Massively Parallel Dynamic Programming on Trees

TL;DR

This paper presents a framework for parallelizing dynamic programming algorithms on trees within the MPC model, enabling efficient solutions for complex graph problems in logarithmic rounds.

Contribution

It introduces classes of tree-based dynamic programming problems that can be parallelized in O(log n) rounds, with techniques applicable to various graph problems.

Findings

Parallel algorithms for tree DP problems in O(log n) rounds

Framework applicable to problems like minimum bisection and maximum independent set

Efficient MPC implementation for dynamic programming on trees

Abstract

Dynamic programming is a powerful technique that is, unfortunately, often inherently sequential. That is, there exists no unified method to parallelize algorithms that use dynamic programming. In this paper, we attempt to address this issue in the Massively Parallel Computations (MPC) model which is a popular abstraction of MapReduce-like paradigms. Our main result is an algorithmic framework to adapt a large family of dynamic programs defined over trees. We introduce two classes of graph problems that admit dynamic programming solutions on trees. We refer to them as "(polylog)-expressible" and "linear-expressible" problems. We show that both classes can be parallelized in $O(\log n)$ rounds using a sublinear number of machines and a sublinear memory per machine. To achieve this result, we introduce a series of techniques that can be plugged together. To illustrate the generality of our framework, we implement in $O(\log n)$ rounds of MPC, the dynamic programming solution of graph problems such as minimum bisection, $k$-spanning tree, maximum independent set, longest path, etc., when the input graph is a tree.