Even Simpler Distance Labeling for (Sparse) Graphs

We propose a labeling scheme for unweighted undirected graphs, that is an assignment of binary labels to vertices of a graph such that the distance between any pair of vertices can be decoded solely from their labels. Recently, Alstrup, Dahlgaard, Knudsen, and Porat (2015) have shown a labeling scheme for sparse graphs with labels of size $O(\frac{n}{D} \log^2 D)$ where $D = \frac{\log n}{\log \frac{m+n}{n}}$. We present a simpler approach achieving size $O(\frac{n}{D} \log D)$ with $O(n^\varepsilon)$ decoding time. Based on similar techniques we also present a scheme for general graphs with labels of size $\frac{\log 3}{2}n + o(n)$ with small decoding time (any $\omega(1)$ decoding time is possible), or labels of size $(\frac{\log 3}{2}+\varepsilon) n$ and decoding time $O(1)$. This almost matches the currently best scheme of Alstrup, Gavoille, Halvorsen, and Petersen (2015) which uses labels of size $\frac{\log 3}{2}n + o(n)$ with decoding time $O(1)$. We believe our simple techniques are of independent value and provide a desirable simplification of the previous approaches.