GeographicLib  1.48
Finding nearest neighbors
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The problem of finding the maritime boundary defined by the "median line" is discussed in Section 14 of

Figure 14 shows the median line which is equidistant from Britain and mainland Europe. Determining the median line involves finding, for any given P, the closest points on the coast of Britain and on the coast of mainland Europe. The operation of finding the closest in a set of points is usually referred to as the nearest neighbor problem and the NearestNeighbor class implements an efficient algorithm for solving it.

The NearestNeighbor class implements nearest-neighbor calculations using the vantage-point tree described by

Given a set of points x, y, z, …, in some space and a distance function d satisfying the metric conditions,

\[ \begin{align} d(x,y) &\ge 0,\\ d(x,y) &= 0, \ \text{iff $x = y$},\\ d(x,y) &= d(y,x),\\ d(x,z) &\le d(x,y) + d(y,z), \end{align} \]

the vantage-point (VP) tree provides an efficient way of determining nearest neighbors. The geodesic distance (implemented by the Geodesic class) satisfies these metric conditions, while the great ellipse distance and the rhumb line distance do not (they do not satisfy the last condition, the triangle inequality). Typically the cost of constructing a VP tree of N points is N log N, while the cost of a query is log N. Thus a VP tree should be used in situations where N is large and at least log N queries are to be made. The condition, N is large, means that \( N \gg 2^D \), where D is the dimensionality of the space.

The figure below shows the construction of the VP tree for the points making up the coastlines of Britain and Ireland (about 5000 points shown in blue). The set of points is recursively split into 2 equal "inside" and "outside" subsets based on the distance from a "vantage point". The boundaries between the inside and outside sets are shown as green circular arcs (arcs of geodesic circles). At each stage, the newly added vantage points are shown as red dots and the vantage points for the next stage are shown as red plus signs. The data is shown in the Cassini-Soldner projection with a central meridian of 5°W.

Vantage-point tree
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