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Lightweight aerial swarms have potential applications in scenarios where larger drones fail to operate efficiently. The primary foundation for lightweight aerial swarms is efficient relative localization, which enables cooperation and collision avoidance. Computing the real-time position is challenging due to extreme resource constraints. This paper presents an autonomous relative localization technique for lightweight aerial swarms without infrastructure by fusing ultra-wideband wireless distance measurements and the shared state information (e.g., velocity, yaw rate, height) from neighbors. This is the first fully autonomous, tiny, fast, and accurate relative localization scheme implemented on a team of 13 lightweight (33 grams) and resource-constrained (168MHz MCU with 192 KB memory) aerial vehicles. The proposed resource-constrained swarm ranging protocol is scalable, and a surprising theoretical result is discovered: the unobservability poses no issues because the state drift leads to control actions that make the state observable again. By experiment, less than 0.2m position error is achieved at the frequency of 16Hz for as many as 13 drones. The code is open-sourced, and the proposed technique is relevant not only for tiny drones but can be readily applied to many other resource-restricted robots. Video and code can be found at \textnormal{\url{https://shushuai3.github.io/autonomous-swarm/}}.