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New Kernel Design and Kernel Traits

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					2D Arrangements in CGAL: Recent Developments
CGAL Team School of Computer Science Tel Aviv University
Eti Ezra, Eyal Flato, Efi Fogel, Dan Halperin, Shai Hirsch, Eran Leiserowitz, Eli Packer, Tali Zvi, Ron Wein

Outline
• • • • • • Introduction The Packages in Brief Exploiting the Kernel Categorizing the Traits Benchmarking More Work

Outline
• Introduction
• • • • • The Package in Brief Exploiting the Kernel Categorizing the Traits Benchmarking More Work

Introduction
“Bypasses are devices that allow some people to dash from point A to point B very fast while other people dash from point B to point A very fast. People living at point C, being a point directly in between, are often given to wonder what's so great about point A that so many people from point B are so keen to get there and what's so great about point B that so many people from point A are so keen to get there. They often wish that people would just once and for all work out where the hell they wanted to be.”
Douglas Adams

Definitions
Planar Maps Planar graphs that are embedded in the plane

Definitions (cont.)
Planar Arrangements Given a collection Γ of planar curves, the arrangement A(Γ) is the partition of the plane to vertices, edges and faces induced by the curves of Γ

Application: GIS

[Nguyen Dong Ha, et al.]

Application: Robot Motion Planning

[Flato, Halperin]

Outline
• Introduction

• The Package in Brief
• • • • Exploiting the Kernel Categorizing the Traits Benchmarking More Work

The Package in Brief
“A common mistake that people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools.”
Douglas Adams

The Package in Brief
• Goal: Construct, maintain, modify, traverse, query and present subdivisions of the plane • Exact • Generic • Handles all degeneracies • Efficient

• Topological_map
– Maintains topological maps of finite edges

• Planar_map_2
– Maintains planar maps of interior-disjoint xmonotone curves

• Planar_map_with_intersections_2
– Maintains planar maps of general curves (may intersect, may be non-x-monotone)

• Arrangement_2
– Maintains planar maps of intersecting curves along with curve history

The Package in Brief

Functionality
• Creation & Destruction • I/O
– Save, Load, Print (ASCII streams) – Draw (graphic streams) – Flexibility (Adaptable and Extensible, Verbose mode, I/O of specific elements)

• Modification
– Insertion, Removal, Split, Merge

• Traversal • Queries
– Number of Vertices, Halfedges, & Faces – Is Point in Face – Point Location, Vertical ray shoot

Traversal
• Element Traversal
– – – – Vertex Iterator Face Iterator Edge Iterator Halfedge Iterator

• Map Traversal
– Connected Component of the Boundary (CCB) Halfedge Circulator – Around Vertex Halfedge Circulator – Hole Iterator

Point Location Strategies
• • • Naive
– – No preprocessing, no internal data Linear query time

Walk along a line
– – – – No preprocessing, no internal data Linear query time with heuristics Preprocessing, internal data Expected logarithmic query time

Trapezoidal decomposition based

Traits Classes
• Geometric Interface • Parameter of package
– Defines the family of curves in interest – Package can be used with any family of curves for which a traits class is supplied

• Aggregate
– geometric types (points, curves) – Operations over types (accessors, predicates, constructors)

Traits Classes
• Supplied Traits Classes
– Segments, Polylines, Circular arcs and Line segments, Conics (and line segments).

• Other Known Traits Classes
– Circular arcs, Canonical Parabola, Bezier Curves

Insertions
• Non intersecting insert
Halfedge_handle non_intersecting_insert(const X_curve_2 & cv, Change_notification * en = NULL);

• Intersecting insert
Halfedge_handle insert(const X_curve_2 & cv, Change_notification * en = NULL);

Insertions
• Incremental Insert • Aggregate Insert • Often information is known in advance
– Containing face
Insert in face interior

– Incident vertices
Insert from vertex, between vertices

– Order around vertex
Insert from halfedge target, between halfedge targets

Aggregate Insert
• Inserts a container into the map
template <class curve_iterator> Halfedge_iterator insert(const curve_iterator & begin, const curve_iterator & end, Change_notification * en = NULL);

• Two versions
– Simplified - planar map no intersections – General - planar map with intersections

• Sweep based
– If planar map is not empty, use overlay

Outline
• • • • • • Introduction The Package in Brief Exploiting the Kernel Categorizing the Traits Benchmarking More Work

Exploiting the Kernel
“Human beings, who are almost unique in having the ability to learn from the experience of others, are also remarkable for their apparent disinclination to do so.”
Douglas Adams

CGAL Kernel Context
• CGAL consists of three major parts
– Kernel – Basic geometric data structures and algorithms
• Convex Hull, Planar_map, Arrangement, etc.

