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:

$C = X_1\sqcup\cdots\sqcup X_n$

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:

$\arg\min_X G(X_1,\dots,X_n \mid P)$

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 $P_b$ of blue points and a set $P_r$ of red points. Divide the plane into $n$ regions such that each region contains approximately a fraction $1/n$ of the blue points and $1/n$ of the red points. Here:

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

$G(X_1,\dots,X_n) := \max_{i\in \{1,\dots, n\}} \left( \left |\frac{|P_b\cap X_i| - |P_b|} n \right| + \left| \frac{|P_r\cap X_i| - |P_r|} n\right| \right).$

It equals 0 if each region has exactly a fraction $1/n$ 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.