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    "title": "Tarjan: inverse-Ackermann analysis of weighted union and path compression",
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        "description": "The amortized bound concerns the specified sequence of disjoint-set operations and union/path-compression rules in its computational model. It is not a worst-case constant bound for each operation or every variant. Input size, storage and operation accounting remain. Later refinements and implementations retain separate credit.",
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      "statement": "# Tarjan: inverse-Ackermann analysis of weighted union and path compression\n\nTarjan established nearly linear amortized bounds for weighted union with path compression, with inverse-Ackermann dependence, and a matching lower-bound analysis within the studied tree-algorithm model.\n\nA disjoint-set structure keeps track of which objects have been joined into the same group. It can represent each group as a rooted tree. Weighted union attaches a smaller tree to a larger one; path compression shortens a traversed path to its root. These operations had predecessors. Tarjan's contribution was to explain their combined sequence cost much more sharply: the number of repeated expensive traversals is controlled by slowly growing classes related to the inverse Ackermann function. The result gives almost-linear total work over a long sequence, rather than promising that each individual operation always takes constant time. The inspected1974 report also studies lower bounds for the same tree/compression family. It explicitly leaves the possibility of a different general linear-time set-union algorithm open. The familiar1975 journal citation belongs to this contribution lineage, but its complete body has not been read here and its more general two-parameter notation is not substituted into the report proof.\n\n## Scientific scope\n\nThe amortized bound concerns the specified sequence of disjoint-set operations and union/path-compression rules in its computational model. It is not a worst-case constant bound for each operation or every variant. Input size, storage and operation accounting remain. Later refinements and implementations retain separate credit.\n\n## People and systems\n\nRobert Endre Tarjan\n\n\n\n## Original sources\n\n- [Tarjan — original Berkeley ERL-M4341974 report](https://www2.eecs.berkeley.edu/Pubs/TechRpts/1974/Archive/ERL-m-434.pdf)\n- [Tarjan —1975 Journal of the ACM publisher deposit](https://api.crossref.org/works/10.1145%2F321879.321884)\n- [Galler/Fischer report citation —1964 publisher identity and date deposit](https://api.crossref.org/works/10.1145%2F364099.364331)\n",
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