On the bitprobe complexity of two probe adaptive schemes

Abstract

Bitprobe complexity of the static membership problem has been widely investigated since Burhman et al. (2000) studied the worst-case bounds for a whole range of membership problems. Though tremendous progress has been made in recent years, as is evident from the remarkable papers due to Alon and Feige (2009) [1] , Viola (2012) [8] , and Lewenstein et al. (2014), among others, Nicholson et al. (2013) [5] in their survey of the state of the art of the area noted that even the first non-trivial scenario of two probe adaptive schemes storing two elements of the universe is not completely settled, in that the lower bound for this problem is still open. We propose a proof (Kesh and Sharma, 2019) [3] for the lower bound for the same problem, but for a restricted class of schemes. We believe that this proof makes progress over the ideas proposed by Radhakrishnan et al. (2001, 2010) towards the proof. It is our belief that the single restriction imposed on the general class of schemes towards our proof faithfully captures the dependencies among the elements that share the same bit in the datastructure, and we hope that the general lower bound proof can be developed using similar ideas.