Abstract
We establish the first nontrivial lower bounds on time-space trade-offs for the selection problem. We prove that any comparison-based randomized algorithm for finding the median requires Ω(nlog logS n) expected time in the RAM model (or more generally in the comparison branching program model), if we have S bits of extra space besides the read-only input array. This bound is tight for all S log n, and remains true even if the array is given in a random order. Our result thus answers a 16-year-old question of Munro and Raman [1996], and also complements recent lower bounds that are restricted to sequential access, as in the multipass streaming model [Chakrabarti et al. 2008b]. We also prove that any comparison-based, deterministic, multipass streaming algorithm for finding the median requires Ω(n log*(n/s)+ nlog s n) worst-case time (in scanning plus comparisons), if we have s cells of space. This bound is also tight for all s log2 n. We get deterministic lower bounds for I/O-efficient algorithms as well. The proofs in this article are self-contained and do not rely on communication complexity techniques.
Original language | English (US) |
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Article number | 26 |
Journal | ACM Transactions on Algorithms |
Volume | 6 |
Issue number | 2 |
DOIs | |
State | Published - Mar 1 2010 |
Externally published | Yes |
Keywords
- Adversary arguments
- Lower bounds
- Median finding
- RAM
- Randomized algorithms
- Streaming algorithms
- Time-space trade-offs
ASJC Scopus subject areas
- Mathematics (miscellaneous)