arXiv subjects

Lazar Milenković

Publications and source records attributed to Lazar Milenković.

3 recordsLinked to original sources

arXiv2026

Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

A 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.

cs.CG↗

arXiv2025

Tree-Like Shortcuttings of Trees

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 [Yao82, Cha87, AS87, BTS94], 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 [Yao82, Cha87, AS87, BTS94] 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 $\text{hop-diameter}\times \text{treewidth} = Ω((\log\log n)^2)$ for all values of hop-diameter, resolving an open question of [FL22, Le23]. [...]

cs.DS↗

arXiv2025

Covering the Euclidean Plane by a Pair of Trees

A {$t$-stretch tree cover} of a metric space $M = (X,δ)$, for a parameter $t \ge 1$, is a collection of trees such that every pair of points has a $t$-stretch path in one of the trees. Tree covers provide an important sketching tool that has found various applications over the years. The celebrated {Dumbbell Theorem} by Arya et al. [STOC'95] states that any set of points in the Euclidean plane admits a $(1+ε)$-stretch tree cover with $O_ε(1)$ trees. This result extends to any (constant) dimension and was also generalized for arbitrary doubling metrics by Bartal et al. [ICALP'19]. Although the number of trees provided by the Dumbbell Theorem is constant, this constant is not small, even for a stretch significantly larger than $1+ε$. At the other extreme, any single tree on the vertices of a regular $n$-polygon must incur a stretch of $Ω(n)$. Using known results of ultrametric embeddings, one can easily get a stretch of $\tilde{O}(\sqrt{n})$ using two trees. The question of whether a low stretch can be achieved using two trees has remained illusive, even in the Euclidean plane. In this work, we resolve this fundamental question in the affirmative by presenting a constant-stretch cover with a pair of trees, for any set of points in the Euclidean plane. Our main technical contribution is a {surprisingly simple} Steiner construction, for which we provide a {tight} stretch analysis of $\sqrt{26}$. The Steiner points can be easily pruned if one is willing to increase the stretch by a small constant. Moreover, we can bound the maximum degree of the construction by a constant. Our result thus provides a simple yet effective reduction tool -- for problems that concern approximate distances -- from the Euclidean plane to a pair of trees. To demonstrate the potential power of this tool, we present some applications [...]

cs.CG↗