Optimal Distance Labeling Schemes for Trees

Labeling schemes seek to assign a short label to each node in a network, so that a function on two nodes can be computed by examining their labels alone. For the particular case of trees, optimal bounds (up to low order terms) were recently obtained for adjacency labeling [FOCS'15], nearest common ancestor labeling [SODA'14], and ancestry labeling [SICOMP'06]. In this paper we obtain optimal bounds for distance labeling. We present labels of size $1/4\log^2n+o(\log^2n)$, matching (up to low order terms) the recent $1/4\log^2n-O(\log n)$ lower bound [ICALP'16]. Prior to our work, all distance labeling schemes for trees could be reinterpreted as universal trees. A tree $T$ is said to be universal if any tree on $n$ nodes can be found as a subtree of $T$. A universal tree with $|T|$ nodes implies a distance labeling scheme with label size $\log |T|$. In 1981, Chung et al. proved that any distance labeling scheme based on universal trees requires labels of size $1/2\log^2 n-\log n\cdot\log\log n+O(\log n)$. Our scheme is the first to break this lower bound, showing a separation between distance labeling and universal trees. The $\Theta(\log^2 n)$ barrier has led researchers to consider distances bounded by $k$. The size of such labels was improved from $\log n+O(k\sqrt{\log n})$ [WADS'01] to $\log n+O(k^2(\log(k\log n))$ [SODA'03] and then to $\log n+O(k\log(k\log(n/k)))$ [PODC'07]. We show how to construct labels whose size is $\min\{\log n+O(k\log((\log n)/k)),O(\log n\cdot\log(k/\log n))\}$. We complement this with almost tight lower bounds of $\log n+\Omega(k\log(\log n/(k\log k)))$ and $\Omega(\log n\cdot\log(k/\log n))$. Finally, we consider $(1+\varepsilon)$-approximate distances. We show that the labeling scheme of [ICALP'16] can be easily modified to obtain an $O(\log(1/\varepsilon)\cdot\log n)$ upper bound and we prove a $\Omega(\log(1/\varepsilon)\cdot\log n)$ lower bound.