Generating Balanced Parentheses and Binary Trees by Prefix Shifts

We show that the set Bn of balanced parenthesis strings with n left and n right parentheses can be generated by prefix shifts. If b1, b2, . . . , b2n is a member of Bn, then the k-th prefix shift is the string b1, bk, b2, . . . , bk-1, bk+1, . . . , b2n. Prefix shift algorithms are also known for combinations, and permutations of a multiset; the combination algorithm appears in fascicles of Knuth vol 4. We show that the algorithm is closely related to the combination algorithm, and like it, has a loopless implementation, and a ranking algorithm that uses O(n) arithmetic operations. Additionally, the algorithm can be directly translated to generate all binary trees by a loopless implementation that makes a constant number of pointer changes for each successively generated tree.

• Type: Conference
• Year: 2008
• Where: CATS
• Authors: Frank Ruskey, Aaron Williams
• Keywords: Algorithm, Applied Computing, CATS 2008, Combination Algorithm, Prefix Shift