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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Veröffentlicht in:Journal of discrete algorithms (Amsterdam, Netherlands) Jg. 16; S. 287 - 307
Hauptverfasser: Durocher, Stephane, Li, Pak Ching, Mondal, Debajyoti, Ruskey, Frank, Williams, Aaron
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier B.V 01.10.2012
Schlagworte:
Bubble langauges
Catalan structures
Cool-lex order
k-ary Dyck words
k-ary trees
Loopless algorithms
Ranking
Loopless algorithms
Catalan structures
Ranking
k-ary trees
Bubble langauges
Cool-lex order
k-ary Dyck words
ISSN:1570-8667, 1570-8675
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Zusammenfassung: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.
ISSN:1570-8667
1570-8675
DOI:10.1016/j.jda.2012.04.015