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A predictive spatiotemporal join finds all pairs of moving objects satisfying a join condition on future time and space. In this paper, we present CoPST, the first and foremost algorithm for such a join using two spatiotemporal indexes. In a predictive spatiotemporal join, the bounding boxes of the outer index are used to perform window searches on the inner index, and these bounding boxes enclose objects with increasing laxity over time. CoPST constructs globally tightened bounding boxes "on the fly" to perform window searches during join processing, thus significantly minimizing overlap and improving the join performance. CoPST adapts gracefully to large-scale databases, by dynamically switching between main-memory buffering and disk-based buffering, through a novel probabilistic cost model. Our extensive experiments validate the cost model and show its accuracy for realistic data sets. We also showcase the superiority of CoPST over algorithms adapted from state-of-the-art spatial join algorithms, by a speedup of up to an order of magnitude.