Red-blue and standard pebble games : complexity and applications in the sequential and parallel models

Abstract

Pebble games are games played on directed acyclic graphs involving placing and moving pebbles on nodes of the graph according to a certain set of rules. The standard pebble game (also known as the black pebble game) is the first of such games played on DAGs. The game itself involves three simple rules: a pebble can be placed on any leaf node (a node without any predecessors), a pebble can be moved or placed on a non-leaf node if all of its predecessors are pebbled, and a pebble can be deleted from any node at any time. Generally, the standard pebble game is used to model space-bounded computation. Each node represents a result of a computation and placing a pebble on a node represents performing a deterministic computation of a result using previously computed results. The standard pebble game has been used in a variety of applications including register allocation, VLSI design, compilers, and, more recently, propositional proof complexity and memory-hard functions. Much previous research has been done in analyzing the computational complexity of the standard pebble game in a variety of settings. It has been shown previously that computing an optimal strategy using the standard pebbling game on any given DAG is PSPACE-hard [GLT79]. Furthermore, it was more recently shown that the standard pebble game is hard to approximate to any constant additive factor [CLNV15]. In this thesis, we present a simpler proof of the result presented in [CLNV15] and strengthen the result to include any polynomial additive factor. In particular, we strengthen the result to show that it is PSPACE-hard to determine the minimum number of pebbles to any n1-[epsilon] additive factor for any [epsilon]. The red-blue pebble game was introduced by [JWK81] as a model of I/O complexity and is also played on DAGs. Despite its importance in applications such as data access complexity and more recently to I/O-complexity in multi-level memory hierarchies [CRSS16], little is known about the computational complexity of determining the minimum number of red pebbles and transitions used when pebbling a given DAG using the rules of the game. In this thesis, we show that the red-blue pebble game is PSPACE-hard and that the red-blue pebble game with no deletions and allowing overwrites is NP-Complete. We also show that the red-blue pebble game parameterized by the number of transitions, k is ...W[1]-hard. In addition to the stated hardness results, we also introduce a graph family that takes (nk) time to pebble given k constant number of pebbles. This graph family partially answers an open question posed in [Nor15] regarding whether such a family that meets the O(nk) time upper bound exists for constant k pebbles. Furthermore, the graph family can be generalized for any k < [square root of]n Given k pebbles where k = w(1), the pebbling time is ... for any graphs with n nodes in this family. Finally, we present a new complexity measure, called decremental complexity, based on the sequential and parallel pebbling models. This complexity measure is concerned with the decrease in pebbling time when switching from the sequential to the parallel model of pebbling any given graph with n nodes using the minimum number of pebbles. Graphs with low decremental complexity have potential applications in proofs of work/space and memory-hard functions. In this thesis, we determine the decremental complexity of several common families of graphs as well as composite graphs consisting of many copies of instances of these families.