Winding Numbers for Networks with Weak Angular Data

We consider a network of nodes in the plane whose locations are unknown but for which the nodes establish communication links based on proximity, as in an ad hoc wireless network. Given a node in the network and a cycle disjoint from the node, we want the network to compute the winding number and determine, in particular, whether the cycle surrounds the node in the plane. We assume that nodes possess unique IDs, a sharp notion of proximity, and, to various degrees, a weak form of angular ordering between neighbors. We give algorithms for computing winding numbers in networks with varying sensing capabilities and architectures with regards to angular data of neighbors. In all instances, we are motivated to compute the minimal sensing required to compute winding data rigorously. This takes the form of investigating varying amounts of uncertainty in angular readings.