Compact roundtrip routing with topology-independent node names

Marta Arias, Lenore J. Cowen, Kofi A. Laing
2008 Journal of computer and system sciences (Print)  
Consider a strongly connected directed weighted network with n nodes. This paper presents compact roundtrip routing schemes withÕ( √ n) sized local tables 4 and stretch 6 for any strongly connected directed network with arbitrary edge weights. A scheme with local tables of sizeÕ( −1 n 2/k ) and stretch min((2 k/2 − 1)(k + ), 8k 2 + 4k − 4), for any > 0 is also presented in the case where edge weights are restricted to be polynomially-sized. Both results are for the topology-independent
more » ... model. These are the first topology-independent results that apply to routing in directed networks. less sense in a dynamic network, where the network topology is changing over time. There are serious consistency and continuity issues if the identifying label of a node changes as network topology evolves. In such a model, a node's identifying label needs to be decoupled from network topology. In fact, network nodes should be allowed to choose arbitrary names (subject to the condition that node names are unique), and packets destined for a particular node enter the network with the node name only, with no additional topological address information. In the grid example above, the packet would come with a destination name independent of its (x, y) coordinates, and would have to learn how to associate its (x, y) coordinates with its name from the local routing tables as it wandered the network. Below, we call this the TINN model (for topology independent node names). Awerbuch et al., in the same paper where they introduced the TINN model, produced the first TINN compact routing schemes for undirected networks. It achieved a stretch of O(k 2 · 9 k ) using O(kn 1/k log n) space in each node [6] . A paper of Awerbuch and Peleg in the following year [8], presented an alternate scheme with a polynomial space/stretch tradeoff, achieving a stretch of O(k 2 ) using O(kn 1/k log n log D) space in each node, where D is the diameter of the network. In 2003, joint with Rajaraman and Taka [3,4], we presented TINN compact routing schemes that use local routing tables of sizeÕ(n 1/2 ), O(log 2 n)-sized packet headers, and obtained a stretch of 5. For smaller table-size requirements, the ideas in these schemes were generalized to a scheme that uses O(log 2 n)-sized headers andÕ(k 2 n 2/k )sized tables, and achieved a stretch of min{1 + (k − 1)(2 k/2 − 2), 16k 2 − 8k}. The following year Abraham et al. improved the stretch of the Arias et al. scheme to 3, which is optimal [2]. Additionally, for the special case of tree networks, Laing [27,28] presented a TINN compact routing algorithm that usesÕ(n 1/k ) space, O(log n) headers, and achieves stretch 2 k − 1. Abraham, Gavoille and Malkhi [23] recently obtained O(k) stretch withÕ(n 1/k ) table size and arbitrary edge weights which is asymptotically optimal [22]. All the low-stretch schemes cited above, as well as the schemes with exponential tradeoffs are highly dependent on small dominating set landmark selection schemes first pioneered by Awerbuch and Peleg [6]; the polynomial tradeoff schemes are based on ideas from their sparse partitions data structures [8], both developed for compact routing in the name-dependent model. All previous papers in the TINN model, as do our current results, depend heavily on the distributed dictionary ideas pioneered by [6, 30] . Arias et al. [3, 4] also introduced a novel coloring idea that was subsequently improved and extended by Laing [27, 28] and Abraham et al. [2] . However, all previous work in the TINN model has been only for undirected networks. In fact, no results are known for constructing (one-way, topology-dependent) compact routing schemes on directed networks; and it appears that it is hard to design "compact" routing schemes when the network is directed. For example, it is shown in [16] that distinguishing between pairs of vertices at distances 2 and ∞ even in unweighted directed graphs is at least as hard as Boolean matrix multiplication. Roditty et al. [35] observe that sparse spanners do not exist for all digraphs, and there is further discussion in [13, 41] . Cowen and Wagner [11, 13] made the observation that in directed graphs, instead of bounding the length of a one-way path from node x to node y in terms of the shortest distance d(x, y), we could bound the length of a roundtrip from node x through node y in terms of a shortest cycle between the two nodes, which is of length d(x, y) + d (y, x). This would account for a packet and its acknowledgment, for example. As observed by Cowen and Wagner [11, 13] , sparse roundtrip spanners do exist in this model, and can be used as a basis for compact roundtrip routing schemes in directed networks where they presented the first sublinear space universal compact routing schemes for directed networks, obtaining a stretch of 2 k+1 for tables of sizẽ O(kn 3 k+1 2 · 3 k ). The (name-dependent) roundtrip routing scheme of [11, 13] was subsequently improved by Roditty et al. [35] to a stretch of 4k + forÕ( k 2 n 1/k ) using ideas of Awerbuch et al. [7], Cohen [10] and Thorup and Zwick [38]. Here we present the first universal compact roundtrip routing schemes for (positive weighted) directed networks in the TINN model. A preliminary version of this paper appeared in PODC 2003 [5].
doi:10.1016/j.jcss.2007.09.001 fatcat:cbmtt73q5vg63cyatup7k4wnnm