2026-09-05T11:57:54Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:25877 2026-06-23T13:00:03Z ddc:004 open_access

Tree-Like Shortcuttings of Trees Le, Hung Milenković, Lazar Solomon, Shay Than, Cuong spanner tree shortcutting arboricity treewidth Sparse shortcuttings of trees - equivalently, sparse 1-spanners for tree metrics with bounded hop-diameter - have been studied extensively (under different names and settings), since the pioneering works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994], initially motivated by applications to range queries, online tree product, and MST verification, to name a few. These constructions were also lifted from trees to other graph families using known low-distortion embedding results. The works of [Andrew Chi-Chih Yao, 1982; Chazelle, 1987; Noga Alon and Baruch Schieber, 1987; Hans L. Bodlaender et al., 1994] establish a tight tradeoff between hop-diameter and sparsity (or average degree) for tree shortcuttings and imply constant-hop shortcuttings for n-node trees with sparsity O(log^* n). Despite their small sparsity, all known constant-hop shortcuttings contain dense subgraphs (of sparsity Ω(log n)), which is a significant drawback for many applications. We initiate a systematic study of constant-hop tree shortcuttings that are "tree-like". We focus on two well-studied graph parameters that measure how far a graph is from a tree: arboricity and treewidth. Our contribution is twofold. - New upper and lower bounds for tree-like shortcuttings of trees, including an optimal tradeoff between hop-diameter and treewidth for all hop-diameter up to O(log log n). We also provide a lower bound for larger values of k, which together yield hop-diameter× treewidth = Ω((log log n)²) for all values of hop-diameter, resolving an open question of [Arnold Filtser and Hung Le, 2022; H. Le, 2023]. - Applications of these bounds, focusing on low-dimensional Euclidean and doubling metrics. A seminal work of Arya et al. [S. Arya et al., 1995] presented a (1+ε)-spanner with constant hop-diameter and sparsity O(log^* n), but with large arboricity. We show that constant hop-diameter is sufficient to achieve arboricity O(log^*{n}). Furthermore, we present a (1+ε)-stretch routing scheme in the fixed-port model with 3 hops and a local memory of O(log²n / log log n) bits, resolving an open question of [Omri Kahalon et al., 2022]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Hung Le and Lazar Milenković and Shay Solomon and Cuong Than 2026 Is Part Of LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2026.70 urn:nbn:de:0030-drops-258776 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.70 eng https://creativecommons.org/licenses/by/4.0/legalcode