  Dynamic time warping ($\texttt{DTW}$) is one of the most used distance
functions to compare time series, e.$\,$g. in nearest neighbor classifiers.
Yet, fast state of the art algorithms only compare 1-dimensional time series
efficiently. One of these state of the art algorithms uses a lower bound
($\texttt{LB}_\texttt{Keogh}$) introduced by E. Keogh to prune $\texttt{DTW}$
computations. We introduce $\texttt{LB}_\texttt{Box}$ as a canonical extension
to $\texttt{LB}_\texttt{Keogh}$ on multi-dimensional time series. We evaluate
its performance conceptually and experimentally and show that an alternative to
$\texttt{LB}_\texttt{Box}$ is necessary for multi-dimensional time series. We
also propose a new algorithm for the dog-keeper distance ($\texttt{DK}$) which
is an alternative distance function to $\texttt{DTW}$ and show that it
outperforms $\texttt{DTW}$ with $\texttt{LB}_\texttt{Box}$ by more than one
order of magnitude on multi-dimensional time series.
