Temporal data are naturally everywhere, especially in the digital era that sees the advent of big data and internet of things. One major challenge that arises during temporal data analysis and mining is the comparison of time series or sequences, which requires to determine a proper distance or (dis)similarity measure. In this context, the Dynamic Time Warping (DTW) has enjoyed success in many domains, due to its 'temporal elasticity', a property particularly useful when matching temporal data. Unfortunately this dissimilarity measure suffers from a quadratic computational cost, which prohibits its use for large scale applications. This work addresses the sparsification of the alignment path search space for DTW-like measures, essentially to lower their computational cost without loosing on the quality of the measure. As a result of our sparsification approach, two new (dis)similarity measures, namely SP-DTW (Sparsified-Paths search space DTW) and its kernelization SP-K rdtw (Sparsified-Paths search space K rdtw kernel) are proposed for time series comparison. A wide range of public datasets is used to evaluate the efficiency (estimated in term of speed-up ratio and classification accuracy) of the proposed (dis)similarity measures on the 1-Nearest Neighbor (1-NN) and the Support Vector Machine (SVM) classification algorithms. Our experiment shows that our proposed measures provide a significant speed-up without loosing on accuracy. Furthermore, at the cost of a slight reduction of the speedup they significantly outperform on the accuracy criteria the old but well known Sakoe-Chiba approach that reduces the DTW path search space using a symmetric corridor.