# Wireless Connectivity and Capacity

Abstract:

Given $n$ wireless transceivers located in a plane, a fundamental problem in wireless communications is to construct a strongly connected digraph on them such that the constituent links can be scheduled in fewest possible time slots, assuming the SINR model of interference.
In this paper, we provide an algorithm that connects an arbitrary point set in $O(\log n)$ slots, improving on the previous best bound of $O(\log^2 n)$ due to Moscibroda. This is complemented with a super-constant lower bound on our approach to connectivity. An important feature is that the algorithms allow for bi-directional (half-duplex) communication.
One implication of this result is an improved bound of $\Omega(1/\log n)$ on the worst-case capacity of wireless networks, matching the best bound known for the extensively studied average-case.
We explore the utility of oblivious power assignments, and show that essentially all such assignments result in a worst case bound of $\Omega(n)$ slots for connectivity. This rules out a recent claim of a $O(\log n)$ bound using oblivious power. On the other hand, using our result we show that $O(\min(\log \Delta, \log n \cdot (\log n + \log \log \Delta)))$ slots suffice, where $\Delta$ is the ratio between the largest and the smallest links in a minimum spanning tree of the points.
Our results extend to the related problem of minimum latency aggregation scheduling, where we show that aggregation scheduling with $O(\log n)$ latency is possible, improving upon the previous best known latency of $O(\log^3 n)$. We also initiate the study of network design problems in the SINR model beyond strong connectivity, obtaining similar bounds for biconnected and $k$-edge connected structures.

Guest: Magnus M. Halldorsson

Host: Merav Parter

# Misleading Stars: What Cannot Be Measured in the Internet?

## Yvonne Anne Pignolet, Stefan Schmid, Gilles Tredan

Abstract:
Traceroute measurements are one of our main instruments to shed light onto the structure and properties of today’s complex networks such as the Internet. This paper studies the feasibility and infeasibility of inferring the network topology given traceroute data from a worst-case perspective, i.e., without any probabilistic assumptions on, e.g., the nodes’ degree distribution. We attend to a scenario where some of the routers are anonymous, and propose two fundamental axioms that model two basic assumptions on the traceroute data: (1) each trace corresponds to a real path in the network, and (2) the routing paths are at most a factor 1/alpha off the shortest paths, for some parameter alpha in (0,1]. In contrast to existing literature that focuses on the cardinality of the set of (often only minimal) inferrable topologies, we argue that a large number of possible topologies alone is often unproblematic, as long as the networks have a similar structure. We hence seek to characterize the set of topologies inferred with our axioms. We introduce the notion of star graphs whose colorings capture the differences among inferred topologies; it also allows us to construct inferred topologies explicitly. We find that in general, inferrable topologies can differ significantly in many important aspects, such as the nodes’ distances or the number of triangles. These negative results are complemented by a discussion of a scenario where the trace set is best possible, i.e., “complete”. It turns out that while some properties such as the node degrees are still hard to measure, a complete trace set can help to determine global properties such as the connectivity.

Guest: Stefan Schmid
Host: Zvi Lotker