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Cerebral X-ray digital subtraction angiography (DSA) is a widely used imaging technique in patients with neurovascular disease, allowing for vessel and flow visualization with high spatio-temporal resolution. Automatic artery-vein segmentation in DSA plays a fundamental role in vascular analysis with quantitative biomarker extraction, facilitating a wide range of clinical applications. The widely adopted U-Net applied on static DSA frames often struggles with disentangling vessels from subtraction artifacts. Further, it falls short in effectively separating arteries and veins as it disregards the temporal perspectives inherent in DSA. To address these limitations, we propose to simultaneously leverage spatial vasculature and temporal cerebral flow characteristics to segment arteries and veins in DSA. The proposed network, coined CAVE, encodes a 2D+time DSA series using spatial modules, aggregates all the features using temporal modules, and decodes it into 2D segmentation maps. On a large multi-center clinical dataset, CAVE achieves a vessel segmentation Dice of 0.84 ($\pm$0.04) and an artery-vein segmentation Dice of 0.79 ($\pm$0.06). CAVE surpasses traditional Frangi-based K-means clustering (P<0.001) and U-Net (P<0.001) by a significant margin, demonstrating the advantages of harvesting spatio-temporal features. This study represents the first investigation into automatic artery-vein segmentation in DSA using deep learning. The code is publicly available at https://github.com/RuishengSu/CAVE_DSA.