New Parallel Order Maintenance Data Structure

The \emph{Order-Maintenance} (OM) data structure maintains a total order list of items for insertions, deletions, and comparisons. As a basic data structure, OM has many applications, such as maintaining the topological order, core numbers, and truss in graphs, and maintaining ordered sets in Unified Modeling Language (UML) Specification. The prevalence of multicore machines suggests parallelizing such a basic data structure. This paper proposes a new parallel OM data structure that supports insertions, deletions, and comparisons in parallel. Specifically, parallel insertions and deletions are synchronized by using locks efficiently, which achieve up to $7$x and $5.6$x speedups with $64$ workers. One big advantage is that the comparisons are lock-free so that they can execute highly in parallel with other insertions and deletions, which achieve up to $34.4$x speedups with $64$ workers. Typical real applications maintain order lists that always have a much larger portion of comparisons than insertions and deletions. For example, in core maintenance, the number of comparisons is up to 297 times larger compared with insertions and deletions in certain graphs. This is why the lock-free order comparison is a breakthrough in practice.