{"paper":{"title":"Shallow, Low, and Light Trees, and Tight Lower Bounds for Euclidean Spanners","license":"","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Michael Elkin, Shay Solomon, Yefim Dinitz","submitted_at":"2008-01-23T13:57:00Z","abstract_excerpt":"We show that for every $n$-point metric space $M$ there exists a spanning tree $T$ with unweighted diameter $O(\\log n)$ and weight $\\omega(T) = O(\\log n) \\cdot \\omega(MST(M))$. Moreover, there is a designated point $rt$ such that for every point $v$, $dist_T(rt,v) \\le (1+\\epsilon) \\cdot dist_M(rt,v)$, for an arbitrarily small constant $\\epsilon > 0$. We extend this result, and provide a tradeoff between unweighted diameter and weight, and prove that this tradeoff is \\emph{tight up to constant factors} in the entire range of parameters. These results enable us to settle a long-standing open que"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0801.3581","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}