Thursday, 1 May 2014

How to reduce a spatial problem to an easier one

The reduction of the computation of a problem A to the computation of an easier problem B is a traditional method of building a more efficient algorithm for the resolution of A, especially useful when A is computationally hard. This method is efficient when it comes to spatial analysis and spatial query problems.
Spatial query problems regard the spatial relationships of some set of objects S in a dataset E with some other set of objects S’ in dataset E’. Of course E and E’ may be the same set and in this case we talk about detection of spatial autocorellation or of overlapping features in E. One reason why such problems are sometimes hard to solve, especially for large datasets, is the nature of the spatial objects. More complex spatial objects require more complex spatial queries. The complexity of a spatial object is proportional to its dimensions and geometry.
In a paper of mine that you may find in here I discuss a method that I have developed to work around this issue and it has been very useful to me in GIS applications. I call this method Geometrical Reduction. It is based in the notion of mapping reduction and it regards the reduction of the computation of a spatial query in dataset E to the computation of a spatial query in dataset J where the objects in J have simpler geometry than those in E. When applied in GIS, it is required to produce dataset J analyzing the geometry of the objects in E
For instance, if the objects in E are polygons then J may consist of lines produced from the edges of the polygons. Then the problem of detection of overlapping edges in E is reduced to the problem of intersecting lines in J.
Next, let us define which reductions are valid between object classes.

Geometrical Reduction of object classes
A computable function g reduces geometrically n-dimensional object class A to d-dimensional object class B, AG B, if for every object o in A <=> object g(o) in B.

Then Geometrical Reduction is defined as follows.

Geometrical Reduction of spatial relationships
Spatial relationship S(A, C) is geometrically reducible to spatial relationship P(B, D), written S(A, C) ≤G P(B, D), if AG B and CG D and if S(A, C) is satisfied when P(B, D) is satisfied and P(B, D) is satisfied when S(A, C) is satisfied.

Geometrical reduction theorem
The problem R of deciding if a spatial relationship P(B, D) is valid in dataset E is mapping reducible to problem V of deciding if a spatial relationship S(A, C) is valid in dataset J, if P(B, D) ≤G S(A, C) and if there is a computable function h that decides V.

In my paper I study the problem of detection of connection errors in networks of linear features and especially in hierarchical networks and I use geometrical reduction in order to build efficient algorithms even for large datasets.