Optimal hierarchical layouts for cache-oblivious search trees

Abstract

This paper proposes a general framework for generating cache-oblivious layouts for binary search trees. A cache-oblivious layout attempts to minimize cache misses on any hierarchical memory, independent of the number of memory levels and attributes at each level such as cache size, line size, and replacement policy. Recursively partitioning a tree into contiguous subtrees and prescribing an ordering amongst the subtrees, Hierarchical Layouts generalize many commonly used layouts for trees such as in-order, pre-order and breadthfirst. They also generalize the various flavors of the van Emde Boas layout, which have previously been used as cache-oblivious layouts. Hierarchical Layouts thus unify previous attempts at deriving layouts for search trees.The paper then derives a new locality measure (the Weighted Edge Product) that mimics the probability of cache misses at multiple levels, and shows that layouts that reduce this measure perform better. We analyze the various degrees of freedom in the construction of Hierarchical Layouts, and investigate the relative effect of each of these decisions in the construction of cacheoblivious layouts. Optimizing the Weighted Edge Product for complete binary search trees, we introduce the MINWEP layout, and show that it outperforms previously used cache-oblivious layouts by almost 20%.

Cache-Oblivious Representation of B-Tree Structures

“…Finally, there is a data structure similar to PMA which permits some segments to be swapped improving the I/O complexity of updates to O((log log N ) 2+ε ) [BCD + 02]; replacing PMA with this data structure may possibly also improve the I/O complexities of updates, but it will make analysis much more complicated. As a minor improvement, using the hierarchical memory layouts of Lindstrom and Rajan[LR14] instead of vEB layout may improve search complexity by a constant factor.…”