Improved algorithms for latency minimization in wireless networks

Alexander Fanghänel, Thomas Kesselheim, Berthold Vöcking

Abstract: In the interference scheduling problem, one is given a set of n<math><mi is="true">n</mi></math> communication requests described by source–destination pairs of nodes from a metric space. The nodes correspond to devices in a wireless network. Each pair must be assigned a power level and a color such that the pairs in each color class can communicate simultaneously at the specified power levels. The feasibility of simultaneous communication within a color class is defined in terms of the Signal to Interference plus Noise Ratio (SINR) that compares the strength of a signal at a receiver to the sum of the strengths of other signals. The objective is to minimize the number of colors as this corresponds to the time needed to schedule all requests.We introduce an instance-based measure of interference, denoted by I<math><mi is="true">I</mi></math>, that enables us to improve on previous results for the interference scheduling problem. We prove the upper and lower bounds in terms of I<math><mi is="true">I</mi></math> on the number of steps needed for scheduling a set of requests. For general power assignments, we prove a lower bound of Ω(I/(logΔlogn))<math><mi is="true">Ω</mi><mrow is="true"><mo is="true">(</mo><mi is="true">I</mi><mo is="true">/</mo><mrow is="true"><mo is="true">(</mo><mo is="true">log</mo><mi is="true">Δ</mi><mo is="true">log</mo><mi is="true">n</mi><mo is="true">)</mo></mrow><mo is="true">)</mo></mrow></math> steps, where Δ<math><mi is="true">Δ</mi></math> denotes the aspect ratio of the metric. When restricting to the two-dimensional Euclidean space (as in the previous work) the bound improves to Ω(I/logΔ)<math><mi is="true">Ω</mi><mrow is="true"><mo is="true">(</mo><mi is="true">I</mi><mo is="true">/</mo><mo is="true">log</mo><mi is="true">Δ</mi><mo is="true">)</mo></mrow></math>. Alternatively, when restricting to linear power assignments, the lower bound improves even to Ω(I)<math><mi is="true">Ω</mi><mrow is="true"><mo is="true">(</mo><mi is="true">I</mi><mo is="true">)</mo></mrow></math>. The lower bounds are complemented by an efficient algorithm computing a schedule for linear power assignments using only O(Ilogn)<math><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><mi is="true">I</mi><mo is="true">log</mo><mi is="true">n</mi><mo is="true">)</mo></mrow></math> steps. A more sophisticated algorithm computes a schedule using even only O(I+log2n)<math><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><mi is="true">I</mi><mo is="true">+</mo><msup is="true"><mrow is="true"><mo is="true">log</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mi is="true">n</mi><mo is="true">)</mo></mrow></math> steps. For dense instances in the two-dimensional Euclidean space, this gives a constant factor approximation for scheduling under linear power assignments, which shows that the price for using linear (and, hence, energy-efficient) power assignments is bounded by a factor of O(logΔ)<math><mi mathvariant="script" is="true">O</mi><mrow is="true"><mo is="true">(</mo><mo is="true">log</mo><mi is="true">Δ</mi><mo is="true">)</mo></mrow></math>.In addition, we extend these results for single-hop scheduling to multi-hop scheduling and combined scheduling and routing problems, where our analysis generalizes the previous results towards general metrics and improves on the previous approximation factors.