Comparison-based time-space lower bounds for selection

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( n log log S 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 )+ n log s n ) worst-case time (in scanning plus comparisons), if we have s cells of space. This bound is also tight for all s >log 2 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.