Saving probe bits by cube domination

Paper i proceeding, 2018

We consider the problem of storing a single element from an m-element set as
a binary string of optimal length, and comparing any queried string to the
stored string without reading all bits. This is the one-element version of the
problem of membership testing in the bit probe model, and solutions can serve
as building blocks of general membership testers. Our principal contribution is
the equivalence of saving probe bits with some generalized notion of domination
in hypercubes. This domination variant requires that every vertex outside the
dominating set belongs to a sub-hypercube, of fixed dimension, in which all
other vertices belong to in the dominating set. This fixed dimension equals the
number of saved probe bits. We give specific constructions showing that up to
three probe bits can be ignored when m is far enough from the next larger
power of 2. The main technical idea is to use low-dimensional (grid)
relaxations of the problem. The design of optimal schemes remains an open
problem, however one has to notice that even usual domination in hypercubes is
far from being completely understood.