The batched predecessor problem in external memory

We give lower and upper bounds for the batched predecessor problem in external memory. We study tradeoffs between the I/O budget to preprocess a dictionary S versus the I/O requirement to find the predecessor in S of each element in a query set Q. For Q polynomially smaller than S, we give lower bounds in three external-memory models: the I/O comparison model, the I/O pointer-machine model, and the indexability model. In the comparison I/O model, we show that the batched predecessor problem needs Ω(log _B n) I/Os per query element (n=|S|) when the preprocessing is bounded by a polynomial. With exponential preprocessing, the problem can be solved faster, in Θ((log ₂ n)/B) per element. We give the tradeoff that quantifies the minimum preprocessing required for a given searching cost. In the pointer-machine model, we show that with O(n ^4/3-ε ) preprocessing for any constant ε>0, the optimal algorithm cannot perform asymptotically faster than a B-tree. In the indexability model, we exhibit the tradeoff between the redundancy r and access overhead α of the optimal indexing scheme, showing that to report all query answers in α(x/B) I/Os, log r=Ω((B/α ²)log (n/B)). Our lower bounds have matching or nearly matching upper bounds.