← 研踪 PaperTrailarXiv:2611.05713 · v1
Incremental Minimum Spanning Trees on a Napkin
首次提交 2026-11-04 06:39
版本更新 2026-11-04 06:39
雷达观测 2026-11-05 10:08
摘要
In the incremental minimum spanning forest problem, weighted edges arrive one at a time. After each insertion we must report the edge that leaves the minimum spanning forest, and queries ask for the bottleneck distance of two vertices: the smallest possible maximum edge cost on a path between them. Link-cut trees solve both in $O(\log n)$ amortized time. In 2021 we wrote the folklore BottleneckTree, a short competitive-programming structure that stores one parent pointer and one cost per vertex, and conjectured that it also runs in $O(\log n)$ amortized time. We show that this conjecture is false: on a family of inputs, its amortized cost per operation is $Ω(\sqrt{n})$. We then present a structure with the same state, two arrays and a single loop, and prove that it runs in $O(\log n)$ amortized time per insertion and per query. Unlike the AM-tree of Ding, Gu and Sun, which attains the same bound, it stores no subtree sizes and maintains no balance invariant. For a connected graph with distinct costs, the structure can be viewed as an implicit link-cut tree on the Kruskal reconstruction tree: each rotation corresponds to exactly one preferred-child change, and each skip acts on a stale path-parent pointer. In experiments it is 1.3-2.1 times faster than the folklore BottleneckTree on insertions, up to 480 times faster on its adversarial input, and 6.3-16 times faster than an explicit link-cut tree.
作者备注：20 pages, 5 figures, 3 tables
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v1 · 2026-11-04 06:39 · 最新观测版本
Incremental Minimum Spanning Trees on a Napkin
In the incremental minimum spanning forest problem, weighted edges arrive one at a time. After each insertion we must report the edge that leaves the minimum spanning forest, and queries ask for the bottleneck distance of two vertices: the smallest possible maximum edge cost on a path between them. Link-cut trees solve both in $O(\log n)$ amortized time. In 2021 we wrote the folklore BottleneckTree, a short competitive-programming structure that stores one parent pointer and one cost per vertex, and conjectured that it also runs in $O(\log n)$ amortized time. We show that this conjecture is false: on a family of inputs, its amortized cost per operation is $Ω(\sqrt{n})$. We then present a structure with the same state, two arrays and a single loop, and prove that it runs in $O(\log n)$ amortized time per insertion and per query. Unlike the AM-tree of Ding, Gu and Sun, which attains the same bound, it stores no subtree sizes and maintains no balance invariant. For a connected graph with distinct costs, the structure can be viewed as an implicit link-cut tree on the Kruskal reconstruction tree: each rotation corresponds to exactly one preferred-child change, and each skip acts on a stale path-parent pointer. In experiments it is 1.3-2.1 times faster than the folklore BottleneckTree on insertions, up to 480 times faster on its adversarial input, and 6.3-16 times faster than an explicit link-cut tree.
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