{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8708","name":"Deterministic Time-Space Trade-Offs for k-SUM","abstract":"Given a set of numbers, the k-SUM problem asks for a subset of k numbers that sums to zero. When the numbers are integers, the time and space complexity of k-SUM is generally studied in the word-RAM model; when the numbers are reals, the complexity is studied in the real-RAM model, and space is measured by the number of reals held in memory at any point. We present a time and space efficient deterministic self-reduction for the k-SUM problem which holds for both models, and has many interesting consequences. To illustrate:\r\n\r\n- 3-SUM is in deterministic time O(n^2*lg(lg(n))\/lg(n)) and space O(sqrt(n*lg(n)\/lg(lg(n)))). In general, any polylogarithmic-time improvement over quadratic time for 3-SUM can be converted into an algorithm with an identical time improvement but low space complexity as well.\r\n\r\n- 3-SUM is in deterministic time O(n^2) and space O(sqrt(n)), derandomizing an algorithm of Wang.\r\n\r\n- A popular conjecture states that 3-SUM requires n^{2-o(1)} time on the word-RAM. We show that the 3-SUM Conjecture is in fact equivalent to the (seemingly weaker) conjecture that every O(n^{.51})-space algorithm for 3-SUM requires at least n^{2-o(1)} time on the word-RAM.\r\n\r\n- For k >= 4, k-SUM is in deterministic O(n^{k-2+2\/k}) time and O(sqrt(n)) space.","keywords":["3SUM","kSUM","time-space tradeoff","algorithm"],"author":[{"@type":"Person","name":"Lincoln, Andrea","givenName":"Andrea","familyName":"Lincoln"},{"@type":"Person","name":"Vassilevska Williams, Virginia","givenName":"Virginia","familyName":"Vassilevska Williams"},{"@type":"Person","name":"Wang, Joshua R.","givenName":"Joshua R.","familyName":"Wang"},{"@type":"Person","name":"Williams, R. Ryan","givenName":"R. Ryan","familyName":"Williams"}],"position":58,"pageStart":"58:1","pageEnd":"58:14","dateCreated":"2016-08-23","datePublished":"2016-08-23","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Lincoln, Andrea","givenName":"Andrea","familyName":"Lincoln"},{"@type":"Person","name":"Vassilevska Williams, Virginia","givenName":"Virginia","familyName":"Vassilevska Williams"},{"@type":"Person","name":"Wang, Joshua R.","givenName":"Joshua R.","familyName":"Wang"},{"@type":"Person","name":"Williams, R. Ryan","givenName":"R. Ryan","familyName":"Williams"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2016.58","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6258","volumeNumber":55,"name":"43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)","dateCreated":"2016-08-23","datePublished":"2016-08-23","editor":[{"@type":"Person","name":"Chatzigiannakis, Ioannis","givenName":"Ioannis","familyName":"Chatzigiannakis"},{"@type":"Person","name":"Mitzenmacher, Michael","givenName":"Michael","familyName":"Mitzenmacher"},{"@type":"Person","name":"Rabani, Yuval","givenName":"Yuval","familyName":"Rabani"},{"@type":"Person","name":"Sangiorgi, Davide","givenName":"Davide","familyName":"Sangiorgi"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8708","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6258"}}}