{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:33:45Z","timestamp":1787340825653,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[1994,2]]},"abstract":"<jats:p>An extension of the disjoint set union problem is considered, where the extra primitive backtrack $(i)$ can undo the last i unions not yet undone. Let n be the total number of elements in all the sets. A data structure is presented that supports each union and find in $O({{\\log n} \/ {\\log \\log n}})$ worst-case time and each backtrack$(i)$ in $O(1)$ worst-case time, irrespective of i. The total space required by the data structure is $O(n)$. A byproduct of this construction is a partially persistent data structure for the standard set union problem, capable of supporting union, find, and find-in-the-past operations, each in $O({{\\log n} \/ {\\log \\log n}})$ worst-case time. All these upper bounds are tight for the class of separable-pointer algorithms as well as in the cell probe model of computation.<\/jats:p>","DOI":"10.1137\/s0097539789173597","type":"journal-article","created":{"date-parts":[[2005,2,24]],"date-time":"2005-02-24T06:39:46Z","timestamp":1109227186000},"page":"50-70","source":"Crossref","is-referenced-by-count":5,"title":["The Set Union Problem with Unlimited Backtracking"],"prefix":"10.1137","volume":"23","author":[{"given":"Alberto","family":"Apostolico","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giuseppe F.","family":"Italiano","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giorgio","family":"Gambosi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maurizio","family":"Talamo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","volume-title":"The Design and Analysis of Computer Algorithms","author":"Aho A. 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