Heap-based algorithm for one-dimensional particle systems

A fast algorithm to study one-dimensional self-gravitating systems, and, more generally, systems that are Lagrangian integrable between collisions, is presented. The algorithm is event-driven, and uses a heap-ordered set of predicted future events. In the limit of large number of particles $N$, the operation count is dominated by the cost of reordering the heap after each event, which goes asymptotically as $\log N$. Some applications are discussed in detail.