 
Summary: On the power of two, three and four probes
Noga Alon
Uriel Feige
December 4, 2008
Abstract
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 nonadaptive (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
4n2
+ 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
nonadaptive (n, m, s, 4)schemes with s = o(m) for all n = o(m). Therefore, three adaptive
probes or four nonadaptive 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.
Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences,
