More efficient algorithms for closest string and substring problems
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We investigate the optimality of (1+\in )-approximation algorithms obtained via the dimensionality reduction method. We show that: --Any data structure for the (1+\in )-approximate nearest neighbor problem in Hamming space, which uses constant number of probes to answer each query, must use n^{\Omega \left( {1/ \in ^2 } \right)}space. --Any algorithm for the (1+\in )-approximate closest substring problem must run in time exponential in 1/ \in ^{2 - \gamma } for any \gamma 0 (unless 3SAT can be solved in subexponential time) Both lower bounds are (essentially) tight.