{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7630","name":"On Multiple Input Problems in Property Testing","abstract":"We consider three types of multiple input problems in the context of property testing. Specifically, for a property Pi (of n-bit long strings), a proximity parameter epsilon, and an integer m, we consider the following problems:\r\n\r\n(1) Direct m-Sum Problem for Pi and epsilon: Given a sequence of m inputs, output a sequence of m bits such that for each i in [m] the i-th bit satisfies the requirements from an epsilon-tester for Pi regarding the i-th input; that is, for each i, the i-th output bit should be 1 (w.p. at least 2\/3) if the i-th input is in Pi, and should be 0 (w.p. at least 2\/3) if the i-th input is epsilon-far from Pi.\r\n\r\n(2) Direct m-Product Problem for Pi and epsilon: Given a sequence of m inputs, output 1 (w.p. at least 2\/3) if all inputs are in Pi, and output 0 (w.p. at least 2\/3) if at least one of the inputs is epsilon-far from Pi. \r\n\r\n(3) The m-Concatenation Problem for Pi and epsilon: Here one is required to epsilon-test the m-product of Pi; that is, the property that consists of the m-wise Cartesian product of Pi.\r\n\r\nWe show that the query complexity of the first two problems\r\nis Theta(m) times the query complexity of epsilon-testing Pi,\r\nwhereas (except in pathological cases) the query complexity\r\nof the third problem is almost of the same order of magnitude\r\nas the query complexity of the problem of epsilon-testing Pi.\r\nAll upper bounds are shown via efficient reductions.\r\n\r\nWe also consider the nonadaptive and one-sided error versions of these problems. The only significant deviation from the picture in the general (adaptive and two-sided error) model is that the one-sided error query complexity of the Direct Product Problem equals Theta(m) times the (two-sided error) query complexity of epsilon-testing Pi plus Theta(1) times the one-sided error query complexity of epsilon-testing Pi.","keywords":["Property Testing","Direct Sum Theorems","Direct Product Theorems","Adaptive vs Nonadaptive queries","One-Sided Error vs Two-Sided Error"],"author":{"@type":"Person","name":"Goldreich, Oded","givenName":"Oded","familyName":"Goldreich"},"position":49,"pageStart":704,"pageEnd":720,"dateCreated":"2014-09-04","datePublished":"2014-09-04","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Goldreich, Oded","givenName":"Oded","familyName":"Goldreich"},"copyrightYear":"2014","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2014.704","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6231","volumeNumber":28,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2014)","dateCreated":"2014-09-04","datePublished":"2014-09-04","editor":[{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"Rolim, Jos\u00e9","givenName":"Jos\u00e9","familyName":"Rolim"},{"@type":"Person","name":"Devanur, Nikhil R.","givenName":"Nikhil R.","familyName":"Devanur"},{"@type":"Person","name":"Moore, Cristopher","givenName":"Cristopher","familyName":"Moore"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7630","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":"#volume6231"}}}