Miller and Reif's FOCS'85 classic and fundamental tree contraction algorithm
is a broadly applicable technique for the parallel solution of a large number
of tree problems. Additionally it is also used as an algorithmic design
technique for a large number of parallel graph algorithms. In all previously
explored models of computation, however, tree contractions have only been
achieved in Ω(logn) rounds of parallel run time. In this work, we not
only introduce a generalized tree contraction method but also show it can be
computed highly efficiently in O(1/ϵ3) rounds in the Adaptive
Massively Parallel Computing (AMPC) setting, where each machine has
O(nϵ) local memory for some 0<ϵ<1. AMPC is a practical
extension of Massively Parallel Computing (MPC) which utilizes distributed hash
tables. In general, MPC is an abstract model for MapReduce, Hadoop, Spark, and
Flume which are currently widely used across industry and has been studied
extensively in the theory community in recent years. Last but not least, we
show that our results extend to multiple problems on trees, including but not
limited to maximum and maximal matching, maximum and maximal independent set,
tree isomorphism testing, and more.Comment: 35 pages, 3 figures, to be published in Innovations in Theoretical
Computer Science (ITCS