Universally Optimal Decremental Tree Minima

TL;DR

A data structure is presented that has optimal total running time for every fixed initial forest and every fixed number of operations/queries of length $m$ when taking the worst case over all weight assignments and operation sequences of length $m$.

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.

Figures (9)

• Figure 1: A graph (right), an elimination tree on that graph (center), and the eight priority-orderings of vertices inducing that elimination tree (right).
• Figure 2: A tree partitioned into chains and the corresponding compressed tree (top left), and the effect an edge deletion between or within super-nodes (right). Chains are light blue rectangles, and super-nodes are blue.
• Figure 3: cut and split in a rooted tree.
• Figure 4: A maximal-chain compression of a forest, with priorities omitted. Chains are overlaid in light blue.
• Figure 5: A cut operation in $F$ causing a canonical cut (left) or split(right) in the compression $F'$.

Theorems & Definitions (113)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Definition 2.4