Undirected st-connectivity is important both for its applications in network problems, and for its theoretical connections with logspace complexity. Classically, a long line of work led to a time-space tradeoff of T = Õ(n²/S) for any S such that S = Ω(log(n)) and S = O(n²/m). Surprisingly, we show that quantumly there is no nontrivial time-space tradeoff: there is a quantum algorithm that achieves both optimal time Õ(n) and space O(log(n)) simultaneously. This improves on previous results, which required either O(log(n)) space and Õ(n^{1.5}) time, or Õ(n) space and time. To complement this, we show that there is a nontrivial time-space tradeoff when given a lower bound on the spectral gap of a corresponding random walk.
@InProceedings{apers_et_al:LIPIcs.ESA.2023.10, author = {Apers, Simon and Jeffery, Stacey and Pass, Galina and Walter, Michael}, title = {{(No) Quantum Space-Time Tradeoff for USTCON}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {10:1--10:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.10}, URN = {urn:nbn:de:0030-drops-186636}, doi = {10.4230/LIPIcs.ESA.2023.10}, annote = {Keywords: Undirected st-connectivity, quantum walks, time-space tradeoff} }
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