Algorithm 2 DTW-based subsequence search algorithm

Input reference pattern $X=(x_1,\dots ,x_N)$, a longer data stream $Y=(y_1,\dots ,y_M)$, with $M\gg N$, and a threshold $\tau$ Output a list $\mathcal{L}$ of repetitions of X within Y having, individually, a DTW lower than $\tau$. The list is ranked depending on the individual DTW 1:$\mathcal{L}=\varnothing$ 2:compute the accumulated cost matrix D between X and Y 3:compute the distance function $\Delta$ 4:find $b^{*}={arg\; min}_{b\in \{1,\dots ,M\}}\Delta \left(b\right)$ 5:If$\Delta \left(b\right)>\tau$then STOP 6:find $a^{*}$ by using Optimal warping path algorithm but initialized in $j=b$ instead of $j=M$ 7:updating $\mathcal{L}$ as: $\mathcal{L}=\mathcal{L}\cup Y_{a^{*}:b^{*}}$ 8:Set $\Delta \left(b\right)=\infty$ for every b in a suitable neighborhood of $b^{*}$ 9:GO TO STEP 3