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

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Growing and Shrinking Polygons 
 for Random Testing 


  • f Computational Geometry Algorithms

Ilya Sergey

Experience Report

ICFP 2016 September 20th, 2016

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Polygons

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Polygons

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Polygons

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Polygons

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Polygons are Trees

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Polygons are Trees

  • We can grow them;
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Polygons are Trees

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

  • We can grow them;
  • We can also trim them;
  • We can use QuickCheck to test their properties.
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Would you like to run a scenario project this year?

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Sure, 
 why not.

Would you like to run a scenario project this year?

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Scenario Project Requirements

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Scenario Project Requirements

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

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

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

  • For 2nd year undergrads, team work,
  • One week-long,
  • Should involve math and programming,
  • Challenging for students, but easy to assess,
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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.
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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.
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How many guards do we really need?

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How many guards do we really need?

The answer depends on the shape of the gallery.

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How many guards do we really need?

The answer depends on the shape of the gallery.

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How many guards do we really need?

The answer depends on the shape of the gallery.

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How many guards do we really need?

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How many guards do we really need?

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How many guards do we really need?

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How many guards do we really need?

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For a given gallery (polygon), 
 find the minimal set of guards’ positions, 
 so together the guards can “see” the whole interior.

Art Gallery Problem

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For a given gallery (polygon), 
 find the minimal set of guards’ positions, 
 so together the guards can “see” the whole interior.

NP-hard

Art Gallery Problem

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Project: Art Gallery Competition

Find the best solutions for a collection of large polygons.

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Project: Art Gallery Competition

Find the best solutions for a collection of large polygons.

  • 58 vertices
  • 5 guards
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Organizers’ TODO

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Organizers’ TODO

  • Problem generator;
  • Polygons with different “features” (convex, rectangular, etc.)
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Organizers’ TODO

  • Problem generator;
  • Polygons with different “features” (convex, rectangular, etc.)
  • Solution checker with online feedback
  • geometric machinery (triangulation, visibility, …)
  • web-server
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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.
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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.
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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.
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  • 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.

Organizers’ TODO

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Growing polygons

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Growing polygons

Primitive polygons with specific “features”

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Growing polygons

Primitive polygons with specific “features”

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Seed

Growing polygons

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Seed

Growing polygons

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Growing polygons

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Growing polygons

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Growing polygons

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Growing polygons

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Growing polygons

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Growing polygons

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Growing polygons

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Growing polygons

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Growing polygons

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Growing polygons

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Growing polygons

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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
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Abstract polygon generator

trait PolygonGenerator extends GeneratorPrimitives {
 
 val seeds : List[Polygon] val primitives : List[(Int) => Polygon] val locate : Double => Option[(Double, Double)] val seedFreqs : List[Int] val primFreqs : List[Int] val generations : Int ... }

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Abstract polygon generator

trait PolygonGenerator extends GeneratorPrimitives {
 
 val seeds : List[Polygon] val primitives : List[(Int) => Polygon] val locate : Double => Option[(Double, Double)] val seedFreqs : List[Int] val primFreqs : List[Int] val generations : Int ... }

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Abstract polygon generator

trait PolygonGenerator extends GeneratorPrimitives {
 
 val seeds : List[Polygon] val primitives : List[(Int) => Polygon] val locate : Double => Option[(Double, Double)] val seedFreqs : List[Int] val primFreqs : List[Int] val generations : Int ... }

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Abstract polygon generator

trait PolygonGenerator extends GeneratorPrimitives {
 
 val seeds : List[Polygon] val primitives : List[(Int) => Polygon] val locate : Double => Option[(Double, Double)] val seedFreqs : List[Int] val primFreqs : List[Int] val generations : Int ... }

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Generating contest problems

Rectilinear

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Quasi-convex

Generating contest problems

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Crazy

Generating contest problems

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  • 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.

Organizers’ TODO

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  • 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.

Organizers’ TODO

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Testing with ScalaCheck

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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;
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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;
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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;
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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.
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How well did that work?

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How well did that work?

  • Multiple bugs (>20) due to inaccurate treatment of

floating points;

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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;

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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;
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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).

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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).

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Testing basic visibility algorithm

  • Main component for checking arbitrary solutions;
  • Original description (1986) has a number of simplifications and

is given in pseudocode.

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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)
 }
 }

Testing basic visibility algorithm

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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)
 }
 }

Testing basic visibility algorithm

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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)
 }
 }

Testing basic visibility algorithm

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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)
 }
 }

Testing basic visibility algorithm

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Bug in Joe-Simpson algorithm

Randomly generated, 260 vertices, guards in every node

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Bug in Joe-Simpson algorithm

??! Randomly generated, 260 vertices, guards in every node

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Shrinking Polygons

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Shrinking Polygons Trimming

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Reconstructing polygon generation

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Reconstructing polygon generation

1 2 3 4 5 6 7 1 2 3 4 5 6 7

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Reconstructing polygon generation

1 2 3 4 5 6 7 1 2 3 4 5 6 7

“Attachment tree”

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Trimming polygons

1 3 4 5 6 7 1 2 4 5 6 7 3 2

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1 2 3 4 5 6 7 1 2 3 4 5 6 7

Trimming polygons

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Shrinking strategy

  • Build the attachment tree while constructing a

random polygon;

  • Construct all its subtrees;
  • “Render” the corresponding trimmed polygons.
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Bug in Joe-Simpson algorithm

Randomly generated, 260 vertices, guards in every node

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Bug in Joe-Simpson algorithm

Randomly generated, 260 vertices, guards in every node

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Bug in Joe-Simpson algorithm

After trimming, 20 vertices

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Bug in Joe-Simpson algorithm

Removed 
 irrelevant guards

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Bug in Joe-Simpson algorithm

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Bug in Joe-Simpson algorithm

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... 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)
 
 } ...

Bug in Joe-Simpson algorithm

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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)
 
 } ...

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Implementation effort

  • Geometric primitives and procedures: 1450 LOC
  • Server infrastructure: 1500 LOC
  • Testing framework: 350 LOC
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Running the Art Gallery Competition

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Running the Art Gallery Competition

  • 94 participants, 24 teams, 2360 submissions in five days
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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)
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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;
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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;

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Mon Tue Wed The Fri

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To take away

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To take away

  • FP works great for rapid development of 


non-trivial and robust geometric applications;

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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;

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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.

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To take away

Thanks!

  • 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.