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 (such as distance or adjacency) can be computed by examining their labels alone. For the particular case of trees, following a long line of research, optimal bounds (up to loworder 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 log² n + o(log² n), matching (up to low order terms) the recent 1/4 log² n - 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 log² n - log n · 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 Θ(log² n) barrier for distance labeling in trees has led researchers to consider distances bounded by k. The size of such labels was shown to be log n + O(k √log n) in [WADS '01], and then improved to log n + O(k² log(k log n)) in [SODA '03] and finally to log n + O(k log(k log(n/k))) in [PODC '07]. We show how to construct labels whose size is the minimum between log n + O(k log((log n)/k)) and O(log n·log(k/ log n)). We complement this with almost tight lower bounds of log n + Ω(k log(log n/(k log k))) and Ω(log n·log(k/ log n)). Finally, we consider (1+ϵ)-approximate distances. We show that the recent labeling scheme of [ICALP '16] can be easily modified to obtain an O(log(1/ϵ) · log n) upper bound and we prove a matching Ω(log(1/ϵ) · log n) lower bound.