This DCH code is based on the paper
	Deformable Spanners and Applications
	By J. Gao, L.J. Guibas, and A. Nguyen
	Symposium on Computational Geometry SoCG'04, pp 179--199, 2004
and partially on
	Distributed Proximity Maintenance in Ad Hoc Mobile Networks
	By J. Gao, L.J. Guibas, and A. Nguyen
	Distributed Computing in Sensor System DCOSS'05, 2005

The code currently implements proximity query, finding all pairs of points
that are within a certain threshold distance.  There must be some bound
on the motion of the points (say velocity or acceleration restriction) so
that the motion of the points is not too arbitrary.

Suppose you store your point in a class called "MovingPoint".  You need to
implement the following 3 functions (see the sample class BVParticle).

  float MovingPoint::dist2(const MovingPoint& other)
  float MovingPoint::firstTimeFurther(float now, const MovingPoint& other, 
    float threshold2)
  float MovingPoint::firstTimeNearer(float now, const MovingPoint& other, 
    float threshold2)

You can then create the following discrete center hierarchy type as follows:
  typedef DiscreteHierarchy<MovingPoint> DCH;

A typical use of the data structure is:

1. Create an instance of the structure
	DCH dch(beta, c, minradius);
  beta is the expansion ratio of the hierarchy.  beta = 2.0 in the 
  paper, though it can take any value > 1.0. c must satisfy 
  c > max(beta, 2*beta/(beta-1)).  The minradius depends on the maximum
  motion allowed between two successive time steps at must be larger
  than (4*maxmovement)/(c*beta-c-2*beta).

  Notice that the minimum pairwise distance between the points may be 
  smaller than minradius.  As the result, the bottom level is not fully
  maintained.  Nodes in level 1 and up in the hierarchy are maintained
  normally, and for nodes in the bottom level, only their parents are 
  maintained.  Edges in the bottom level, which contains all pairs
  of nodes within c*minradius are not maintained and can be easily
  extracted from the structure, see function numBottomEdges();

  The parameter beta, c, and minradius should be chosen so that
  the edges at the bottom give the pairs of points of interest.
  That means if we want to find out pairs of points within a certain
  threshold r, minradius should be at least r/c.

2. Insert/delete points to the hierarchy
	MovingPoint *mp = new MovingPoint(...);
	...
	dch.insert(mp); 
	...
	dch.remove(mp); 

3. Move the points and from time to time call dch.update()
  Between calls to dch.update(), the points must not move more than
  a certain threshhold distance dictated by the minradius used during
  the creation of the dch.

An Nguyen
anguyen@graphics.stanford.edu
yanguyen@gmail.com