Algorithms and Data Structures

Volume 2748 of the series Lecture Notes in Computer Science pp 451-461

Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries

  • David BremnerAffiliated withFaculty of Computer Science, University of New Brunswick
  • , Erik DemaineAffiliated withMIT Laboratory for Computer Science
  • , Jeff EricksonAffiliated withComputer Science Department, University of Illinois
  • , John IaconoAffiliated withPolytechnic University
  • , Stefan LangermanAffiliated withChargé de recherches du FNRS, Université Libre de Bruxelles
  • , Pat MorinAffiliated withSchool of Computer Science, Carleton University
  • , Godfried ToussaintAffiliated withSchool of Computer Science, McGill University

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Given a set R of red points and a set B of blue points, the nearest-neighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R ∪ B comes from R (respectively, B). This rule implicitly partitions space into a red set and a blue set that are separated by a red-blue decision boundary. In this paper we develop output-sensitive algorithms for computing this decision boundary for point sets on the line and in ℝ2. Both algorithms run in time O(n log k), where k is the number of points that contribute to the decision boundary. This running time is the best possible when parameterizing with respect to n and k.