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  "path": "/abs/2602.17801v1",
  "publishedAt": "2026-02-23T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Sujoy Bhore",
    "Sándor Kisfaludi-Bak",
    "Lazar Milenković",
    "Csaba D. Tóth",
    "Karol Węgrzycki",
    "Sampson Wong"
  ],
  "textContent": "**Authors:** Sujoy Bhore, Sándor Kisfaludi-Bak, Lazar Milenković, Csaba D. Tóth, Karol Węgrzycki, Sampson Wong\n\nA Euclidean noncrossing Steiner $(1+ε)$-spanner for a point set $P\\subset\\mathbb{R}^2$ is a planar straight-line graph that, for any two points $a, b \\in P$, contains a path whose length is at most $1+ε$ times the Euclidean distance between $a$ and $b$. We construct a Euclidean noncrossing Steiner $(1+ε)$-spanner with $O(n/ε^{3/2})$ edges for any set of $n$ points in the plane. This result improves upon the previous best upper bound of $O(n/ε^{4})$ obtained nearly three decades ago. We also establish an almost matching lower bound: There exist $n$ points in the plane for which any Euclidean noncrossing Steiner $(1+ε)$-spanner has $Ω_μ(n/ε^{3/2-μ})$ edges for any $μ>0$. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.",
  "title": "Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity"
}