Tight space bounds for ℓ-exclusion

The ℓ-exclusion problem is to design an algorithm which guarantees that up to ℓprocesses and no more may simultaneously access identical copies of the same non-sharable resource when there are several competing processes. For ℓ = 1, the 1-exclusion problem is the familiar mutual exclusion problem. The simplest deadlock-free algorithm for mutual exclusion requires only one single-writer non-atomic bit per process (Burns in SIGACT News 10(2):42-47, 1978; Burns and Lynch in Inf Comput 107(2):171-184, 1993; Lamport in J ACM 33:327-348, 1986). This algorithm is known to be space optimal (Burns and Lynch in 18th Annual Allerton conference on communication, control and computing, pp 833-842, 1980; Burns and Lynch in Inf Comput 107(2):171-184, 1993). 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 ℓand n n, we provide a tight space bound on the number of single-writer bits required to solve ℓ-exclusion for n 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 solved using one single-writer non-atomic bit per process.