Faster Approximate Distance Queries and Compact Routing in Sparse Graphs

A distance oracle is a compact representation of the shortest distance matrix of a graph. It can be queried to retrieve approximate distances and corresponding paths between any pair of vertices. A lower bound, due to Thorup and Zwick, shows that a distance oracle that returns paths of worst-case stretch (2k − 1) must require space Ω(n) for graphs over n nodes. The hard cases that enforce this lower bound are, however, rather dense graphs with average degree Ω(n). We present distance oracles that, for sparse graphs, substantially break the lower bound barrier at the expense of higher query time. For any 1 ≤ α ≤ n, our distance oracles can return stretch 2 paths using O(m + n/α) space and stretch 3 paths using O(m + n/α) space, at the expense of O(αm/n) query time. By setting appropriate values of α, we get the first distance oracles that have size linear in the size of the graph, and return constant stretch paths in non-trivial query time. The query time can be further reduced to O(α), by using an additional O(mα) space for all our distance oracles, or at the cost of a small constant additive stretch. We use our stretch 2 distance oracle to design a compact routing scheme that requires Õ(n)memory at each node and, after a handshaking phase, routes along paths with worst-case stretch 2. Moreover, supported by large-scale simulations on graphs including the AS-level Internet graph, we argue that our stretch-2 scheme would be simple and efficient to implement as a distributed compact routing protocol. ∗An earlier version of this paper appeared in INFOCOM 2011 [1]. The extended version presents results that improve upon the results presented in the conference version; significantly more simplified presentation and proofs for the results in the conference version; and in addition, distance oracles for unweighted graphs. University of Illinois, Urbana, IL 61801. Email: [email protected]. University of Illinois, Urbana, IL 61801. Email: [email protected]. University of Illinois, Urbana, IL 61801. Email: [email protected].