Tight Bounds for Deterministic h-Shot Broadcast in Ad-Hoc Directed Radio Networks

We consider the classical broadcast problem in ad-hoc (that is, unknown topology) directed radio networks with no collision detection, under the additional assumption that at most h transmissions (shots) are available per node. We focus on adaptive deterministic protocols for small values of h. We provide asymptotically matching lower and upper bounds for the cases h=2 and h=3. While for h=2 our bound is quadratic, similar to the bound obtained for oblivious protocols, for h=3 we prove a sub-quadratic bound of Theta(n^2 log log n / log n), where n is the number of nodes in the network. The latter is the first result showing an adaptive algorithm which is asymptotically faster than oblivious h-shot broadcast protocols, for which a tight quadratic bound is known for every constant h. Our upper bound for h=3 is constructive, making use of constructions of graphs with large girth. We also show an improved upper bound of O(n^(1+alpha/sqrt{h})) for h >= 4, where alpha is an absolute constant independent of h. Our upper bound for h >= 4 is non-constructive.