Reducing the Randomness Complexity of Property Testing, with an Emphasis on Testing Bipartiteness

Property testers are algorithms whose goal is distinguishing between inputs that have a certain property and inputs which are far from all instances with this property. We show that for a wide variety of properties, there exists no deterministic tester that queries only a sublinear number of input entries. Therefore, most sublinear property testers must be probabilistic algorithms. Nevertheless, we aspire to reduce the randomness complexity of property testers without increasing their query complexity. We also introduce and motivate the Q-R complexity of a tester, which is the number of queries that a tester makes multiplied by its randomness complexity. We reduce the randomness complexity of testers mostly by using randomness-efficient hitters and samplers, rather than the ones that use total independence. This alone achieves a great improvement in the Q-R complexity of many testers. In addition, we focus on the bipartiteness tester presented in [GGR98], and reduce its Q-R complexity. To that end, we not only use randomness-efficient hitters and samplers, but also modify the original tester, in more than one fashion. We also present some general results regarding property testers. In particular, we present a general, non-explicit, scheme to reduce the randomness complexity of all property testers that share the same basic outline.