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Compact routing on tree shortcuttings with o(log^2 n) memory and sublogarithmic hop-diameter

Develop a stretch‑1 compact routing scheme operating on a shortcutting of an n‑vertex tree T that uses o(log^2 n) bits of memory per node and attains o(log n) hop‑diameter, preferably a constant number of hops.

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Background

Kahalon et al. proposed a compact routing scheme on tree shortcuttings with hop-diameter 2 that uses O(log2 n) bits per node, a bound tied to arboricity/treewidth Θ(log n) in that setting. They left open whether larger (but sublogarithmic, ideally constant) hop-diameter could reduce the per-node memory below Θ(log2 n). The authors here restate and focus this question in the context of stretch‑1 routing on tree metrics via shortcuttings.

References

Breaking the bound of Θ(log2 n) bits is a main open question left in their work. Quoting [KLMS22]: “Whether or not one can use a spanner of larger (sublogarithmic and preferably constant) hop-diameter for designing compact routing schemes with o(log2 n) bits is left here as an intriguing open question.” Given an n-vertex tree T, is there a compact routing scheme (operating on a shortcutting of T) with stretch 1 which uses o(log2 n) bits of space for every node and achieves an o(log n), and ideally constant hop-diameter?

— Tree-Like Shortcuttings of Trees (2510.14918 - Le et al., 16 Oct 2025) in Question [KLMS22], Section 1 (Introduction)