The answer depends on the shape of the gallery.

29 How many guards do we really need? The answer depends on the shape of the gallery.

30 How many guards do we really need?

31 How many guards do we really need?

32 How many guards do we really need?

33 How many guards do we really need?

34 Art Gallery Problem For a given gallery (polygon), find the minimal set of guards positions, so together the guards can see the whole interior.

35 Art Gallery Problem For a given gallery (polygon), find the minimal set of guards positions, so together the guards can see the whole interior. NP-hard

36 Project: Art Gallery Competition Find the best solutions for a collection of large polygons.

37 Project: Art Gallery Competition Find the best solutions for a collection of large polygons. 58 vertices 5 guards

38 Organizers TODO

39 Organizers TODO Problem generator; Polygons with different features (convex, rectangular, etc.)

40 Organizers TODO Problem generator; Polygons with different features (convex, rectangular, etc.) Solution checker with online feedback geometric machinery (triangulation, visibility, ) web-server

41 Organizers TODO Problem generator; Polygons with different features (convex, rectangular, etc.) Solution checker with online feedback geometric machinery (triangulation, visibility, ) web-server Make sure that it all works.

42 Organizers TODO Problem generator; Polygons with different features (convex, rectangular, etc.) Solution checker with online feedback geometric machinery (triangulation, visibility, ) web-server Make sure that it all works.

) Solution checker with online feedback geometric

43 Organizers TODO Problem generator; Polygons with different features (convex, rectangular, etc.) Solution checker with online feedback geometric machinery (triangulation, visibility, ) web-server Make sure that it all works.

44 Organizers TODO Problem generator; Polygons with different features (convex, rectangular, etc.) Solution checker with online feedback geometric machinery (triangulation, visibility, ) web-server Make sure that it all works.

45 Growing polygons

46 Growing polygons Primitive polygons with specific features

47 Growing polygons Primitive polygons with specific features

48 Seed Growing polygons

49 Seed Growing polygons

50 Growing polygons

51 Growing polygons

52 Growing polygons

53 Growing polygons

54 Growing polygons

55 Growing polygons

56 Growing polygons

57 Growing polygons

58 Growing polygons

59 Growing polygons

60 Growing polygons

61 Algorithm for growing polygons 1. Pick a primitive polygon; 2. Locate a segment on a host polygon edge; 3. Scale, rotate and attach the primitive; check for self-intersections 4. Repeat

62 Abstract polygon generator trait PolygonGenerator extends GeneratorPrimitives { val seeds : List[Polygon] val primitives : List[(Int) => Polygon] val locate : Double => Option[(Double, Double)] val seedfreqs val primfreqs : List[Int] : List[Int] val generations : Int }...

63 Abstract polygon generator trait PolygonGenerator extends GeneratorPrimitives { val seeds : List[Polygon] val primitives : List[(Int) => Polygon] val locate : Double => Option[(Double, Double)] val seedfreqs val primfreqs : List[Int] : List[Int] val generations : Int }...

64 Abstract polygon generator trait PolygonGenerator extends GeneratorPrimitives { val seeds : List[Polygon] val primitives : List[(Int) => Polygon] val locate : Double => Option[(Double, Double)] val seedfreqs val primfreqs : List[Int] : List[Int] val generations : Int }...

65 Abstract polygon generator trait PolygonGenerator extends GeneratorPrimitives { val seeds : List[Polygon] val primitives : List[(Int) => Polygon] val locate : Double => Option[(Double, Double)] val seedfreqs val primfreqs : List[Int] : List[Int] val generations : Int }...

66 Generating contest problems Rectilinear

67 Generating contest problems Quasi-convex

68 Generating contest problems Crazy

69 Organizers TODO Problem generator; Polygons with different features (convex, rectangular, etc.) Solution checker with online feedback geometric machinery (triangulation, visibility, ) web-server Make sure that it all works.

70 Organizers TODO Problem generator; Polygons with different features (convex, rectangular, etc.) Solution checker with online feedback geometric machinery (triangulation, visibility, ) web-server Make sure that it all works.

