A universal bound on the space complexity of Directed Acyclic Graph computations

TL;DR

This paper establishes a new universal upper bound on the space complexity for pebbling any DAG, improving previous bounds by introducing novel decomposition and scheduling techniques.

Contribution

It introduces the B-budget decomposition and challenging vertices technique, enabling tighter space bounds for DAG pebbling problems.

Findings

Achieves an $O(m/\log m + d)$ pebbling space bound for DAGs.

Provides improved bounds for DAGs with bounded genus and topological depth.

Introduces efficient algorithms for DAG decomposition and pebbling schedule construction.

Abstract

It is shown that $S(G) = O\left(m/\log_2 m + d\right)$ pebbles are sufficient to pebble any DAG $G=(V,E)$, with $m$ edges and maximum in-degree $d$. It was previously known that $S(G) = O\left(d n/\log n\right)$. The result builds on two novel ideas. The first is the notion of $B-budget\ decomposition$ of a DAG $G$, an efficiently computable partition of $G$ into at most $2^{\lfloor \frac{m}{B} \rfloor}$ sub-DAGs, whose cumulative space requirement is at most $B$. The second is the challenging vertices technique, which constructs a pebbling schedule for $G$ from a pebbling schedule for a simplified DAG $G'$, obtained by removing from $G$ a selected set of vertices $W$ and their incident edges. This technique also yields improved pebbling upper bounds for DAGs with bounded genus and for DAGs with bounded topological depth.