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.