71 Testing with ScalaCheck

72 Testing with ScalaCheck Triangulation of a polygon of size N: centre of each triangle lies within a polygon; triangulation generates N 2 (possibly degenerate) triangles;

73 Testing with ScalaCheck Triangulation of a polygon of size N: centre of each triangle lies within a polygon; triangulation generates N 2 (possibly degenerate) triangles; Joe-Simpson algorithm for visibility polygons (VPs): a vertex of a VP is also within the original polygon; every VP s edge is within the original polygon; a random point within a VP is indeed visible from its origin;

74 Testing with ScalaCheck Triangulation of a polygon of size N: centre of each triangle lies within a polygon; triangulation generates N 2 (possibly degenerate) triangles; Joe-Simpson algorithm for visibility polygons (VPs): a vertex of a VP is also within the original polygon; every VP s edge is within the original polygon; a random point within a VP is indeed visible from its origin; Solution visibility checker: a refutation (if it exists) is within the original polygon; a refutation for a set of guards is not within any of their VPs;

75 Testing with ScalaCheck Triangulation of a polygon of size N: centre of each triangle lies within a polygon; triangulation generates N 2 (possibly degenerate) triangles; Joe-Simpson algorithm for visibility polygons (VPs): a vertex of a VP is also within the original polygon; every VP s edge is within the original polygon; a random point within a VP is indeed visible from its origin; Solution visibility checker: a refutation (if it exists) is within the original polygon; a refutation for a set of guards is not within any of their VPs; Basic algorithm for solving AGP by Fisk: delivers a solution of size within the boundary N/3 ; the solution checker finds no refutations for its result.

76 How well did that work?

77 How well did that work? Multiple bugs (>20) due to inaccurate treatment of floating points;

78 How well did that work? Multiple bugs (>20) due to inaccurate treatment of floating points; Several bugs (~10) due to misreading of the textbook algorithms or simplifications in their descriptions;

79 How well did that work? Multiple bugs (>20) due to inaccurate treatment of floating points; Several bugs (~10) due to misreading of the textbook algorithms or simplifications in their descriptions; Two bugs in the Joe-Simpson algorithm itself;

80 How well did that work? Multiple bugs (>20) due to inaccurate treatment of floating points; Several bugs (~10) due to misreading of the textbook algorithms or simplifications in their descriptions; Two bugs in the Joe-Simpson algorithm itself; Bonus: an undocumented behavior in the state-of-the-art CGAL library (written in C++ using exact arithmetics).

81 How well did that work? Multiple bugs (>20) due to inaccurate treatment of floating points; Several bugs (~10) due to misreading of the textbook algorithms or simplifications in their descriptions; Two bugs in the Joe-Simpson algorithm itself; Bonus: an undocumented behavior in the state-of-the-art CGAL library (written in C++ using exact arithmetics).

82 Testing basic visibility algorithm Main component for checking arbitrary solutions; Original description (1986) has a number of simplifications and is given in pseudocode.

83 Testing basic visibility algorithm import RandomRectilinearPolygonGenerator._ property("all visibility polygons lie within the original polygon") = forall { (p : Polygon) => val guards = p.vertices val vps = guards.map(visibilitypolygon(p, _)) "Every edge of a visibility polygon is within ${p}" : { val edges = for (vp <- vps; e <- vp.edges) yield e edges.forall(p.containssegment) } }

84 Testing basic visibility algorithm import RandomRectilinearPolygonGenerator._ property("all visibility polygons lie within the original polygon") = forall { (p : Polygon) => val guards = p.vertices val vps = guards.map(visibilitypolygon(p, _)) "Every edge of a visibility polygon is within ${p}" : { val edges = for (vp <- vps; e <- vp.edges) yield e edges.forall(p.containssegment) } }

85 Testing basic visibility algorithm import RandomRectilinearPolygonGenerator._ property("all visibility polygons lie within the original polygon") = forall { (p : Polygon) => val guards = p.vertices val vps = guards.map(visibilitypolygon(p, _)) "Every edge of a visibility polygon is within ${p}" : { val edges = for (vp <- vps; e <- vp.edges) yield e edges.forall(p.containssegment) } }

