Abstract
A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. We propose a series of new labeling schemes within the framework of so-called hub labeling (HL, also known as landmark labeling or 2-hop-cover labeling), in which each node u stores its distance to all nodes from an appropriately chosen set of hubs \(S(u) \subseteq V\). For a queried pair of nodes (u, v), the length of a shortest \(u\!-\!v\)-path passing through a hub node from \(S(u)\cap S(v)\) is then used as an upper bound on the distance between u and v.
We present a hub labeling which allows us to decode exact distances in sparse graphs using labels of size sublinear in the number of nodes. For graphs with at most n nodes and average degree \(\varDelta \), the tradeoff between label bit size L and query decoding time T for our approach is given by \(L = \mathcal {O}(n \log \log _\varDelta T / \log _\varDelta T)\), for any \(T \le n\). Our simple approach is thus the first sublinear-space distance labeling for sparse graphs that simultaneously admits small decoding time (for constant \(\varDelta \), we can achieve any \(T=\omega (1)\) while maintaining \(L=o(n)\)), and it also provides an improvement in terms of label size with respect to previous slower approaches.
By using similar techniques, we then present a 2-additive labeling scheme for general graphs, i.e., one in which the decoder provides a 2-additive-approximation of the distance between any pair of nodes. We achieve almost the same label size-time tradeoff \(L = \mathcal {O}(n \log ^2 \log T / \log T)\), for any \(T \le n\). To our knowledge, this is the first additive scheme with constant absolute error to use labels of sublinear size. The corresponding decoding time is then small (any \(T=\omega (1)\) is sufficient).
We believe all of our techniques are of independent value and provide a desirable simplification of previous approaches.
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Notes
- 1.
For the sake of sanity of the notation, we define \(\log x = \max (1, \log _2(x))\).
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Acknowledgments
Most of the work was done while PU was affiliated to Aalto University, Finland. Research partially supported by the National Science Centre, Poland - grant number 2015/17/B/ST6/01897.
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Gawrychowski, P., Kosowski, A., Uznański, P. (2016). Sublinear-Space Distance Labeling Using Hubs. In: Gavoille, C., Ilcinkas, D. (eds) Distributed Computing. DISC 2016. Lecture Notes in Computer Science(), vol 9888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53426-7_17
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