Farthest-first traversal

In

Farthest-point traversals may be characterized by the following properties. Fix a number k, and consider the prefix formed by the first k points of the farthest-first traversal of any metric space. Let r be the distance between the final point of the prefix and the other points in the prefix. Then this subset has the following two properties:

• All pairs of the selected points are at distance at least r from each other, and
• All points of the metric space are at distance at most r from the subset.

Conversely any sequence having these properties, for all choices of k, must be a farthest-first traversal. These are the two defining properties of a Delone set, so each prefix of the farthest-first traversal forms a Delone set.^[3]

Applications

As well as for clustering, the farthest-first traversal can also be used in another type of facility location problem, the max-min facility dispersion problem, in which the goal is to choose the locations of k different facilities so that they are as far apart from each other as possible. More precisely, the goal in this problem is to choose k points from a given metric space or a given set of candidate points, in such a way as to maximize the minimum pairwise distance between the selected points. Again, this can be approximated by choosing the first k points of a farthest-first traversal. If r denotes the distance of the kth point from all previous points, then every point of the metric space or the candidate set is within distance r of the first k − 1 points. By the pigeonhole principle, some two points of the optimal solution (whatever it is) must both be within distance r of the same point among these first k − 1 chosen points, and (by the triangle inequality) within distance 2r of each other. Therefore, the heuristic solution given by the farthest-first traversal is within a factor of two of optimal.^[8][9][10]

Other applications of the farthest-first traversal include color quantization (clustering the colors in an image to a smaller set of representative colors),^[11] progressive scanning of images (choosing an order to display the pixels of an image so that prefixes of the ordering produce good lower-resolution versions of the whole image rather than filling in the image from top to bottom),^[12] point selection in the

The farthest-first traversal of a finite point set may be computed by a greedy algorithm that maintains the distance of each point from the previously selected points, performing the following steps:^[3]

• Initialize the sequence of selected points to the empty sequence, and the distances of each point to the selected points to infinity.
• While not all points have been selected, repeat the following steps:
  • Scan the list of not-yet-selected points to find a point p that has the maximum distance from the selected points.
  • Remove p from the not-yet-selected points and add it to the end of the sequence of selected points.
  • For each remaining not-yet-selected point q, replace the distance stored for q by the minimum of its old value and the distance from p to q.

For a set of n points, this algorithm takes O(n²) steps and O(n²) distance computations.^[3]

Approximations

A faster approximation algorithm, given by Har-Peled & Mendel (2006), applies to any subset of points in a metric space with bounded doubling dimension, a class of spaces that include the Euclidean spaces of bounded dimension. Their algorithm finds a sequence of points in which each successive point has distance within a 1 − ε factor of the farthest distance from the previously-selected point, where ε can be chosen to be any positive number. It takes time ${\displaystyle O(n\log n)}$.^[2]

The results for bounded doubling dimension do not apply to high-dimensional Euclidean spaces, because the constant factor in the big O notation for these algorithms depends on the dimension. Instead, a different approximation method based on the Johnson–Lindenstrauss lemma and locality-sensitive hashing has running time ${\displaystyle O(\varepsilon ^{-2}n^{1+1/(1+\varepsilon )^{2}+o(1)}).}$ For metrics defined by shortest paths on weighted undirected graphs, a randomized incremental construction based on Dijkstra's algorithm achieves time ${\displaystyle O(\varepsilon ^{-1}m\log n\log {\tfrac {n}{\varepsilon }})}$, where n and m are the numbers of vertices and edges of the input graph, respectively.^[26]

Incremental Voronoi insertion

For selecting points from a continuous space such as the Euclidean plane, rather than from a finite set of candidate points, these methods will not work directly, because there would be an infinite number of distances to maintain. Instead, each new point should be selected as the center of the largest empty circle defined by the previously-selected point set.^[12] This center will always lie on a vertex of the Voronoi diagram of the already selected points, or at a point where an edge of the Voronoi diagram crosses the domain boundary. In this formulation the method for constructing farthest-first traversals has also been called incremental Voronoi insertion.^[27] It is similar to Delaunay refinement for finite element mesh generation, but differs in the choice of which Voronoi vertex to insert at each step.^[28]

See also

• Lloyd's algorithm, a different method for generating evenly spaced points in geometric spaces

References