\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - Kinetic Data Structures

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User Manual

Author
  Daniel Russel

Lets say you want to maintain a sorted list of items (each item is associate with a real number key). You can imagine placing each of the items on the point on the real line corresponding to its key. Now, let the key for each item change continuously (i.e. no jumps are allowed). As long as no two (consecutive) items cross, the sorted order is intact. When two items cross, they need to be exchanged in the list and then the sorted order is once again correct. This is a trivial example of a kinetic data structure. The key observation is that the combinatorial structure which is maintained changes at discrete times (events) even though the basic building blocks are changing continuously.

This chapter describes a number of such kinetic data structures implemented using the Kinetic framework described in Chapter Kinetic Framework. We first, in Section An Overview of Kinetic Data Structures and Sweep Algorithms introduce kinetic data structures and sweepline algorithms. This section can be skipped if the reader is already familiar with the area. The next sections, Section Terms Used and Section An Overview of the Kinetic Framework introduce the terms and give an overview of the framework. They are recommended reading for all readers, even if you are just using provided kinetic data structures. We then present kinetic data structures for Delaunay triangulations in two and three dimensions in Section Using Kinetic Data Structures.

If you are already familiar with kinetic data structures and know what you want to do, you might want to first take a look at the next section Section Quick Hints which covers quick hints.

# Quick Hints

This section gives quick answers to some questions people might have. It presumes knowledge of kinetic data structures and this framework.

## How do I store extra information allow with, for example, a kinetic Point_2?

See the example Kinetic_data_structures/defining_a_simulation_traits.cpp to see how to define a new `SimulationTraits` class where the `Active-Objects-Table` contains extra data along with the point.

## Where is the best place to look if I want to write my own kinetic data structure?

We provide two simple kinetic data structures, first most trivial is Kinetic_data_structures/trivial_kds.cpp and a slightly more complicated one is:

Defined in `CGAL/Kinetic/Sort.h`

## How can I use kinetic data structures to update Delaunay triangulations?

We are working on that one, but you will have to wait.

# An Overview of Kinetic Data Structures and Sweep Algorithms

Kinetic data structures were first introduced in by Basch et. al. in 1997 [1]. The idea stems from the observation that most, if not all, computational geometry structures are built using predicates - functions on quantities defining the geometric input (e.g. point coordinates), which return a discrete set of values. Many predicates reduce to determining the sign of a polynomial on the defining parameters of the primitive objects. For example, to test whether a point lies above or below a plane we compute the dot product of the point with the normal of the plane and subtract the plane's offset along the normal. If the result is positive, the point is above the plane, zero on the plane, negative below. The validity of many combinatorial structures built on top of geometric primitives can be verified by checking a finite number of predicates of the geometric primitives. These predicates, which collectively certify the correctness of the structure, are called certificates. For a Delaunay triangulation in three dimensions, for example, the certificates are one `InCircle` test per facet of the triangulation, plus a point plane orientation test for each facet or edge of the convex hull.

The kinetic data structures approach is built on top of this view of computational geometry. Let the geometric primitives move by replacing each of their defining quantities with a function of time (generally a polynomial). As time advances, the primitives trace out paths in space called trajectories. The values of the polynomial functions of the defining quantities used to evaluate the predicates now also become functions of time. We call these functions certificate functions. Typically, a geometric structure is valid when all predicates have a specific non-zero sign. In the kinetic setting, as long as the certificate functions maintain the correct sign as time varies, the corresponding predicates do not change values, and the original data structure remains correct. However, if one of the certificate functions changes sign, the original structure must be updated, as well as the set of certificate functions that verify it. We call such occurrences events.

Maintaining a kinetic data structure is then a matter of determining which certificate function changes sign next, i.e. determining which certificate function has the first real root that is greater than the current time, and then updating the structure and the set of certificate functions. In addition, the trajectories of primitives are allowed to change at any time, although \( C^0\)-continuity of the trajectories must be maintained. When a trajectory update occurs for a geometric primitive, all certificates involving that primitive must be updated. We call the collection of kinetic data structures, primitives, event queue and other support structures a simulation.

Sweepline algorithms for computing arrangements in \( d\) dimensions easily map on to kinetic data structures by taking one of the coordinates of the ambient space as the time variable. The kinetic data structure then maintains the arrangement of a set of objects defined by the intersection of a hyperplane of dimension \( d-1\) with the objects whose arrangement is being computed.

