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In mobile-beacon assisted sensor localization, beacon mobility scheduling aims to determine the best beacon trajectory so that each sensor receives sufficient beacon signals and becomes localized with minimum delay. We propose a novel DeteRministic dynamic bEAcon Mobility Scheduling (DREAMS) algorithm, without requiring any prior knowledge of the sensory field. In this algorithm, the beacon trajectory is defined as the track of Depth-First Traversal (DFT) of the network graph, thus deterministic. The mobile beacon performs DFT dynamically, under the instruction of nearby sensors on the fly. It moves from sensor to sensor in an intelligent heuristic manner according to Received Signal Strength (RSS)-based distance measurements. We prove that DREAMS guarantees full localization (every sensor is localized) when the measurements are noise-free, and derive the upper bound of beacon total moving distance in this case. Then, we suggest to apply node elimination and Local Minimum Spanning Tree (LMST) to shorten beacon tour and reduce delay. Further, we extend DREAMS to multibeacon scenarios. Beacons with different coordinate systems compete for localizing sensors. Loser beacons agree on winner beacons' coordinate system, and become cooperative in subsequent localization. All sensors are finally localized in a commonly agreed coordinate systems. Through simulation we show that DREAMS guarantees full localization even with noisy distance measurements. We evaluate its performance on localization delay and communication overhead in comparison with a previously proposed static path-based scheduling method.