Geometric Algorithms in Business Geographics

Presented by T. N. Badri

• Geometric algorithms came to the frontline in the early 1970s in the field of algorithm design.

• They are useful in graphics, motion planning of robots, CAD/CAM, Geographic Information Systems and also molecular biology.

• These applications can be used in some business situations, to make more informed decisions.

This presentation describes three geometric ideas and then addresses their usefulness in the corporate world

• Computational Geometry Algorithms

• Their uses in GIS/Molecular Biology/CAD/Robotics

• Three Examples• Voronoi Diagram and Delaunay Triangulation

•Use of the Voronoi Diagram in Product mix for stores

•Use of Delaunay Triangulation in finding Steiner Tree

• Convex Hull algorithms•Use in range and gap detection

•Segment Intersection•Use of segment intersection in overlaid maps

Voronoi Diagram for a set of vertices in the plane• For a given set of vertices the Voronoi Diagram gives us

cells identified with each vertex

• All points in the Voronoi cell are closer to the vertex than to any other vertex

Voronoi Diagram for Hardware Stores in a neighbourhood

What sort of items should a Home Store have in stock? That depends on the modernity of houses in a sub-area.

The colour gives some idea of the age of houses in an area. Darker greens indicate more recent houses.

Source of map: Altavision Geographics

Stacked barchart for sub-areas

Source of map: Altavision Geographics

• The Delaunay Triangulation is the dual of the Voronoi Diagram• It can be obtained from the Voronoi Diagram• The Delaunay is special among meshes because it maximizes the the minimum angle among the triangles in the mesh.

What is a Delaunay Triangulation?

Length = 4 kms

2/3 (3)

Length = 3(2/3)3 = 3.4641 kms

a saving of about 13.4%

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What is a Steiner Minimal Tree?

• The Delaunay Triangulation can be used as the scaffolding for the Steiner Tree

Steiner Tree on the Delaunay Triangulation

Steiner Tree and Minimum Spanning Tree for 100 random points in unit cube

We can use a Delaunay Triangulation with Steiner Tree methods

Convex hull can be used to determine gaps in Network range

• What is a Convex Hull?

• Suppose there is a set P of n vertices in the plane: P = (x1, x2, …., xn).

• The smallest convex set enclosing all the vertices is called the convex hull.

The Convex hull gives an empire map

We can use discs of various diameters instead of vertices

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If the discs representbroadcast range, thispicture shows the gapsin the overall range.

Segment Intersection and Subregion Intersection

• Given a map of roads and a map of rivers one may be interested in the intersection points, as possible locations for bridges

• Overlay of two or more geographical maps can be as helpful as querying a relational database

References

1. Voronoi Diagrams and Delaunay Triangulations, Steven Fortune, Computing in Euclidean Geometry, pp. 193-233

2. Computational Geometry, Mark de Berg, Cheong, Marc Van Kreveld and Mark Overmars, 1997, Third edition Springer Verlag.

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