Abstract
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.
This research was supported in part by NSF grants CCF 1114809, CCF 1114930, CCF 1217708, IIS 1247726, IIS 1247750, and IIS 1251137.
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Bender, M.A., Farach-Colton, M., Goswami, M., Medjedovic, D., Montes, P., Tsai, MT. (2014). The Batched Predecessor Problem in External Memory. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_10
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