Deterministic tourist walk (DTW) has attracted increasing interest in computer vision. In the last years, different methods for analysis of dynamic and static textures were proposed. So far, all works based on the DTW for texture analysis use all image pixels as initial point of a walk. However, this requires much runtime. In this paper, we conducted a study to verify the performance of the DTW method according to the number of initial points to start a walk. The proposed method assigns a unique code to each image pixel, then, the pixels whose code is not divisible by a given $k$ value are ignored as initial points of walks. Feature vectors were extracted and a classification process was performed for different percentages of initial points. Experimental results on the Brodatz and Vistex datasets indicate that to use fewer pixels as initial points significantly improves the runtime compared to use all image pixels. In addition, the correct classification rate decreases very little.