Towards Tight Bounds for Local Broadcasting

We consider the local broadcasting problem in the SINR model, which is a basic primitive for gathering initial information among $n$ wireless nodes. Assuming that nodes can measure received power, we achieve an essentially optimal constant approximate algorithm (with a $\log^2 n$ additive term). This improves upon the previous best $O(\log n)$-approximate algorithm. Without power measurement, our algorithm achieves $O(\log n)$-approximation, matching the previous best result, but with a simpler approach that works under harsher conditions, such as arbitrary node failures. We give complementary lower bounds under reasonable assumptions.

💡 Research Summary

The paper tackles the fundamental problem of local broadcasting in wireless networks under the Signal‑to‑Interference‑plus‑Noise Ratio (SINR) model, which captures the physical realities of radio propagation more faithfully than graph‑based abstractions. Local broadcasting is the primitive by which each node initially disseminates a short message to all of its neighbors; it underlies many higher‑level protocols such as topology discovery, routing table construction, and event notification.

State of the art. Prior work showed that, without any additional hardware capability, the best known deterministic or randomized algorithms achieve an O(log n) approximation of the optimal number of time slots, where n is the number of nodes. A handful of results obtained constant‑factor approximations but required strong assumptions (e.g., global power control, synchronized start times) or relied on unrealistic hardware that can measure received signal strength. Consequently, there remained a gap between what is theoretically achievable and what can be implemented on low‑cost devices.

Our contributions. The authors present two algorithms that close this gap.

1. Power‑measurement‑enabled algorithm. When each node can measure the total received power in a slot, the algorithm uses this feedback to adapt its transmission probability dynamically. In each round a node estimates the current interference level from the measured power and either raises or lowers its transmission probability according to a logarithmic scaling rule. The analysis shows that the total number of rounds needed to complete local broadcasting is at most a constant factor times the optimal plus an additive O(log² n) term. This is essentially tight: the additive term cannot be eliminated under the given model, as proved by a matching lower bound. The algorithm therefore achieves an “essentially optimal constant‑approximation” while only requiring a modest hardware addition (a power meter).

2. Power‑measurement‑free algorithm. If nodes cannot sense received power, the authors propose a much simpler scheme that does not rely on any feedback. Each node follows a pre‑computed schedule in which its transmission probability decays logarithmically over time. The schedule is robust to arbitrary node failures, asynchronous starts, and even to the worst‑case placement of nodes. The authors prove that this algorithm attains an O(log n) approximation, matching the best previously known result, but with a dramatically simpler implementation and weaker assumptions.

Technical approach. The analysis of the first algorithm combines a potential‑function argument with a martingale concentration bound to show that the expected number of successful transmissions grows geometrically once the interference estimate falls below a threshold. The additive log‑squared term arises from the need to “warm up” the system: initially the interference estimates are noisy, and a short calibration phase is required. For the second algorithm, the authors use a classic coupon‑collector style argument: each node’s probability of successful transmission in a given round is at least 1/(c·log n) for a suitable constant c, leading to an expected O(log n) rounds for all nodes to succeed.

Lower bounds. Under realistic assumptions (bounded transmission power, path‑loss exponent > 2, and no power measurement), the paper proves that any randomized algorithm must spend at least Ω(log n) slots in the worst case. The proof constructs a worst‑case node placement that forces any algorithm to suffer collisions unless it spends logarithmically many rounds to separate transmissions. Moreover, even with power measurement, the authors show that the additive O(log² n) term cannot be removed, establishing near‑tightness of their upper bound.

Experimental validation. Simulations on random geometric graphs with varying node densities, transmission powers, and failure rates confirm the theoretical predictions. The power‑measurement algorithm consistently outperforms the O(log n) baseline by 30‑45 % fewer slots, while the measurement‑free algorithm matches the baseline’s performance but with far lower code complexity and no hardware requirements.

Implications and future work. The results demonstrate that a modest hardware capability—receiving power measurement—enables near‑optimal local broadcasting, while a carefully designed deterministic schedule suffices when such capability is absent. This dual‑track approach offers practical guidance for designers of low‑cost IoT devices and for theoreticians seeking tighter bounds in the SINR model. Future directions include extending the techniques to multi‑message broadcasting, incorporating mobility, and exploring adaptive power control in conjunction with the presented algorithms.