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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References
Afshani, P., Arge, L., Larsen, K.D.: Orthogonal range reporting: Query lower bounds, optimal structures in 3-d, and higher-dimensional improvements. In: 26th Annual Symposium on Computational Geometry (SoCG), pp. 240–246 (2010)
Afshani, P., Arge, L., Larsen, K.G.: Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. In: 28th Annual Symposium on Computational Geometry (SoCG), pp. 323–332 (2012)
Aggarwal, A., Vitter, J.S.: The input/output complexity of sorting and related problems. Commun. ACM 31, 1116–1127 (1988)
Arge, L.: The buffer tree: A technique for designing batched external data structures. Algorithmica 37(1), 1–24 (2003)
Bollobás, B., Fernandez de la Vega, W.: The diameter of random regular graphs. Combinatorica 2(2), 125–134 (1982)
Brodal, G.S., Fagerberg, R.: Lower bounds for external memory dictionaries. In: 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 546–554 (2003)
Buchsbaum, A.L., Goldwasser, M., Venkatasubramanian, S., Westbrook, J.R.: On external memory graph traversal. In: 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 859–860 (2000)
Dittrich, W., Hutchinson, D., Maheshwari, A.: Blocking in parallel multisearch problems (extended abstract). In: 10th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 98–107 (1998)
Goodrich, M.T., Tsay, J.J., Cheng, N.C., Vitter, J., Vengroff, D.E., Vitter, J.S.: External-memory computational geometry. In: 1993 IEEE 34th Annual Foundations of Computer Science (FOCS), pp. 714–723 (1993)
Hellerstein, J.M., Koutsoupias, E., Papadimitriou, C.H.: On the analysis of indexing schemes. In: 16th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS), pp. 249–256 (1997)
Hellerstein, J.M., Koutsoupias, E., Miranker, D.P., Papadimitriou, C.H., Samoladas, V.: On a model of indexability and its bounds for range queries. J. ACM 49, 35–55 (2002)
Karpinski, M., Nekrich, Y.: Predecessor queries in constant time? In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 238–248. Springer, Heidelberg (2005)
Knudsen, M., Larsen, K.: I/O-complexity of comparison and permutation problems. Master’s thesis, DAIMI (November 1992)
Knuth, D.E.: The Art of Computer Programming: Sorting and Searching, vol. 3. Addison-Wesley (1973)
Pătraşcu, M., Thorup, M.: Time-space trade-offs for predecessor search. In: 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 232–240 (2006)
Rödl, V.: On a packing and covering problem. European Journal of Combinatorics 6(1), 69–78 (1985)
Samoladas, V., Miranker, D.P.: A lower bound theorem for indexing schemes and its application to multidimensional range queries. In: 17th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS), pp. 44–51 (1998)
Subramanian, S., Ramaswamy, S.: The p-range tree: A new data structure for range searching in secondary memory. In: Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 378–387 (1995)
Tamassia, R., Vitter, J.S.: Optimal cooperative search in fractional cascaded data structures. In: Algorithmica, pp. 307–316 (1990)
Tao, T., Vu, V.H.: Additive Combinatorics. Cambridge University Press (2009)
Tarjan, R.E.: A class of algorithms which require nonlinear time to maintain disjoint sets. Journal of Computer and System Sciences 18(2), 110–127 (1979)
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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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