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The need to allow threads to abort an attempt to acquire a lock (sometimes called a timeout) is an interesting new requirement driven by state-of-the-art database applications with soft real-time constraints. This paper presents a new composite abortable lock (CAL), a combination of abortable queue-based (QL) and test-and-set based backoff (BL) lock mechanisms, which provides non-blocking aborts while ensuring low space requirements without need for a memory reclamation scheme. The key observation motivating our approach is that the fast lock hand-off achieved by QLs only requires the first few threads to be queued (not all waiting threads), and that the remaining threads can run as in a BL. We developed an algorithm that uses only a short fixed size structure for queueing, allowing most threads to back-off. This reduces worst-case space overhead dramatically, and improves performance by eliminating the need for expensive and complicated memory management mechanisms. Experimental results show that our new CAL algorithm not only saves on space, it actually outperforms Scott's state-of-the-art nonblocking abortable QL under contention, and even more so when there are more threads than processors. Moreover, as the rate of lock aborts increases, the CAL continues to perform well, while Scott's algorithm deteriorates rapidly.