86 Testing basic visibility algorithm import RandomRectilinearPolygonGenerator._ property("all visibility polygons lie within the original polygon") = forall { (p : Polygon) => val guards = p.vertices val vps = guards.map(visibilitypolygon(p, _)) "Every edge of a visibility polygon is within ${p}" : { val edges = for (vp <- vps; e <- vp.edges) yield e edges.forall(p.containssegment) } }

87 Bug in Joe-Simpson algorithm Randomly generated, 260 vertices, guards in every node

88 Bug in Joe-Simpson algorithm Randomly generated, 260 vertices, guards in every node??!

89 Shrinking Polygons

90 Trimming Shrinking Polygons

91 Reconstructing polygon generation

92 Reconstructing polygon generation

93 Reconstructing polygon generation Attachment tree

94 Trimming polygons

95 Trimming polygons

96 Shrinking strategy Build the attachment tree while constructing a random polygon; Construct all its subtrees; Render the corresponding trimmed polygons.

97 Bug in Joe-Simpson algorithm Randomly generated, 260 vertices, guards in every node

98 Bug in Joe-Simpson algorithm Randomly generated, 260 vertices, guards in every node

99 Bug in Joe-Simpson algorithm After trimming, 20 vertices

100 Bug in Joe-Simpson algorithm Removed irrelevant guards

101 Bug in Joe-Simpson algorithm

102 Bug in Joe-Simpson algorithm

103 Bug in Joe-Simpson algorithm... intersectwithwindow(v(i), v(i + 1), s.top, windowend) match { } case Some(p) => s.push(p) advance(v, s, i) case _ => scan(v, s, i, windowend, ccw)...

104 Bug in Joe-Simpson algorithm... intersectwithwindow(v(i), v(i + 1), s.top, windowend) match { } case Some(p) if!(windowend.isdefined && p =~= windowend) => s.push(p) advance(v, s, i) case _ => scan(v, s, i, windowend, ccw)...

105 Implementation effort Geometric primitives and procedures: 1450 LOC Server infrastructure: 1500 LOC Testing framework: 350 LOC

106 Running the Art Gallery Competition

107 Running the Art Gallery Competition 94 participants, 24 teams, 2360 submissions in five days

108 Running the Art Gallery Competition 94 participants, 24 teams, 2360 submissions in five days Server running 24/5 for different teams, solution processed concurrently no crashes during the week one non-critical bug in server logic (fixed without restarting)

109 Running the Art Gallery Competition 94 participants, 24 teams, 2360 submissions in five days Server running 24/5 for different teams, solution processed concurrently no crashes during the week one non-critical bug in server logic (fixed without restarting) Most of the solutions: textbook algorithm with optimizations;

110 Running the Art Gallery Competition 94 participants, 24 teams, 2360 submissions in five days Server running 24/5 for different teams, solution processed concurrently no crashes during the week one non-critical bug in server logic (fixed without restarting) Most of the solutions: textbook algorithm with optimizations; The winning team implemented the state-of-the art algorithm (2014) using linear programming;

111 Fri The Wed Tue Mon

112 To take away

113 To take away FP works great for rapid development of non-trivial and robust geometric applications;

114 To take away FP works great for rapid development of non-trivial and robust geometric applications; QuickCheck-ing CG with polygons is feasible and efficient in practice;

115 To take away FP works great for rapid development of non-trivial and robust geometric applications; QuickCheck-ing CG with polygons is feasible and efficient in practice; Polygons can be grown and trimmed, just like trees.

116 To take away FP works great for rapid development of non-trivial and robust geometric applications; QuickCheck-ing CG with polygons is feasible and efficient in practice; Polygons can be grown and trimmed, just like trees. Thanks!

Scenario Week 4: Art Gallery Competition

Scenario Week 4: Art Gallery Competition Organisers' Report scenario@cs.ucl.ac.uk 26 February 2016 The Week 94 participants 24 teams 1892 468 submissions for Part 2 One submissions for Part 1 subtle server