## Abstract

This paper gives tight bounds on the cost of cache-oblivious searching. The paper shows that no cache-oblivious search structure can guarantee a search performance of fewer than*lg e log*memory transfers between any two levels of the memory hierarchy. This lower bound holds even if all of the block sizes are limited to be powers of 2. The paper gives modiﬁed versions of the van Emde Boas layout, where the expected number of memory transfers between any two levels of the memory hierarchy is arbitrarily close to [

_{B}N*lg e+O(lg lgB= lgB)*]

*log*. This factor approaches

_{B}N +O(1)*lg e ≈1.443*as

*B*increases. The expectation is taken over the random placement in memory of the ﬁrst element of the structure. Because searching in the disk-access machine (DAM) model can be performed in logB

*N +O(1)*block transfers, this result establishes a separation between the (2-level) DAM model and cache-oblivious model. The DAM model naturally extends to

*k*levels. The paper also shows that as

*k*grows, the search costs of the optimal

*k*-level DAM search structure and the optimal cache-oblivious search structure rapidly converge. This result demonstrates that for a multilevel memory hierarchy, a simple cache-oblivious structure almost replicates the performance of an optimal parameterized k-level DAM structure.