– Non-geometric support facilities

CGAL Kernel
• Encapsulates
– Constant-size non-modifiable geometric primitive object representations
• Point, Segments, hopefully Conics, etc

– operations (and predicates) on these objects

• Adaptable and Extensible • Efficient • Used as a traits class for algorithms

Adapting the kernel
• Exchange of representation classes
– Representation classes are parameterized by a number type – Geometric objects are extracted from a representation class
template <class Kernel> class Pm_segment_traits_2 : public Kernel { public typedef typename Kernel::Point_2 Point_2; typedef typename Kernel::Segment_2 X_curve_2; … };

Adapting the kernel
• Functors provide the functionality
– Functor – a class that define an appropriate operator()

• Object for functors are obtained through access member functions
template <class Kernel> class Pm_segment_traits_2 : public Kernel { Comparison_result compare_x(const Point_2 & p1, const Point_2 & p2) const { return compare_x_2_object()(p1, p2); } };

Adapting the kernel
• Code reduction
– Implementation is simple and concise

• Traits reduction
– Matthias Baesken LEDA Kernel makes the dedicated LEDA Traits obsolete
#if defined(USE_LEDA_KERNEL) typedef CGAL::leda_rat_kernel_traits #else typedef leda_rational typedef CGAL::Cartesian<NT> #endif typedef CGAL::Pm_segment_traits_2<Kernel>
Kernel; NT; Kernel; Traits;

Outline
• • • • • • Introduction The Package in Brief Exploiting the Kernel Categorizing the Traits Benchmarking More Work

Categorizing the Traits
“It is a mistake to think you can solve any major problems just with potatoes.”
Douglas Adams

Categorizing the Traits
• In the past – 2 levels of refinements
– Planar map Traits – Planar map of intersecting curves Traits

• In the future – multiple categories
– Each category identifies a behavior
• Multiple Tags

– All categories identify the Traits

Dispatching Algorithms
• Tailored Algorithms
– Curve category
• Segments, Circular Arcs, Conics
template <class Kernel> class Arr_segment_traits_2 { typedef Segment_tag Curve_category; };

template <class Kernel> class Arr_conic_traits_2 { typedef Conic_tag Curve_category; };

Dispatching Algorithms
• Trading between efficiency and complexity
– Intersection Category
• Lazy, Efficient
typedef Lazy_intersection_tag Intersection_category; Point_2 reflect_point(const Point_2 & pt) const; X_curve_2 reflect_curve(const X_curve_2 & cv) const; Bool nearest_intersection_to_right(…) const;

typedef Efficient_intersection_tag Intersection_category; Bool nearest_intersection_to_right(…) const; Bool nearest_intersection_to_left(…) const;

Tightening the Traits
• Different operations may have
– Different requirements – Different preconditions

• Minimal set of requirements
– Sweep has less requirement
bool do_intersect_to_left(c1, c2, pt) bool do_intersect_to_right(c1, c2, pt) bool nearest_intersection_to_left(c1, c2, pt, …) bool nearest_intersection_to_right(c1, c2, pt, …)

result curve_compare_at_x_left(cv1, cv2, pt) result curve_compare_at_x_right(cv1, cv2, pt)

Specialization
• Caching
– Avoid computations (intersection points) – Avoid construction (extreme end-points) – Code Reuse
• Caching of intersection points is currently implemented as part of the conic traits

– Requires redefinition of some classes (e.g., halfedge)
Work in progress

Outline
• • • • Introduction The Package in Brief Exploiting the Kernel Categorizing the Traits

• Benchmarking
• More Work

Insert Multiplications
Non intersecting vs. intersecting Incremental vs. aggregate
Point location strategies

2 2
3

CGAL cartesian parameterized with 2 LEDA rational number type vs. Matthias LEDA Kernel Segments, Conics 2 Traits categories Total 2 96

Benchmarks
onebig 100 8 7
operation time (sec)
operation time (sec)

onebig 100 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

6 5 4 3 2 1 0 31-Oct 10-Nov 20-Nov 30-Nov 10-Dec date Inc,Leda traits Agg, Leda traits Inc, Leda Kernel Agg, Leda Kernel

0 5-Nov 10Nov

15Nov

20Nov date

25Nov

30Nov

5Dec

Agg, Leda traits

Agg, Leda Kernel

Benchmarks
random 100 3.5 3
operation time (sec)

random 100 0.8 0.7
operation time (sec)

2.5 2 1.5 1 0.5 0 5-Nov 10Nov

0.6 0.5 0.4 0.3 0.2 0.1

15Nov

20Nov date

25Nov

305Nov Dec

0 5-Nov 10Nov

15Nov

20Nov date

25Nov

30Nov

5Dec

Inc,Leda traits Agg, Leda traits

Inc, Leda Kernel Agg, Leda Kernel

Agg, Leda traits

Agg, Leda Kernel

Outline
• • • • • Introduction The Package in Brief Exploiting the Kernel Categorizing the Traits Benchmarking

• More Work

More Work
“Capital letters were always the best way of dealing with things you didn't have a good answer to.”
Douglas Adams

More Work
• Consolidate Pm and Pmwx into a unified class Planar_map_2 • Introduce more Specialization categories and options • Introduce more Point Location Strategies • Introduce Traits classes for complex curves • Move up to higher dimensions

End


				
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posted:11/16/2009
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