On efficient distributed construction of near optimal routing schemes

Given a distributed network represented by a weighted undirected graph G=(V,E) on n vertices, and a parameter k , we devise a randomized distributed algorithm that whp computes a routing scheme in O(n^1/2+1/k+D)· n^o(1) rounds, where D is the hop-diameter of the network. Moreover, for odd k , the running time of our algorithm is O(n^1/2 + 1/(2k) + D) · n^o(1) . Our running time nearly matches the lower bound of Ω̃(n^1/2+D) rounds (which holds for any scheme with polynomial stretch). The routing tables are of size Õ(n^1/k) , the labels are of size O(klog ^2n) , and every packet is routed on a path suffering stretch at most 4k-5+o(1) . Our construction nearly matches the state-of-the-art for routing schemes built in a centralized sequential manner. The previous best algorithms for building routing tables in a distributed small messages model were by Lenzen and Patt-Shamir (In: Symposium on theory of computing conference, STOC’13, Palo Alto, CA, USA, 2013 ) and Lenzen and Patt-Shamir (In: Proceedings of the 2015 ACM symposium on principles of distributed computing, PODC 2015, Donostia-San Sebastián, Spain, 2015 ). The former has similar properties but suffers from substantially larger routing tables of size O(n^1/2+1/k) , while the latter has sub-optimal running time of Õ(min{(nD)^1/2· n^1/k,n^2/3+2/(3k)+D}) .