Saving probe bits by cube domination

Paper in 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.