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In mobile computing environments, the continuous range query is one of the most important types of queries required to support various location-based services. With a large number of queries, the real-time response to query answers and the concurrent execution are two major challenges. In this paper, we propose a novel approach cGridex for efficient processing of continuous range queries over moving objects. cGridex abstracts the dynamic relations between moving objects and concurrent queries, and introduces the concept of coloring on a high level. With colored grids, it forms a direct mapping between moving objects and continuous queries, and can process both of them in a unified way, without redundant evaluation. By precomputing and materializing the color transferring relations, it avoids frequent operations between query sets and spatial join between objects and queries at each time interval, thus leading to fast query evaluation. The experimental results show that cGridex has good performance for the evaluation of continuous range queries.