2025/10/16 by Le, Hung, Milenković, Lazar, Solomon, Shay +1
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2510.14918
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(loglog n). We also provide a lower bound for larger values of k, which together yield hop-diameter× treewidth = Ω((loglog n)2) for all values of hop-diameter, resolving an open question of [FL22, Le23]. [...]