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:
- 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.
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:
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:
where the minimization is on the set of all partitions of C.
1. Red-blue partitioning: there is a set of blue points and a set of red points. Divide the plane into regions such that each region contains approximately a fraction of the blue points and of the red points. Here:
- The cake C is the entire plane ;
- The parameters P are the two sets of points;
- The goal function G is
- It equals 0 if each region has exactly a fraction of the points of each color.
- 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.
- 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.
- 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.