  K-Means++ and its distributed variant K-Means$\|$ have become de facto tools
for selecting the initial seeds of K-means. While alternatives have been
developed, the effectiveness, ease of implementation, and theoretical grounding
of the K-means++ and $\|$ methods have made them difficult to "best" from a
holistic perspective. By considering the limited opportunities within seed
selection to perform pruning, we develop specialized triangle inequality
pruning strategies and a dynamic priority queue to show the first acceleration
of K-Means++ and K-Means$\|$ that is faster in run-time while being
algorithmicly equivalent. For both algorithms we are able to reduce distance
computations by over $500\times$. For K-means++ this results in up to a
17$\times$ speedup in run-time and a $551\times$ speedup for K-means$\|$. We
achieve this with simple, but carefully chosen, modifications to known
techniques which makes it easy to integrate our approach into existing
implementations of these algorithms.
