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Let $X$ be an arbitrary set. Metric is a function $$d: X \times X \rightarrow [0, \infty)$$ such that for all $x,y,z \in X$:

  1. $d(x,y) = 0 \iff x = y,$
  2. $d(x,y) = d(y,x),$
  3. $d(x,y) + d(y,z) \geq d(x,z).$

In Dynamic Time Warping Algorithm, $d$ is a local dissimilarity measure used to measure the distance between points $x_i, y_j$ in compared sequences $x_1, ... x_n$ and $y_1, ... y_m$. In this paper $d$ is defined simply as function with non-negative values.I know that DTW still works without triangle inequality, but what about other metric properties? Aren't they neccessary?

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