Finding the median requires 2n comparisons
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
A time-space tradeoff for element distinctness
SIAM Journal on Computing
Upper bounds for time-space trade-offs in sorting and selection
Journal of Computer and System Sciences
No. 318 on SWAT 88: 1st Scandinavian workshop on algorithm theory
Journal of the ACM (JACM)
A general sequential time-space tradeoff for finding unique elements
SIAM Journal on Computing
Near-Optimal Time-Space Tradeoff for Element Distinctness
SIAM Journal on Computing
Randomized algorithms
Selection from read-only memory and sorting with minimum data movement
Theoretical Computer Science
SIAM Journal on Computing
Expected time bounds for selection
Communications of the ACM
Time-space tradeoffs for branching programs
Journal of Computer and System Sciences
Median Selection Requires $(2+\epsilon)n$ Comparisons
SIAM Journal on Discrete Mathematics
Time-space trade-off lower bounds for randomized computation of decision problems
Journal of the ACM (JACM)
Determinism versus nondeterminism for linear time RAMs with memory restrictions
Journal of Computer and System Sciences - STOC 1999
Improved upper bounds for time-space trade-offs for selection
Nordic Journal of Computing
Optimal Time-Space Trade-Offs for Sorting
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
A Non-Linear Time Lower Bound for Boolean Branching Programs
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Multi-Pass Geometric Algorithms
Discrete & Computational Geometry
Tight lower bounds for selection in randomly ordered streams
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Robust lower bounds for communication and stream computation
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Selection and sorting with limited storage
SFCS '78 Proceedings of the 19th Annual Symposium on Foundations of Computer Science
The recognition problem for the set of perfect squares
SWAT '66 Proceedings of the 7th Annual Symposium on Switching and Automata Theory (swat 1966)
Lower bounds for quantile estimation in random-order and multi-pass streaming
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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We establish the first nontrivial lower bounds on time-space tradeoffs for the selection problem. We prove that any comparison-based randomized algorithm for finding the median requires Ω(n log logsn) 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, and also complements recent lower bounds that are restricted to sequential access, as in the multi-pass streaming model [Chakrabarti et al., SODA 2008]. We also prove that any comparison-based, deterministic, multi-pass streaming algorithm for finding the median requires Ω(n log*(n/s) + n logs 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. All proofs in this paper involve "elementary" techniques only.