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      "statement": "# Arlazarov, Dinitz, Kronrod and Faradzhev: faster graph reachability through subset tabulation\n\nArlazarov, Dinitz, M.A.Kronrod and Faradzhev tabulated unions over small blocks of intermediate vertices to compute Boolean relation products and acyclic graph closure in O(n^3/ln n) elementary operations.\n\nA directed network answers questions such as whether one place can reach another by following arrows. Computing every answer by checking every possible intermediate vertex takes cubic work in a familiar baseline. This1970 note groups intermediate vertices into small blocks. For each block it precomputes the possible unions of outgoing destinations, so a later row can select a stored result instead of rebuilding the same union. Choosing blocks of logarithmic size keeps those tables manageable and saves a logarithmic factor in the authors’ operation count. The note applies the composition lemma along the ranks of an acyclic graph. A graph with a directed cycle cannot be ranked this way directly: the paper relies on existing reductions through a graph whose vertices represent strongly connected components. The construction later became associated with the name method of Four Russians. That nickname is not a nationality attribution. The original identifies M.Kronrod as the source of its lemma, while all four authors remain responsible for the article. This is a historical algorithmic improvement under a declared representation and operation model, not a claim that it is always the fastest practical method today.\n\n## Scientific scope\n\nThe original direct result concerns acyclic graph closure with O(n^3/log n) elementary operations and table lookup; general graphs require the cited condensation/rank reductions. Lookup, storage and bit-operation assumptions remain part of the computational model. This account follows a transcription of the short original; it does not supply a certified formula transcription or independent proof reconstruction. Finite examples do not certify the complete asymptotic argument or a modern runtime benchmark. M. A. Kronrod is Mikhail Kronrod.\n\n## People and systems\n\nVladimir L'vovich Arlazarov; Yefim Dinitz; Mikhail Aleksandrovich Kronrod; Igor Aleksandrovich Faradzhev\n\n\n\n## Original sources\n\n- [Arlazarov, Dinitz, M.A.Kronrod and Faradzhev, economical transitive closure (1970) — original Russian web extraction](https://www.mathnet.ru/php/getFT.phtml?jrnid=dan&option_lang=eng&paperid=35675&what=fullt)\n- [Modern English transcription of the1970 two-page closure note — auxiliary reading aid](https://minorfree.github.io/assets/figs/Method_of_Four_Russians.pdf)\n- [Math-Net original bibliographic entry and full byline](https://www.mathnet.ru/eng/dan35675)\n- [Math-Net linked author identities: Mikhail Aleksandrovich Kronrod](https://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=98440)\n",
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