Map segmentation

In mathematics, the map segmentation problem is a kind of optimization problem. It involves a certain geographic region that has to be partitioned into smaller sub-regions in order to achieve a certain goal. Typical optimization objectives include: ^[1]

• Minimizing the workload of a fleet of vehicles assigned to the sub-regions;
• Balancing the consumption of a resource, as in fair cake-cutting.
• Determining the optimal locations of supply depots;
• Maximizing the surveillance coverage.

Fair division of land has been an important issue since ancient times, e.g. in ancient Greece. ^[2]

Notation

There is a geographic region denoted by C ("cake").

A partition of C, denoted by X, is a list of disjoint subregions whose union is C:

${\displaystyle C=X_{1}\sqcup \cdots \sqcup X_{n}}$

There is a certain set of additional parameters (such as: obstacles, fixed points or probability density functions), denoted by P.

There is a real-valued function denoted by G ("goal") on the set of all partitions.

The map segmentation problem is to find:

${\displaystyle \arg \min _{X}G(X_{1},\dots ,X_{n}\mid P)}$

where the minimization is on the set of all partitions of C.

Often, there are geometric shape constraints on the partitions, e.g., it may be required that each part be a convex set or a connected set or at least a measurable set.

Examples

1. Red-blue partitioning: there is a set ${\displaystyle P_{b}}$ of blue points and a set ${\displaystyle P_{r}}$ of red points. Divide the plane into ${\displaystyle n}$ regions such that each region contains approximately a fraction ${\displaystyle 1/n}$ of the blue points and ${\displaystyle 1/n}$ of the red points. Here:

• The cake C is the entire plane ${\displaystyle \mathbb {R} ^{2}}$;
• The parameters P are the two sets of points;
• The goal function G is

${\displaystyle G(X_{1},\dots ,X_{n}):=\max _{i\in \{1,\dots ,n\}}\left(\left|{\frac {|P_{b}\cap X_{i}|-|P_{b}|}{n}}\right|+\left|{\frac {|P_{r}\cap X_{i}|-|P_{r}|}{n}}\right|\right).}$

It equals 0 if each region has exactly a fraction ${\displaystyle 1/n}$ of the points of each color.

Related problems

• A Voronoi diagram is a specific type of map-segmentation problem.
• Fair cake-cutting, when the cake is two-dimensional, is another specific map-segmentation problem when the cake is two-dimensional, like in the Hill–Beck land division problem.
• The Stone–Tukey theorem is related to a specific map-segmentation problem.

References

1. Raghuveer Devulapalli (Advisor: John Gunnar Carlsson) (2014). Geometric Partitioning Algorithms for Fair Division of Geographic Resources. A Ph.D. thesis submitted to the faculty of university of Minnesota. 
2. Boyd, Thomas D.; Jameson, Michael H. (1981). "Urban and Rural Land Division in Ancient Greece". Hesperia. 50 (4): 327. doi:10.2307/147876. JSTOR   147876.