An analysis of budgeted parallel search on conditional Galton-Watson trees

Recently Avis and Jordan have demonstrated the efficiency of a simple technique called budgeting for the parallelization of a number of tree search algorithms. The idea is to limit the amount of work that a processor performs before it terminates its search and returns any unexplored nodes to a master process. This limit is set by a critical budget parameter which determines the overhead of the process. In this paper we study the behaviour of the budget parameter on conditional Galton-Watson trees obtaining asymptotically tight bounds on this overhead. We present empirical results to show that this bound is surprisingly accurate in practice.Comment: 15 pages, 3 figures, 2 tables Minor revisions including an extended description of the Q-process with additional figur

mts: A light framework for parallelizing tree search codes

We describe mts, a generic framework for parallelizing certain types of tree search programs including reverse search, backtracking, branch and bound and satisfiability testing. It abstracts and generalizes the ideas used in parallelizing lrs, a reverse search code for vertex enumeration. mts supports sharing information between processes which is important for applications such as satisfiability testing and branch-and-bound. No parallelization is implemented in the legacy single processor programs minimizing the changes needed and simplying debugging. mts is written in C, uses MPI for parallelization and can be used on a network of computers. We give four examples of reverse search codes parallelized by using mts: topological sorts, spanning trees, triangulations and Galton-Watson trees. We also give a parallelization of two codes for satisfiability testing. We give experimental results comparing the parallel codes with other codes for the same problems.Comment: This complements the earlier tutorial (arXiv:1610.07735) on how to use mts, with which it shares some material. The tutorial should be consulted for specific details of the programming changes to legacy code required. This version included two more application

Parallel Enumeration of Triangulations

We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing full triangulations only; this case is particularly relevant for tropical geometry. Most importantly, down-flip reverse search allows for massive parallelization, i.e., it scales well even for many cores. Our implementation allows to compute the triangulations of much larger point sets than before.Comment: 27 pages, 5 figure

mplrs: A scalable parallel vertex/facet enumeration code

We describe a new parallel implementation, mplrs, of the vertex enumeration code lrs that uses the MPI parallel environment and can be run on a network of computers. The implementation makes use of a C wrapper that essentially uses the existing lrs code with only minor modifications. mplrs was derived from the earlier parallel implementation plrs, written by G. Roumanis in C++. plrs uses the Boost library and runs on a shared memory machine. In developing mplrs we discovered a method of balancing the parallel tree search, called budgeting, that greatly improves parallelization beyond the bottleneck encountered previously at around 32 cores. This method can be readily adapted for use in other reverse search enumeration codes. We also report some preliminary computational results comparing parallel and sequential codes for vertex/facet enumeration problems for convex polyhedra. The problems chosen span the range from simple to highly degenerate polytopes. For most problems tested, the results clearly show the advantage of using the parallel implementation mplrs of the reverse search based code lrs, even when as few as 8 cores are available. For some problems almost linear speedup was observed up to 1200 cores, the largest number of cores tested.Comment: Revision incorporating additional suggested change