Adaptive Massively Parallel Constant-round Tree Contraction

TL;DR

This paper introduces a new tree contraction algorithm that operates in constant rounds within the Adaptive Massively Parallel Computing model, significantly improving efficiency over previous methods and applicable to various tree problems.

Contribution

It presents a generalized tree contraction technique that runs in $O(1/\epsilon^3)$ rounds in the AMPC model, extending the applicability to multiple tree problems.

Findings

Achieves tree contraction in constant rounds in AMPC

Extends to multiple tree problems like matching and isomorphism

Applicable to industry-relevant parallel computing frameworks

Abstract

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 $\Omega(\log n)$ 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 $O(1/\epsilon^3)$ rounds in the Adaptive Massively Parallel Computing (AMPC) setting, where each machine has $O(n^\epsilon)$ local memory for some $0 < \epsilon < 1$. 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.