On the power of two, three and four probes

An adaptive (n, m, s, t)-scheme is a deterministic scheme for encoding a vector X of m bits with at most n ones by a vector Y of s bits, so that any bit of X can be determined by t adaptive probes to Y. A non-adaptive (n, m, s, t)-scheme is defined analogously. The study of such schemes arises in the investigation of the static membership problem in the bitprobe model. Answering a question of Buhrman, Miltersen, Radhakrishnan and Venkatesh [SICOMP 2002] we present adaptive (n, m, s, 2) schemes with s < m for all n satisfying 4n ² + 4n < m and adaptive (n, m, s, 2) schemes with s = o(m) for all n = o(log m). We further show that there are adaptive (n, m, s, 3)-schemes with s = o(m) for all n = o(m), settling a problem of Radhakrishnan, Raman and Rao [ESA 2001], and prove that there are non-adaptive (n, m, s, 4)-schemes with s = o(m) for all n = o(m). Therefore, three adaptive probes or four non-adaptive probes already suffice to obtain a significant saving in space compared to the total length of the input vector. Lower bounds are discussed as well.