Universally Optimal Decremental Tree Minima

2026-02-17

AI Summary

This work addresses the problem of efficiently maintaining the minimum-weight vertex in each tree of a weighted forest subject to dynamic edge deletions. To this end, the authors propose a novel data structure based on path decomposition and splay trees, which introduces the notion of “universal optimality” to the design of data structures for the first time. The structure achieves worst-case optimal total update time across all possible initial forests, sequences of operations, and weight assignments. Furthermore, it supports semigroup-sum queries, enabling not only theoretically optimal performance under the universal optimality framework but also practical applications to classical problems such as Cartesian tree construction, path minimum queries, and bottleneck edge queries.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

An algorithm on weighted graphs is called universally optimal if it is optimal for every input graph, in the worst case taken over all weight assignments. Informally, this means the algorithm is competitive even with algorithms that are optimized for only one specific input graph. Universal optimality was recently introduced [Haeupler et al. 2024] as an alternative to the stronger, but often unachievable instance optimality. In this paper, we extend the concept of universal optimality to data structures. In particular, we investigate the following dynamic graph problem: Given a vertex-weighted forest, maintain the minimum-weight vertex of every tree under edge deletions. The problem requires $Θ(\log n)$ amortized time per operation in general, but only $O(1)$ time if the initial forest is a path. We present a data structure that has optimal total running time for every fixed initial forest and every fixed number of operations/queries $m$, when taking the worst case over all weight assignments and operation sequences of length $m$. This definition of universal optimality is easily adapted to other data structure problems. Our result combines two techniques: (1) A decomposition of the input into paths, to take advantage of the $O(1)$-time path-specific data structure; and (2) splay trees [Sleator and Tarjan 1985], which, informally speaking, are used to optimally handle a certain sorting-related subproblem. We apply our data structure to solve problems related to Cartesian trees, path minimum queries, and bottleneck vertex/edge queries, each with a certain universal optimality guarantee. Our data structure also can be modified to support edge weights instead of vertex weights. Further, it generalizes to support semigroup sum queries instead of minimum queries, in universally optimal time.

Problem

• dynamic graph
• vertex-weighted forest
• edge deletions
• tree minima
• universal optimality

Innovation

• universal optimality
• decremental tree minima
• splay trees
• path decomposition
• dynamic graph data structures