Tight space bounds for l-exclusion

The simplest deadlock-free algorithm for mutual exclusion requires only one single-writer non-atomic bit per process [4,6,13]. This algorithm is known to be space optimal [5,6]. For over 20 years now it has remained an intriguing open problem whether a similar type of algorithm, which uses only one single-writer bit per process, exists also for l-exclusion for some l ≥ 2. We resolve this longstanding open problem. For any l and n, we provide a tight space bound on the number of single-writer bits required to solve l-exclusion for n processes. It follows from our results that it is not possible to solve l-exclusion with one single-writer bit per process, for any l ≥ 2. In an attempt to understand the inherent difference between the space complexity of mutual exclusion and that of l-exclusion for l ≥ 2, we define a weaker version of l-exclusion in which the liveness property is relaxed, and show that, similarly to mutual exclusion, this weaker version can be solve using one single-writer non-atomic bit per process.