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 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 n` oplog nq, matching (up to low order terms) the recent 1{4 log n ́Oplog nq 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 n ́ log n ̈ log log n`Oplog nq. Our scheme is the first to break this lower bound, showing a separation between distance labeling and universal trees. The Θplog nq barrier for distance labeling in trees has led researchers to consider distances bounded by k. The size of such labels was improved from log n`Opk ? log nq [WADS ’01] to log n ` Opk logpk log nqq [SODA ’03] and finally to log n ` Opk logpk logpn{kqqq [PODC ’07]. We show how to construct labels whose size is the minimum between log n ` Opk logpplog nq{kqq and Oplog n ̈ logpk{ log nqq. We complement this with almost tight lower bounds of log n ` Ωpk logplog n{pk log kqqq and Ωplog n ̈ logpk{ log nqq. Finally, we consider p1` εq-approximate distances. We show that the recent labeling scheme of [ICALP ’16] can be easily modified to obtain an Oplogp1{εq ̈ log nq upper bound and we prove a matching Ωplogp1{εq ̈ log nq lower bound. ̊The research was supported in part by Israel Science Foundation grant 794/13.