Cool-lex order and k-ary Catalan structures
For any given k, the sequence of k-ary Catalan numbers, Ct,k=1kt+1(ktt), enumerates a number of combinatorial objects, including k-ary Dyck words of length n=kt and k-ary trees with t internal nodes. We show that these objects can be efficiently ordered using the same variation of lexicographic orde...
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| Vydáno v: | Journal of discrete algorithms (Amsterdam, Netherlands) Ročník 16; s. 287 - 307 |
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| Jazyk: | angličtina |
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Elsevier B.V
01.10.2012
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| ISSN: | 1570-8667, 1570-8675 |
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| Abstract | For any given k, the sequence of k-ary Catalan numbers, Ct,k=1kt+1(ktt), enumerates a number of combinatorial objects, including k-ary Dyck words of length n=kt and k-ary trees with t internal nodes. We show that these objects can be efficiently ordered using the same variation of lexicographic order known as cool-lex order. In particular, we provide loopless algorithms that generate each successive object in O(1) time. The algorithms are also efficient in terms of memory, with the k-ary Dyck word algorithm using O(1) additional index variables, and the k-ary tree algorithm using O(t) additional pointers and index variables. We also show how to efficiently rank and unrank k-ary Dyck words in cool-lex order using O(kt) arithmetic operations, subject to an initial precomputation. Our results are based on the cool-lex successor rule for sets of binary strings that are bubble languages. However, we must complement and reverse 1/k-ary Dyck words to obtain the stated efficiency. |
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| AbstractList | For any given k, the sequence of k-ary Catalan numbers, Ct,k=1kt+1(ktt), enumerates a number of combinatorial objects, including k-ary Dyck words of length n=kt and k-ary trees with t internal nodes. We show that these objects can be efficiently ordered using the same variation of lexicographic order known as cool-lex order. In particular, we provide loopless algorithms that generate each successive object in O(1) time. The algorithms are also efficient in terms of memory, with the k-ary Dyck word algorithm using O(1) additional index variables, and the k-ary tree algorithm using O(t) additional pointers and index variables. We also show how to efficiently rank and unrank k-ary Dyck words in cool-lex order using O(kt) arithmetic operations, subject to an initial precomputation. Our results are based on the cool-lex successor rule for sets of binary strings that are bubble languages. However, we must complement and reverse 1/k-ary Dyck words to obtain the stated efficiency. |
| Author | Li, Pak Ching Ruskey, Frank Williams, Aaron Durocher, Stephane Mondal, Debajyoti |
| Author_xml | – sequence: 1 givenname: Stephane surname: Durocher fullname: Durocher, Stephane email: durocher@cs.umanitoba.ca organization: Department of Computer Science, University of Manitoba, Canada – sequence: 2 givenname: Pak Ching surname: Li fullname: Li, Pak Ching email: lipakc@cs.umanitoba.ca organization: Department of Computer Science, University of Manitoba, Canada – sequence: 3 givenname: Debajyoti surname: Mondal fullname: Mondal, Debajyoti email: jyoti@cs.umanitoba.ca organization: Department of Computer Science, University of Manitoba, Canada – sequence: 4 givenname: Frank surname: Ruskey fullname: Ruskey, Frank email: ruskey@cs.uvic.ca organization: Department of Computer Science, University of Victoria, Canada – sequence: 5 givenname: Aaron surname: Williams fullname: Williams, Aaron email: haron@uvic.ca organization: Department of Mathematics and Statistics, McGill University, Canada |
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| Cites_doi | 10.1016/S0020-0190(01)00141-7 10.1137/0207039 10.1016/S0020-0190(00)00155-1 10.1090/S0025-5718-1963-0159764-2 10.1137/0207034 10.1145/368637.368660 10.1016/0020-0190(82)90140-5 10.1007/BF03024089 10.1016/j.jda.2005.07.003 10.1016/j.mcm.2010.12.019 10.1093/jigpal/jzp097 10.1093/comjnl/bxp015 10.1080/00207160310001624214 10.1016/j.disc.2007.11.048 10.1016/j.tcs.2012.01.013 10.1145/321765.321781 10.1016/0020-0190(88)90027-0 10.1016/j.jcta.2011.07.005 10.1137/0404037 10.1007/BF01108590 |
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| Keywords | Loopless algorithms Catalan structures Ranking k-ary trees Bubble langauges Cool-lex order k-ary Dyck words |
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