The Batched Predecessor Problem in External Memory

We give lower bounds for the batched predecessor problem in external memory. Given the underlying sorted set S of size n, and a sorted query Q of size x ≤ n, 0 ≤ c < 1, we study tradeoffs between the searching cost, and the cost to preprocess S. We give lower bounds in three external memory models: the I/O comparison model, I/O pointer-machine model, and the indexability model. Our results show that in the comparison I/O model, the batched predecessor problem needs Ω(logB n + 1/B) I/Os per element if the preprocessing is bounded by a polynomial; however, with exponential preprocessing, the problem can be solved faster, in Θ((log2 n+ 1)/B). We also present the tradeoff that quantifies the minimum preprocessing required for a given searching cost. In the pointer-machine model, we show that with O(n4/3−ε) preprocessing for any ε bounded above 0 by a constant, the optimal algorithm cannot perform asymptotically faster than a B-tree. In the more general 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, r = (n/B) 2). We also devise algorithms and indexing schemes that either always match or nearly match the stated lower bounds.