Comparison-based time-space lower bounds for selection

We establish the first nontrivial lower bounds on timespace tradeoffs for the selection problem. We prove that any comparison-based randomized algorithm for finding the median requires Ω(n log 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, 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/...

• Type: Journal
• Year: 2010
• Where: TALG
• Authors: Timothy M. Chan
• Keywords: Deterministic Lower Bounds, Lower Bounds, Nontrivial Lower Bounds, TALG 2010