eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
7:1
7:23
10.4230/LIPIcs.FSTTCS.2022.7
article
The DAG Visit Approach for Pebbling and I/O Lower Bounds
Bilardi, Gianfranco
1
De Stefani, Lorenzo
2
https://orcid.org/0000-0001-9569-2086
Department of Information Engineering, University of Padova, Italy
Department of Computer Science, Brown University, Providence, RI, USA
We introduce the notion of an r-visit of a Directed Acyclic Graph DAG G = (V,E), a sequence of the vertices of the DAG complying with a given rule r. A rule r specifies for each vertex v ∈ V a family of r-enabling sets of (immediate) predecessors: before visiting v, at least one of its enabling sets must have been visited. Special cases are the r^(top)-rule (or, topological rule), for which the only enabling set is the set of all predecessors and the r^(sin)-rule (or, singleton rule), for which the enabling sets are the singletons containing exactly one predecessor. The r-boundary complexity of a DAG G, b_r(G), is the minimum integer b such that there is an r-visit where, at each stage, for at most b of the vertices yet to be visited an enabling set has already been visited. By a reformulation of known results, it is shown that the boundary complexity of a DAG G is a lower bound to the pebbling number of the reverse DAG, G^R. Several known pebbling lower bounds can be cast in terms of the r^{(sin)}-boundary complexity. The main contributions of this paper are as follows:
- An existentially tight 𝒪(√{d_{out} n}) upper bound to the r^(sin)-boundary complexity of any DAG of n vertices and out-degree d_{out}.
- An existentially tight 𝒪(d_{out}/(log₂ d_{out}) log₂ n) upper bound to the r^(top)-boundary complexity of any DAG. (There are DAGs for which r^(top) provides a tight pebbling lower bound, whereas r^(sin) does not.)
- A visit partition technique for I/O lower bounds, which generalizes the S-partition I/O technique introduced by Hong and Kung in their classic paper "I/O complexity: The Red-Blue pebble game". The visit partition approach yields tight I/O bounds for some DAGs for which the S-partition technique can only yield a trivial lower bound.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol250-fsttcs2022/LIPIcs.FSTTCS.2022.7/LIPIcs.FSTTCS.2022.7.pdf
Pebbling
Directed Acyclic Graph
Pebbling number
I/O complexity