Error resilient space partitioning

In this paper we consider a new type of space partitioning which bridges the gap between continuous and discrete spaces in an error resilient way. It is motivated by the problem of rounding noisy measurements from some continuous space such as ℝd to a discrete subset of representative values, in which each tile in the partition is defined as the preimage of one of the output points. Standard rounding schemes seem to be inherently discontinuous across tile boundaries, but in this paper we show how to make it perfectly consistent (with error resilience ∈) by guaranteeing that any pair of consecutive measurements X1 and X2 whose L2 distance is bounded by ∈ will be rounded to the same nearby representative point in the discrete output space. We achieve this resilience by allowing a few bits of information about the first measurement X1 to be unidirectionally communicated to and used by the rounding process of the second measurement X2. Minimizing this revealed information can be particularly important in privacy-sensitive applications such as COVID-19 contact tracing, in which we want to find out all the cases in which two persons were at roughly the same place at roughly the same time, by comparing cryptographically hashed versions of their itineraries in an error resilient way. The main problem we study in this paper is characterizing the achievable tradeoffs between the amount of information provided and the error resilience for various dimensions. We analyze the problem by considering the possible colored tilings of the space with k available colors, and use the color of the tile in which X1 resides as the side information. We obtain our upper and lower bounds with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Sperner's lemma, and Cech cohomology. In particular, we show that when Xi ∈ ℝd, communicating log2(d + 1) bits of information is both sufficient and necessary (in the worst case) to achieve positive resilience, and when d=3 we obtain a tight upper and lower asymptotic bound of (0.561⋯)k1/3 on the achievable error resilience when we provide log2(k) bits of information about X1's color.