eed evaluation, the triangular inequality is used to lower and upper bound distances after a center update.</p>
<p>Elkan algorithms uses two key observations. First, one has</p>
<p class="formulaDsp">
\[ \|\bx_i - \bc_{q_i}\|_p \leq \|\bc - \bc_{q_i}\|_p / 2 \quad\Rightarrow\quad \|\bx_i - \bc_{q_i}\|_p \leq \|\bx_i - \bc\|_p. \]
</p>
<p>Thus if the distance between \(\bx_i\) and its current center \(\bc_{q_i}\) is less than half the distance of the center \(\bc_{q_i}\) to another center \(\bc\), then \(\bc