Time is one of the central concepts in a kinetic simulation. Just as static geometric data structures divide the continuous space of all possible inputs (as defined by sets of coordinates) into a discrete set of combinatorial structures, kinetic data structures divide the continuous time domain into a set of disjoint intervals. In each interval the combinatorial structure does not change, so, in terms of the combinatorial structure, all times in the interval are equivalent. We capitalize on this equivalence in the framework in order to simplify computations. If the primitives move on polynomial trajectories and the certificates are polynomials in the coordinates, then events occur at real roots of polynomials of time. Real numbers, which define the endpoints of the interval, are more expensive to compute with than rational numbers, so performing computations at a rational number inside the interval is preferable whenever possible. See Section The Kinetic::InstantaneousKernel for an example of where this equivalence is exploited.

## Terms Used

primitive
  The basic geometric types, e.g., the points of a triangulation. A primitive has a set of coordinates.
combinatorial structure
  A structure built on top of the primitives. The structure does not depend directly on the coordinates of the primitives, only on relationships between them.
trajectory
  The path traced out by a primitive as time passes. In other words how the coordinates of a primitive change with time.
snapshot
  The position of all the primitives at a particular moment in time.
static
  Having to do with geometric data structures on non-moving primitives.
predicate
  A function which takes the coordinates of several primitives from a snapshot as input and produces one of a discrete set of outputs.
certificate
  One of a set of predicates which, when all having the correct values, ensure that the combinatorial structure is correct.
certificate function
  A function of time which is positive when the corresponding certificate has the correct value. When the certificate function changes sign, the combinatorial structure needs to be updated.
event
  When a certificate function changes sign and the combinatorial structure needs to be updated.

# An Overview of the Kinetic Framework

The provided kinetic data structures are implemented on top of the Kinetic framework presented in Chapter Kinetic Framework. It is not necessary to know the details of the framework, but some familiarity is useful. Here we presented a quick overview of the framework.

The framework is structured around five main concepts. See Figure 84.1 for a schematic of how a kinetic data structure interacts with the various parts. The main concepts are

- the `Kinetic::Simulator`. Models of this concept process events in the correct order and audit kinetic data structures. There should be one instance of a model of this concept per simulation.
- the `Kinetic::Kernel`. The structure of a `Kinetic::Kernel` is analogous to the static CGAL (i.e., non-kinetic) kernels in that it defines a set of primitives and functors which generate certificates from the primitives.
- the `Kinetic::ActiveObjectsTable`. Models of this concept hold a collection of kinetic primitives in a centralized manner. This structure centralizes management of the primitives in order to properly disseminate notifications when trajectories change, new primitives are added or primitives are deleted. There is generally one instance of a model of this concept per simulation.
- the `Kinetic::InstantaneousKernel`. Models of this concept allow existing non-kinetic CGAL data structures to be used on a snapshot of kinetic data. As a result, pre-existing static structures can be used to initialize and audit kinetic data structures.
- the `Kinetic::FunctionKernel`. This concept is the computational kernel of our framework. Models of this concept are responsible for representing, generating and manipulating the motional and certificate functions and their roots. It is this concept that provides the kinetic data structures framework with the necessary algebraic operations for manipulating event times. The `Kinetic::FunctionKernel` is discussed in detail in Section Algebraic Kernel.

Figure 84.1 The figure shows the interaction between the `Kinetic::Sort<Traits, Visitor>` kinetic data structure and the various pieces of our package. Other, more complicated, kinetic data structures will also use the `Kinetic::InstantaneousKernel` in order to insert/remove geometric primitives and audit themselves. `Kinetic::Sort<Traits, Visitor>` uses the sorting functionality in the STL instead.

For simplicity, we added an additional concept, that of `Kinetic::SimulationTraits`, which wraps together a particular set of choices for the above concepts and is responsible for creating instances of each of the models. As a user of existing kinetic data structures, this is the only framework object you will have to create. The addition of this concept reduces the choices the user has to make to picking the dimension of the ambient space and choosing between exact and inexact computations. The model of `Kinetic::SimulationTraits` creates an instance each of the `Kinetic::Simulator` and `Kinetic::ActiveObjectsTable`. Handles for these instances as well as instances of the `Kinetic::Kernel` and `Kinetic::InstantaneousKernel` can be requested from the simulation traits class. Both the `Kinetic::Kernel` and the `Kinetic::Simulator` use the `Kinetic::FunctionKernel`, the former to find certificate failure times and the later to operate on them. For technical reasons, each supplied model of `Kinetic::SimulationTraits` also picks out a particular type of kinetic primitive which will be used by the kinetic data structures.

