Low-Depth Spatial Tree Algorithms
arxiv(2024)
摘要
Contemporary accelerator designs exhibit a high degree of spatial
localization, wherein two-dimensional physical distance determines
communication costs between processing elements. This situation presents
considerable algorithmic challenges, particularly when managing sparse data, a
pivotal component in progressing data science. The spatial computer model
quantifies communication locality by weighting processor communication costs by
distance, introducing a term named energy. Moreover, it integrates depth, a
widely-utilized metric, to promote high parallelism. We propose and analyze a
framework for efficient spatial tree algorithms within the spatial computer
model. Our primary method constructs a spatial tree layout that optimizes the
locality of the neighbors in the compute grid. This approach thereby enables
locality-optimized messaging within the tree. Our layout achieves a polynomial
factor improvement in energy compared to utilizing a PRAM approach. Using this
layout, we develop energy-efficient treefix sum and lowest common ancestor
algorithms, which are both fundamental building blocks for other graph
algorithms. With high probability, our algorithms exhibit near-linear energy
and poly-logarithmic depth. Our contributions augment a growing body of work
demonstrating that computations can have both high spatial locality and low
depth. Moreover, our work constitutes an advancement in the spatial layout of
irregular and sparse computations.
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