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On The I/O Complexity of Dynamic Distinct Counting

Authors Xiaocheng Hu, Yufei Tao, Yi Yang, Shengyu Zhang, Shuigeng Zhou



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Xiaocheng Hu
Yufei Tao
Yi Yang
Shengyu Zhang
Shuigeng Zhou

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Xiaocheng Hu, Yufei Tao, Yi Yang, Shengyu Zhang, and Shuigeng Zhou. On The I/O Complexity of Dynamic Distinct Counting. In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 265-276, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.ICDT.2015.265

Abstract

In dynamic distinct counting, we want to maintain a multi-set S of integers under insertions to answer efficiently the query: how many distinct elements are there in S? In external memory, the problem admits two standard solutions. The first one maintains $S$ in a hash structure, so that the distinct count can be incrementally updated after each insertion using O(1) expected I/Os. A query is answered for free. The second one stores S in a linked list, and thus supports an insertion in O(1/B) amortized I/Os. A query can be answered in O(N/B log_{M/B} (N/B)) I/Os by sorting, where N=|S|, B is the block size, and M is the memory size. In this paper, we show that the above two naive solutions are already optimal within a polylog factor. Specifically, for any Las Vegas structure using N^{O(1)} blocks, if its expected amortized insertion cost is o(1/log B}), then it must incur Omega(N/(B log B)) expected I/Os answering a query in the worst case, under the (realistic) condition that N is a polynomial of B. This means that the problem is repugnant to update buffering: the query cost jumps from 0 dramatically to almost linearity as soon as the insertion cost drops slightly below Omega(1).
Keywords
  • distinct counting
  • lower bound
  • external memory

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