We prove an Omega(d/log(sw/nd)) lower bound for the average-case cell-probe complexity of deterministic or Las Vegas randomized algorithms solving approximate near-neighbor (ANN) problem in ddimensional Hamming space in the cell-probe model with w-bit cells, using a table of size s. This lower bound matches the highest known worst-case cell-probe lower bounds for any static data structure problems. This average-case cell-probe lower bound is proved in a general framework which relates the cell-probe complexity of ANN to isoperimetric inequalities in the underlying metric space. A tighter connection between ANN lower bounds and isoperimetric inequalities is established by a stronger richness lemma proved by cell-sampling techniques.
@InProceedings{yin:LIPIcs.ICALP.2016.84, author = {Yin, Yitong}, title = {{Simple Average-Case Lower Bounds for Approximate Near-Neighbor from Isoperimetric Inequalities}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {84:1--84:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.84}, URN = {urn:nbn:de:0030-drops-62010}, doi = {10.4230/LIPIcs.ICALP.2016.84}, annote = {Keywords: nearest neighbor search, approximate near-neighbor, cell-probe model, isoperimetric inequality} }
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