Edward Sapir

The algorithmic geometry is the field of the Algorithmique which treats algorithms handling of geometrical concepts .

The discipline which undoubtedly contributed the most historically to the development of the algorithmic geometry is the Infographie. However, at present, the algorithmic geometry is frequently seen implied in problems of algorithmic general.

Convex envelope

The convex envelope of a whole of points of the plan is smallest convex Polygone containing all the points. This concept can be immediately generalized with dimensions higher than 2.

The best algorithm known currently making it possible to determine the convex envelope of an unspecified whole of $n$ points in 2D (the Course of Graham) or 3D is in $O (N \ log (N))\,$ . The fact of knowing if there exists a better algorithm remains an opened problem, however several algorithms in $O (N) \,$ exist to treat the simple case of polygons (polygons not car-intersecting) given in the order of appearance of the points.

In the case of of an unspecified dimension D ( D > 3) the best known algorithms are in $O (n^ {\ lfloor {\ frac {D} {2}} \ rfloor+1})$ .

Problems of algorithmic general

The algorithmic geometry provides optimal solutions to problems described on a limited universe. Indeed, this one milked of problems stated in terms of points laid out on a limited grid X '' of dimension 2. By extension, it deals with these same problems on grids of higher size and intervals of entireties ( dimension 1 ).

For example, being given a whole of points S in the interval of entireties “, it is possible to benefit from the character enclosed of the universe ” to solve certain problems in lower part of their minimal complexity for a universe not limited. The most commonplace case and most known is that of the tri linear , the bucket leaves . The elements of S can be sorted in time |S|+U by traversing S first once and by storing each element found in the “bucket” corresponding in a series of numbered buckets of 1 to U. Very many problems of algorithmic find solutions optimal in a limited universe:

being given a unit S of cardinal N on a whole interval “,

To recover the element of S nearest to a X” given in time $\ sqrt {\ log (U)}$ instead of log (N) thanks to the structure of Q-fast-sorts .

being given a unit S on a grid x' ,

To recover all the points which dominate a point x' over their two coordinates in time k+log (log (U)) if K is the number of answers, instead of k+log (N) . See also Research by beach (searching arranges).

being given a unit I of intervals on the interval “,

To find all the intervals of I” which pass over a point X given in time k+log (log (U)) instead of k+log (N) if K is the number of answers, thanks to the structure of tree of research with priority.

Some other important problems

Random links: