The DAG Visit Approach for Pebbling and I/O Lower Bounds

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 ?(?{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 ?(d_{out}/(log? d_{out}) log? 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