# Using Kinetic Data Structures

There are five provided kinetic data structures. They are

- `Kinetic::Sort<Traits, Visitor>` maintain a list of pointssorted by x-coordinate.
- `Kinetic::Delaunay_triangulation_2<Traits, Visitor, Triangulation>` maintain the Delaunay triangulation of a set of two dimensional points.
- `Kinetic::Delaunay_triangulation_3<Traits,Visitor, Triangulation>` maintain the Delaunay triangulation of a set of three dimensional points.
- `Kinetic::Regular_triangulation_3<Traits, Visitor, Triangulation>` maintain the regular triangulation of a set of waiting three dimensional points.
- `Kinetic::Enclosing_box_2<Traits>` and `Kinetic::Enclosing_box_3<Traits>` restrict points to stay within a box by bouncing them off the walls.

## A Simple Example

Using a kinetic data structure can be as simple as the following:
File Kinetic_data_structures/Kinetic_sort.cpp

#include <CGAL/Kinetic/basic.h>
#include <CGAL/Kinetic/Exact_simulation_traits.h>
#include <CGAL/Kinetic/Sort.h>
int main()
{
typedef CGAL::Kinetic::Exact_simulation_traits Traits;
typedef CGAL::Kinetic::Sort<Traits> Sort;
Traits tr(0,100000);
Sort sort(tr);
Traits::Simulator::Handle sp= tr.simulator_handle();
std::ifstream in("data/points_1");
in >> *tr.active_points_1_table_handle();
while (sp->next_event_time() != sp->end_time()) {
sp->set_current_event_number(sp->current_event_number()+1);
}
return EXIT_SUCCESS;
}

Using the other kinetic data structures is substantially identical. Please see the appropriate files in the demo/Kinetic_data_structures directory.

In the example, first the `Kinetic::SimulationTraits` object is chosen (in this case one that supports exact computations). Then the kinetic data structure is defined using the chosen traits object and a visitor class which logs changes to the sorted list. Next, instances of the two are created and a set of points is read from a file. Then, the simulator is instructed to process all the events until the end of the simulation. Finally, a record of what happened is printed to the terminal.

Several important things happen behind the scenes in this example. First, the `Kinetic::ActiveObjectsTable` which holds the moving points notifies the kinetic data structure that new points have been added to the simulation. Second, the `Kinetic::Sort<Traits,Visitor>` kinetic data structure registers its events with the `Kinetic::Simulator` by providing a time and a proxy object for each event. When a particular event occurs, the `Kinetic::Simulator` calls a function on the proxy object which in turn updates the kinetic data structure.

The example illustrates how to monitor the supplied data structures as they evolve by using a `Kinetic::SortVisitor` object - a small class whose methods are called whenever the kinetic data structure changes. Hooks for such visitor concepts are provided for all of the shipped kinetic data structures. In the case of kinetic sorting, the visitor's methods are called every time a new point is inserted in the sorted list, when one is removed, or when two points are swapped in the sorted order.

The visitor concept is quite powerful, allowing us, for example, to implement a data structure for computing and storing two-dimensional arrangements of \( x\)-monotone curves on top of the `Kinetic::Sort<Traits, Visitor>` data structure using about 60 lines of code. This sweepline code is presented in Section Extending Kinetic Data Structures.

## Creating Kinetic Primitives

One key part of the framework not shown is how to create kinetic primitives (rather than just reading them in from a file). There are two ways to construction the necessary motion functions (which are models of `Kinetic::FunctionKernel::Function`). The first is to create an array of polynomial coeffients and simply call the constructor as in:

typedef Traits::Kinetic_kernel::Motion_function F;
std::vector<F::NT> coefs;
coefs.push_back(F::NT(1.0));
coefs.push_back(F::NT(2.0));
F x(coefs.begin(), coefs.end());

A slightly more flexible way is to use a `Kinetic::FunctionKernel::ConstructFunction` object. To do this do the following:

typedef Traits::Kinetic_kernel::Function_kernel::Construct_function
CF; typedef Traits::Kinetic_kernel::Motion_function F; CF cf; F
x=cf(F::NT(1.0), F::NT(2.0));

The `Kinetic::FunctionKernel::ConstructFunction` can be passed (almost) an number of arguments and will construct a polynomial with those arguments are coefficients.

