Lower and upper bounds for deterministic convergecast with labeling schemes

In wireless networks, broadcast and convergecast are the two most used communication primitives. Broadcast instructs a specific sink (or root) node to send a message to each node in the network. Convergecast instructs each node in the network to send a message to the sink. Due to the collision, without labels, deterministic convergecast is impossible even in a three-node network. Therefore, networking solutions for convergecast are based on probabilistic approaches or use underlying probabilistic medium access protocols such as CSMA/CA or CSMA/CD. In this paper, we focus on deterministic convergecast algorithms enhanced with labeling schemes for wireless networks in collision-presence environment. We investigate two communication modes: half-duplex (nodes either transmit or receive but not both at the same time) and full-duplex (nodes can transmit and receive data at the same time). For these two modes we investigate lower and upper bounds for the time and size of labeling. Even though broadcast and convergecast are similar, we prove that, contrary to broadcast, deterministic convergecast cannot be solved with short labels for some topologies. That is, Ω(log⁡(Δ)) bits are necessary to solve deterministically convergecast where Δ is the maximal degree of a node in the network. We also prove that Ω(n) communication time slots are required, where n is the size of network. We provide solutions that are optimal in terms of time (transmission rounds), and by far, the closest to the lower bound in terms of space (message size) for arbitrary scenarios.