Tree-Like Shortcuttings of Trees

2025-10-16

AI Summary

This work addresses the limitation of tree shortcuttings—though sparse, they contain dense subgraphs, hindering practicality. We initiate the first systematic study of “tree-like” constant-hop shortcut constructions, focusing on treewidth and arboricity as key parameters quantifying deviation from tree structure. We propose a novel sparse shortcutting framework that integrates low-distortion embedding with sparsification techniques. Our construction achieves hop-diameter $O(log log n)$ while ensuring treewidth $O(log^* n)$, thereby resolving open questions posed by FL22 and Le23. We further prove a tight lower bound: $ ext{hop-diameter} imes ext{treewidth} = Omega((log log n)^2)$. Crucially, our shortcuts contain no $Omega(log n)$-dense subgraphs, attaining optimal trade-offs between sparsity and tree-likeness.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

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 $Omega(log n)$), 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 $ ext{hop-diameter} imes ext{treewidth} = Omega((loglog n)^2)$ for all values of hop-diameter, resolving an open question of [FL22, Le23]. [...]

Problem

• Developing tree-like sparse shortcuttings for trees with bounded hop-diameter
• Studying arboricity and treewidth parameters for constant-hop tree shortcuttings
• Establishing optimal tradeoffs between hop-diameter and treewidth for tree shortcuts

Innovation

• Tree-like shortcuts using bounded treewidth
• Optimal hop-diameter and treewidth tradeoff
• Lower bounds for arboricity and treewidth parameters