Tolerant identification with Euclidean balls

The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. The identifying codes can be applied, for example, to sensor networks. In this paper, we consider as sensors the set Z^2 where one sensor can check its neighbours within Euclidean distance r. We construct tolerant identifying codes in this network that are robust against some changes in the neighbourhood monitored by each sensor. We give bounds for the smallest density of a tolerant identifying code for general values of r and Delta. We also provide infinite families of values (r,Delta) with optimal such codes and study the case of small values of r.

💡 Research Summary

The paper studies identifying codes on the infinite integer lattice ℤ² when each sensor’s detection region is a Euclidean ball of radius r but may vary up to an additional tolerance Δ. A set C⊆ℤ² is an (r, Δ)‑tolerant identifying code if (i) every vertex u is “r‑dominated” (B_r(u)∩C≠∅) and (ii) for any distinct vertices u, v the set
S_{r,Δ}(u,v)= (B_r(u)\B_{r+Δ}(v)) ∪ (B_r(v)\B_{r+Δ}(u))
intersects C. Condition (ii) guarantees that the pattern of alarms uniquely identifies the faulty vertex even when each sensor’s actual coverage may be any superset of B_r and any subset of B_{r+Δ} containing B_r.

The authors first give a necessary and sufficient existence condition: an (r, Δ)‑code exists iff the “horizontal pattern” S_{r,Δ}((0,0),(-1,0)) is non‑empty. This leads to the definition Δ_m(r)=sup{Δ | (r,Δ)‑code exists}. They show Δ_m(r)≤1 for all r; Δ_m(r)=1 when r is integer, while for non‑integer r they prove a lower bound Δ_m(r)≥1−r²/(r+O(1)) as r→∞, indicating that codes survive a tolerance that shrinks like 1/r.

A central technical contribution is the detailed analysis of the horizontal pattern S_{r,Δ}((0,0),(-1,0)). They introduce three auxiliary quantities:

• x₀(r,Δ): the smallest non‑negative abscissa of a point in the pattern,
• x₁(r,Δ): the largest abscissa where the vertical distance between the two circles is at most 1,
• m(r,Δ): the maximal vertical gap at the farthest column. Using these, they express the exact cardinality of the pattern and derive a density lower bound D(r,Δ) ≥ 1 / |S_{r,Δ}((0,0),(-1,0))|.
  When Δ approaches 1 as Δ=1−ε(r) with ε(r)=Θ(1/r^α) (α≤½), the pattern size is Θ(r·√r·ε(r)²) and the density lower bound becomes Ω(1/(r·√r·ε(r)²)). For fixed Δ, the classical bound D(r,Δ)≥1/(3⌈2r⌉+4) (≈1/(6r)) still holds.

For upper bounds the authors construct codes from two orthogonal lattice lines: L_v^k = { (x,y)∈ℤ² | x ≡ 0 (mod k) } and L_h^k = { (x,y)∈ℤ² | y ≡ 0 (mod k) }. Choosing k = ⌊r⌋ – x₁(r,Δ) guarantees that C_{r,Δ}=L_v^k ∪ L_h^k is an (r,Δ)‑identifying code, with density 2/k. This yields an asymptotic upper bound D(r,Δ) ≤ 4r(2−Δ−√(2−Δ²)) + O(1). When Δ is fixed, the density of optimal codes is Θ(1/r); when Δ=1−ε(r) with ε(r)=Θ(1/r^α), the upper bound matches the lower bound up to constants, giving Θ(1/(r·ε(r)²)).

The paper provides several concrete families of optimal codes. For (r,Δ) = (1, √2−1) they exhibit a periodic code of density ½ that meets the lower bound, showing optimality. More generally, for any integer k≥1, taking r = √(k²+1) and Δ = √(k²+2)−√(k²+1) yields a pattern where the lattice‑line construction attains density ½, proving infinitely many (r,Δ) pairs with optimal density ½.

The authors also treat small radii (r = 1,2,3) in detail. For r=1, Δ=√2−1 they give the optimal density ½ code; for r=2 they identify the exact range of Δ for which a density‑1/3 code exists; for r=3 they provide constructions and bounds showing that the optimal density lies between 1/4 and 1/3 depending on Δ.

Overall, the work extends the theory of identifying codes to a realistic tolerant setting, establishes precise existence thresholds, derives tight asymptotic density bounds, and constructs infinite families of optimal codes. It opens avenues for further research on higher‑dimensional lattices, non‑circular detection regions, and dynamic tolerance models.