SINR Diagrams: Towards Algorithmically Usable SINR Models of Wireless Networks

The rules governing the availability and quality of connections in a wireless network are described by physical models such as the signal-to-interference & noise ratio (SINR) model. For a collection of simultaneously transmitting stations in the plane, it is possible to identify a reception zone for each station, consisting of the points where its transmission is received correctly. The resulting SINR diagram partitions the plane into a reception zone per station and the remaining plane where no station can be heard. SINR diagrams appear to be fundamental to understanding the behavior of wireless networks, and may play a key role in the development of suitable algorithms for such networks, analogous perhaps to the role played by Voronoi diagrams in the study of proximity queries and related issues in computational geometry. So far, however, the properties of SINR diagrams have not been studied systematically, and most algorithmic studies in wireless networking rely on simplified graph-based models such as the unit disk graph (UDG) model, which conveniently abstract away interference-related complications, and make it easier to handle algorithmic issues, but consequently fail to capture accurately some important aspects of wireless networks. The current paper focuses on obtaining some basic understanding of SINR diagrams, their properties and their usability in algorithmic applications. Specifically, based on some algebraic properties of the polynomials defining the reception zones we show that assuming uniform power transmissions, the reception zones are convex and relatively well-rounded. These results are then used to develop an efficient approximation algorithm for a fundamental point location problem in wireless networks.

💡 Research Summary

The paper investigates the geometric structure of reception zones defined by the Signal‑to‑Interference‑plus‑Noise Ratio (SINR) model, a physically realistic description of wireless communication. While many algorithmic studies rely on simplified graph abstractions such as the Unit Disk Graph (UDG), these abstractions ignore cumulative interference and often give misleading results. The authors focus on a uniform‑power network in the Euclidean plane, assume a path‑loss exponent α = 2 (the classic inverse‑square law), and consider a reception threshold β > 1. Under these conditions the SINR inequality for a point p and a transmitter s_i can be rewritten as a rational inequality whose numerator is a polynomial in the coordinates of p.

The core technical contribution is a rigorous proof that each reception zone H_i (the set of points where s_i is heard) is convex and “fat”. Convexity is established by analyzing the polynomial boundary using complex analysis and Sturm’s theorem to count real roots; the authors show that the boundary curve is a single, simple, closed curve that never bends inward, guaranteeing that any line segment between two points in H_i stays inside H_i. Fatness is defined via the ratio between the smallest ball centered at s_i that contains H_i and the largest ball centered at s_i that is fully contained in H_i; the authors prove this ratio is bounded by a constant independent of the network size. They also demonstrate that if β ≤ 1 the convexity property can fail, providing a counter‑example.

These geometric insights have algorithmic implications. The paper tackles the point‑location problem: given a query point p, determine which transmitter’s reception zone (if any) contains p. By pre‑computing, for each transmitter, the maximal inscribed ball and the minimal enclosing ball, the algorithm quickly eliminates most transmitters. The remaining candidates are checked against the exact SINR inequality. The authors present an ε‑approximation scheme whose running time is O(n·polylog n / ε) and whose error can be made arbitrarily small by adjusting ε.

Beyond the point‑location algorithm, the work bridges the gap between realistic physical models and tractable algorithmic frameworks. It validates the intuition that, for realistic thresholds, SINR reception zones behave similarly to the disks used in UDG models, but with provable guarantees that account for interference. This opens the door to adapting many graph‑based techniques (e.g., clustering, routing, topology control) to SINR‑aware versions without sacrificing correctness.

In summary, the paper makes three major contributions: (1) a novel proof of convexity and bounded fatness for SINR reception zones in uniform‑power, α = 2 networks with β > 1; (2) an efficient, provably accurate point‑location algorithm that leverages these geometric properties; and (3) a compelling argument that SINR diagrams can serve as a foundational geometric tool—much like Voronoi diagrams—for the design and analysis of wireless network algorithms, thereby advancing both theoretical understanding and practical protocol design.