Different results and performances with different libraries

Some additional information on top of Felipe Mello's informative answer (disclaimer: author of DTAIDistance here).

For the distance results:

• DTAIDistance only uses Euclidean distance (or L2 norm), this is hardcoded. This choice was made to speed up the execution of the C-code (no function calls). The 'fast' refers to using the C-based implementation instead of a pure Python version and both methods thus give the exact same results.
• FastDTW is a different algorithm than DTW. It is a linear approximation. The 'fast' refers to a lower complexity.
• cDTW. I'm not very familiar with this toolbox but it seems to implement L1 norm.

For the speed results:

In general, pure C-based algorithms are ~100 times faster than pure Python ones (in DTAIDistance this is the difference between distance() and distance_fast()). For the C-based methods the differences are mainly due to flexibility of the methods. Passing a custom norm, for example, will slow down the method (more function calls). Also, different methods have different options which cause more or less switch statements in the algorithm. For example, DTAIDistance, offers quite a number of options to tune the method because it prefers early stopping the computation over further optimizations (also observed by Felipe Mello). Furthermore, different methods store different amounts of data. The DTAIDistance distance method does not store the entire matrix to also offer linear space complexity (the full matrix is obtained using the warping_paths method that has quadratic space complexity). In general for DTW it is recommended to use a window to reduce also the time complexity a bit.

For DTAIDistance, all the design choices were made with timeseries clustering applications in mind (the distance_matrix_fast method). This is another reason not to allow custom norms. The DTW code needs to be pure C to support parallelization on the level of C-code and have minimal overhead (it uses OpenMP) to compute all pairwise distances between series.

Edit: what are the units of the time measurements? I believe that you compared them as they were all in the same unit. Probably the dtw.distance is, for example, in microseconds, while the other answers are in milliseconds, and you thought that dtw.distance performed slower, when it is actually the opposite.

There are different methodologies to measure the distance between two points. It could be based on standard deviation or just euclidian distance. Here is a list of many of those distance.

Some of them might be more computational intensive than others, and also have different meanings. Fast dtw, for example, uses as a third input the type of distance that you want, as described on their github

distance3, path3 = fastdtw(s1, s2, dist = euclidean)

Another reason for the speed difference is the underlying code. Some of them are in pure python, while others are in C, which can be easily 100x faster. A way to speed up your dtaidistance is to set a maximum distance threshold. The algorithm will stop the calculation if it realizes that the total distance will be above a certain value:

distance2 = dtw.distance_fast(array.array('d',s1),array.array('d',s2), max_dist = your_threshold)

It is also important to note that some might be optimized for longer or shorter arrays. Looking at the example below and running it in my computer, I find different results:

from cdtw import pydtw
from dtaidistance import dtw
from fastdtw import fastdtw
from scipy.spatial.distance import euclidean
s1=np.array([1,2,3,4],dtype=np.double)
s2=np.array([4,3,2,1],dtype=np.double)

%timeit dtw.distance_fast(s1, s2)
4.1 µs ± 28.6 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
%timeit d2 = pydtw.dtw(s1,s2,pydtw.Settings(step = 'p0sym', window = 'palival', param = 2.0, norm = False, compute_path = True)).get_dist()
45.6 µs ± 3.39 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
%timeit d3,_=fastdtw(s1, s2, dist=euclidean)
901 µs ± 9.95 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

fastdtw is 219 times slower than dtaidistance lib and 20x slower than cdtw