Faster communication in known topology radio networks

This paper concerns the communication primitives of broadcasting (one-to-all communication) and gossiping (all-to-all communication) in known topology radio networks, i.e., where for each primitive the schedule of transmissions is precomputed in advance based on full knowledge about the size and the topology of the network. The first part of the paper examines the two communication primitives in arbitrary graphs. In particular, for the broadcast task we deliver two new results: a deterministic efficient algorithm for computing a radio schedule of length D + O(log3n), and a randomized algorithm for computing a radio schedule of length D + O(log2n). These results improve on the best currently known D + O(log4n) time schedule due to Elkin and Kortsarz (Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 222---231, 2005). Later we propose a new (efficiently computable) deterministic schedule that uses 2D + Δlog n + O(log3n) time units to complete the gossiping task in any radio network with size n, diameter D and max-degree Δ. Our new schedule improves and simplifies the currently best known gossiping schedule, requiring time $$O(D+\sqrt[{i+2}]{D}\Delta\log^{i+1} n)$$, for any network with the diameter D = Ω(logi+4n), where i is an arbitrary integer constant i ý 0, see Gąsieniec et al. (Proceedings of the 11th International Colloquium on Structural Information and Communication Complexity, vol. 3104, pp. 173---184, 2004). The second part of the paper focuses on radio communication in planar graphs, devising a new broadcasting schedule using fewer than 3D time slots. This result improves, for small values of D, on the currently best known D + O(log3n) time schedule proposed by Elkin and Kortsarz (Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 222---231, 2005). Our new algorithm should be also seen as a separation result between planar and general graphs with small diameter due to the polylogarithmic inapproximability result for general graphs by Elkin and Kortsarz (Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, vol. 3122, pp. 105---116, 2004; J. Algorithms 52(1), 8---25, 2004).