59 research outputs found
Inapproximability of the Standard Pebble Game and Hard to Pebble Graphs
Pebble games are single-player games on DAGs involving placing and moving
pebbles on nodes of the graph according to a certain set of rules. The goal is
to pebble a set of target nodes using a minimum number of pebbles. In this
paper, we present a possibly simpler proof of the result in [CLNV15] and
strengthen the result to show that it is PSPACE-hard to determine the minimum
number of pebbles to an additive term for all , which improves upon the currently known additive constant hardness of
approximation [CLNV15] in the standard pebble game. We also introduce a family
of explicit, constant indegree graphs with nodes where there exists a graph
in the family such that using constant pebbles requires moves
to pebble in both the standard and black-white pebble games. This independently
answers an open question summarized in [Nor15] of whether a family of DAGs
exists that meets the upper bound of moves using constant pebbles
with a different construction than that presented in [AdRNV17].Comment: Preliminary version in WADS 201
Model and complexity results for tree traversals on hybrid platforms
International audienceWe study the complexity of traversing tree-shaped workflows whose tasks require large I/O files. We target a heterogeneous architec- ture with two resource of different types, where each resource has its own memory, such as a multicore node equipped with a dedicated accelera- tor (FPGA or GPU). Tasks in the workflow are tagged with the type of resource needed for their processing. Besides, a task can be processed on a given resource only if all its input files and output files can be stored in the corresponding memory. At a given execution step, the amount of data stored in each memory strongly depends upon the ordering in which the tasks are executed, and upon when communications between both memories are scheduled. The objective is to determine an efficient traver- sal that minimizes the maximum amount of memory of each type needed to traverse the whole tree. In this paper, we establish the complexity of this two-memory scheduling problem, provide inapproximability results, and show how to determine the optimal depth-first traversal. Altogether, these results lay the foundations for memory-aware scheduling algorithms on heterogeneous platforms
Pebbling Arguments for Tree Evaluation
The Tree Evaluation Problem was introduced by Cook et al. in 2010 as a
candidate for separating P from L and NL. The most general space lower bounds
known for the Tree Evaluation Problem require a semantic restriction on the
branching programs and use a connection to well-known pebble games to generate
a bottleneck argument. These bounds are met by corresponding upper bounds
generated by natural implementations of optimal pebbling algorithms. In this
paper we extend these ideas to a variety of restricted families of both
deterministic and non-deterministic branching programs, proving tight lower
bounds under these restricted models. We also survey and unify known lower
bounds in our "pebbling argument" framework
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