Once the motion functions are constructed, constructing the primitive is just like constructing the corresponding static object.

typedef Traits::Kinetic_kernel::Point_1 Point_1;
Point_1 p(x);

## Extending Kinetic Data Structures

Here we present a simple example that uses the `Kinetic::Sort<Traits, Visitor>` kinetic data structure to compute an arrangement of algebraic functions. It wraps the sorting data structure and uses a visitor to monitor changes and map them to corresponding features in the arrangement. To see an example using this kinetic data structure read the example at examples/Kinetic_data_structures/Kinetic_sweepline.cpp.

First we define the visitor class. An object of this type is passed to the `Kinetic::Sort<Traits, Visitor>` data structure and turns events into calls on the arrangement structure. This class has to be defined externally since the arrangement will inherit from the sorting structure.

template <class Arrangement>
struct Arrangement_visitor: public Kinetic::Sort_visitor_base
{
Arrangement_visitor(Arrangement *a):p_(a){}
template <class Vertex_handle>
void remove_vertex(Vertex_handle a) {
p_->erase(a);
}
template <class Vertex_handle>
void create_vertex(Vertex_handle a) {
p_->insert(a);
}
template <class Vertex_handle>
void after_swap(Vertex_handle a, Vertex_handle b) {
p_->swap(a, b);
}
Arrangement *p_;
};

Now we define the actual arrangement data structure.

template <class TraitsT>
class Planar_arrangement:
public Kinetic::Sort<TraitsT,
Arrangement_visitor<Planar_arrangement<TraitsT> > > {
typedef TraitsT Traits;
typedef Planar_arrangement<TraitsT> This;
typedef typename Kinetic::Sort<TraitsT,
Arrangement_visitor<This> > Sort;
typedef Arrangement_visitor<This> Visitor;
typedef typename Traits::Active_objects_table::Key Key;
public:
typedef CGAL::Exact_predicates_inexact_constructions_kernel::Point_2 Approximate_point;
typedef std::pair<int,int> Edge;
typedef typename Sort::Vertex_handle Vertex_handle;
// Register this KDS with the MovingObjectTable and the Simulator
Planar_arrangement(Traits tr): Sort(tr, Visitor(this)) {}
Approximate_point vertex(int i) const
{
return approx_coords_[i];
}
size_t vertices_size() const
{
return approx_coords_.size();
}
typedef std::vector<Edge >::const_iterator Edges_iterator;
Edges_iterator edges_begin() const
{
return edges_.begin();
}
Edges_iterator edges_end() const
{
return edges_.end();
}
void insert(Vertex_handle k) {
last_points_[*k]=new_point(*k);
}
void swap(Vertex_handle a, Vertex_handle b) {
int swap_point= new_point(*a);
edges_.push_back(Edge(swap_point, last_points_[*a]));
edges_.push_back(Edge(swap_point, last_points_[*b]));
last_points_[*a]= swap_point;
last_points_[*b]= swap_point;
}
void erase(Vertex_handle a) {
edges_.push_back(Edge(last_points_[*a], new_point(*a)));
}
int new_point(typename Traits::Active_objects_table::Key k) {
double tv= CGAL::to_double(Sort::traits().simulator_handle()->current_time());
double dv= CGAL::to_double(Sort::traits().active_objects_table_handle()->at(k).x()(tv));
approx_coords_.push_back(Approximate_point(tv, dv));
return approx_coords_.size()-1;
}
std::vector<Approximate_point > approx_coords_;
std::map<Key, int> last_points_;
std::vector<Edge> edges_;
};

Finally, we have to set everything up. To do this we use some special event classes: `Kinetic::Insert_event<ActiveObjectsTable>` and `Kinetic::Erase_event<ActiveObjectsTable>`. These are events which can be put in the event queue which either insert a primitive into the set of active objects or remove it. Using these, we can allow curves in the arrangement to begin or end in arbitrary places.

typedef CGAL::Kinetic::Insert_event<Traits::Active_points_1_table> Insert_event;
typedef CGAL::Kinetic::Erase_event<Traits::Active_points_1_table> Erase_event;
do {
NT begin, end;
Point function;
// initialize the function and the beginning and end somewhere
tr.simulator_handle()->new_event(Time(begin),
Insert_event(function, tr.active_points_1_table_handle()));
tr.simulator_handle()->new_event(Time(end),
Erase_event(Traits::Active_points_1_table::Key(num),
tr.active_points_1_table_handle()));
++num;
} while (true);