{
  "id": "2108.10124",
  "version": "v1",
  "published": "2021-08-23T12:48:17.000Z",
  "updated": "2021-08-23T12:48:17.000Z",
  "title": "Projections of Tropical Fermat-Weber Points",
  "authors": [
    "Weiyi Ding",
    "Xiaoxian Tang"
  ],
  "comment": "21 pages, 5 figures, 4 tables",
  "categories": [
    "math.CO"
  ],
  "abstract": "In the tropical projective torus, it is not guaranteed that the projection of a Fermat-Weber point of a given data set is a Fermat-Weber point of the projection of the data set. In this paper, we focus on the projection on the tropical triangle (the three-point tropical convex hull), and we develop one algorithm (Algorithm 1) and its improved version (Algorithm 4), such that for a given data set in the tropical projective torus, these algorithms output a tropical triangle, on which the projection of a Fermat-Weber point of the data set is a Fermat-Weber point of the projection of the data set. We implement these algorithms in R and test how it works with random data sets. The experimental results show that, these algorithms can succeed with a much higher probability than choosing the tropical triangle randomly, the succeed rate of these two algorithms is stable while data sets are changing randomly, and Algorithm 4 can output the results much faster than Algorithm 1 averagely.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-23T12:48:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14T90",
      "62R07",
      "68R01"
    ],
    "keywords": [
      "tropical fermat-weber points",
      "projection",
      "tropical triangle",
      "tropical projective torus",
      "random data sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}