This paper deals with the relationship between cluster analysis and computational geometry describing clustering strategies using a Voronoi diagram approach in general and a line separation approach to improve the efficiency in a special case. We state the following theorems :<list><item>The set of all centralized 2-clusterings (S<subscrpt>1</subscrpt>,S<subscrpt>2</subscrpt>) of a planar point set S with |S<subscrpt>1</subscrpt>|=a and |S<subscrpt>2</subscrpt>|=b is exactly the set of all pairs of labels of opposite Voronoi polygons v<subscrpt>a</subscrpt>(S<subscrpt>1</subscrpt>,S) and v<subscrpt>b</subscrpt>(S<subscrpt>2</subscrpt>,S) of V<subscrpt>a</subscrpt>(S) and V<subscrpt>b</subscrpt>(S) respectively.
</item><item>An optimal centralized 2-clustering [centralized divisive hierarchical 2- clustering] can be constructed in &Ogr;(n n<supscrpt>1/2</supscrpt> log<supscrpt>2</supscrpt>n + U<subscrpt>F</subscrpt>(n) n n<supscrpt>1/2</supscrpt> + P<subscrpt>F</subscrpt>(n)) [&Ogr;(n n<supscrpt>1/2</supscrpt> log<supscrpt>3</supscrpt>n + U<subscrpt>F</subscrpt>(n) n n<supscrpt>1/2</supscrpt> + P<subscrpt>F</subscrpt>(n)) respectively] steps with P<subscrpt>F</subscrpt>(n) and U<subscrpt>F</subscrpt>(n) being the time complexity to compute and update a given clustering measure f.
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