Dynamic ordered sets with approximate queries, approximate heaps and soft heaps

Mikkel Thorup, Or Zamir, Uri Zwick

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We consider word RAM data structures for maintaining ordered sets of integers whose select and rank operations are allowed to return approximate results, i.e., ranks, or items whose rank, differ by less than ∆ from the exact answer, where ∆ = ∆(n) is an error parameter. Related to approximate select and rank is approximate (one-dimensional) nearest-neighbor. A special case of approximate select queries are approximate min queries. Data structures that support approximate min operations are known as approximate heaps (priority queues). Related to approximate heaps are soft heaps, which are approximate heaps with a different notion of approximation. We prove the optimality of all the data structures presented, either through matching cell-probe lower bounds, or through equivalences to well studied static problems. For approximate select, rank, and nearest-neighbor operations we get matching cell-probe lower bounds. We prove an equivalence between approximate min operations, i.e., approximate heaps, and the static partitioning problem. Finally, we prove an equivalence between soft heaps and the classical sorting problem, on a smaller number of items. Our results have many interesting and unexpected consequences. It turns out that approximation greatly speeds up some of these operations, while others are almost unaffected. In particular, while select and rank have identical operation times, both in comparison-based and word RAM implementations, an interesting separation emerges between the approximate versions of these operations in the word RAM model. Approximate select is much faster than approximate rank. It also turns out that approximate min is exponentially faster than the more general approximate select. Next, we show that implementing soft heaps is harder than implementing approximate heaps. The relation between them corresponds to the relation between sorting and partitioning. Finally, as an interesting byproduct, we observe that a combination of known techniques yields a deterministic word RAM algorithm for (exactly) sorting n items in O(n log logw n) time, where w is the word length. Even for the easier problem of finding duplicates, the best previous deterministic bound was O(min{n log log n, n logw n}). Our new unifying bound is an improvement when w is sufficiently large compared with n.

Original languageEnglish
Title of host publication46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
EditorsChristel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi
ISBN (Electronic)9783959771092
StatePublished - 1 Jul 2019
Event46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 - Patras, Greece
Duration: 9 Jul 201912 Jul 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs


Conference46th International Colloquium on Automata, Languages, and Programming, ICALP 2019


  • Lower bounds
  • Order queries
  • Word RAM

All Science Journal Classification (ASJC) codes

  • Software


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