Beep-and-Sleep: Message and Energy Efficient Set Cover

This article considers message and energy-efficient distributed algorithms for the SetCover Problem. Given a ground set U of n elements S m subsets U, we aim to find minimal number these that contain all elements. In default setup this problem, each has bidirected communication link with element it contains. results in graph n+m nodes degree Δ. The value Δ denotes maximal graph, i.e., maximum subsets' sizes sets an is contained in. We present algorithm Beeping model only relies on carrier-sensing. synchronous time step, node can either listen channel or beep. A listening learns if one more its neighbors beeped none beeped. particular, neither which nor how many exactly. model, runs O(k3) expected approximation ratio O(Δ3/klog2⁡Δ). k∈[3,log⁡Δ] parameter lets us trade runtime similar celebrated by Kuhn Wattenhofer [PODC '03]. Our next result O(k2)-time O˜(Δ12+1k(n+m))-message (where O˜(⋅) hides polylogarithmic factors) O(Δ1klog⁡Δ) KT0-Congest model. variant well-known Congest proceeds rounds, send distinct size O(log⁡(n+m)) neighbors. Further, unique identifier O(log⁡(n+m)). However, crucial aspect do not know their neighbors' identifiers. almost optimal concerning complexity as show there are hard instances require Ω(Δ12−ϵm) messages constant ratio.