{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T02:40:16Z","timestamp":1790563216278,"version":"4.1.0"},"reference-count":12,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>Given a permutation $\\pi$, the application of prefix reversal $f^{(i)}$ to $\\pi$ reverses the order of the first i elements of $\\pi$. The problem of sorting by prefix reversals (also known as pancake flipping), made famous by Gates and Papadimitriou (Discrete Math., 27 (1979), pp. 47\u201357), asks for the minimum number of prefix reversals required to sort the elements of a given permutation. In this paper we study a variant of this problem where the prefix reversals act not on permutations but on strings over a fixed size alphabet. We determine the minimum number of prefix reversals required to sort binary and ternary strings, with polynomial-time algorithms for these sorting problems as a result; demonstrate that computing the minimum prefix reversal distance between two binary strings is NP-hard; give an exact expression for the prefix reversal diameter of binary strings; and give bounds on the prefix reversal diameter of ternary strings. We also consider a weaker form of sorting called grouping (of identical symbols) and give polynomial-time algorithms for optimally grouping binary and ternary strings. A number of intriguing open problems are also discussed.<\/jats:p>","DOI":"10.1137\/060664252","type":"journal-article","created":{"date-parts":[[2007,7,25]],"date-time":"2007-07-25T18:00:19Z","timestamp":1185386419000},"page":"592-611","source":"Crossref","is-referenced-by-count":17,"title":["Prefix Reversals on Binary and Ternary Strings"],"prefix":"10.1137","volume":"21","author":[{"given":"Cor","family":"Hurkens","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leo","family":"van Iersel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Judith","family":"Keijsper","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Steven","family":"Kelk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leen","family":"Stougie","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"John","family":"Tromp","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1109\/TCBB.2005.48"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480197331995"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(94)00009-3"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(01)00150-9"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"J. Fischer and S. W. Ginzinger,\n                      A 2-approximation algorithm for sorting by prefix reversals\n                      , in Proceedings of ESA 2005: 13th Annual European Symposium, Lecture Notes in Comput. Sci. 3669, Springer, New York, 2005, pp. 415\u2013425.","DOI":"10.1007\/11561071_38"},{"key":"R6","unstructured":"M. R. Garey and D. S. Johnson,\n                      Computers And Intractability: A Guide To The Theory Of NP-completeness\n                      , W. H. Freeman, San Francisco, CA, 1979."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(79)90068-2"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"A. Goldstein, P. Kolman, and J. Zheng,\n                      Minimum common string partition problem: Hardness and approximations\n                      , Electron. J. Combin., 12 (2005), #R50.","DOI":"10.37236\/1947"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.1997.0874"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"L. Morales and I. H. Sudborough,\n                      Comparing star and pancake networks\n                      , in The Essence of Computation: Complexity, Analysis, Transformation, T. Mogensen, D. Schmidt, and I. H. Sudborough, eds., Lecture Notes in Comput. Sci. 2566, Springer, New York, 2002, pp. 18\u201336.","DOI":"10.1007\/3-540-36377-7_2"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480103433550"},{"key":"R12","unstructured":"A. D. Scott,\n                      personal communication\n                      , Merton College, Oxford, 2006."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060664252","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:54:34Z","timestamp":1787320474000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060664252"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/060664252"],"URL":"https:\/\/doi.org\/10.1137\/060664252","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}