On the Complexity of Worst-Case Blocking Analysis of Nested Critical Sections

Abstract

Accurately bounding the worst-case blocking for finite job sets, a special case of the classic sporadic task model of recurrent real-time systems, using either nested FIFO-or priorityordered locks on multiprocessors is NP-hard. These intractability results are obtained with reductions from the Multiple-Choice Matching problem. The reductions are quite general and do not depend on (1) whether the locks are spin-or suspension-based, or (2) whether global or partitioned scheduling is used, or (3) which scheduling policy is employed (as long as it is work-conserving).Further, we show that, for a special case in which the blocking analysis problem is NP-hard for FIFO-and priority-ordered locks, the problem for unordered spin locks with nested critical sections can be answered in polynomial time by solving a reachability problem on a suitably constructed graph, although (or rather, because) unordered locks do not offer any acquisition-order guarantees.Finally, we identify several challenging open problems, pertaining both to circumventing the hardness results and to classifying the inherent difficulty of the problem more precisely. 109

Migration-Based Synchronization

“…The complexity of such analyses depends both on the used synchronization protocol and the resource access structure of the application. If critical sections are nested, the problem is NP-hard [19].…”

“…Nested locks are considered difficult for timing analysis [19], primarily due to the effect of transitive blocking, where a thread holding resource R A and waiting for another resource R B potentially delays other threads that want to acquire resource R A . Thus, the contention for resource R B affects all threads working with R A .…”

Multiprocessor Real-Time Locking Protocols: A Systematic Review

“…However, even if programmers take care to manually avoid deadlocks by carefully ordering all requests, the blocking analysis problem becomes much more challenging. In fact, in the presence of nested critical sections, the blocking analysis problem is NP-hard even in extremely simplified settings [210], while it can be solved in polynomial time on both uniprocessors (even in the presence of nesting) and multiprocessors in the absence of nesting (at least in simplified settings) [207,210].…”

“…Aiming for this level of accuracy was a major step forward, but unfortunately Faggioli et al's algorithm exhibits superexponential runtime complexity [85]. As already mentioned, unrestricted nesting is inherently difficult to analyze accurately [210].…”

A contention-sensitive fine-grained locking protocol for multiprocessor real-time systems

“…Intuitively, the RNLP achieves optimality by delaying the satisfaction of some lock requests to ensure that earlier-issued outermost requests are never blocked by later-issued ones. Wieder and Brandenburg [15] recently showed that the problem of obtaining precise worst-case pi-blocking bounds for FIFO-or priority-ordered nested locks is NP-hard. This result does not directly apply to a protocol such as the RNLP that can opt to delay the satisfaction of some requests.…”