Balanced parentheses strike back

A Simple Optimal Representation for Balanced Parentheses

We consider succinct, or highly space-efficient, representations of a (static) string consisting of n pairs of balanced parentheses, that support natural operations such as finding the matching parenthesis for a given parenthesis, or finding the pair of parentheses that most tightly enclose a given pair. This problem was considered by Jacobson, [Proc. 30th FOCS, 549–554, 1989] and Munro and Ram...

The microbes strike back.

Charming Penguins Strike Back

By using the recent experimental measurements of B → ππ and B → Kπ branching ratios, we find that factorization is unable to reproduce the observed BRs even taking into account the uncertainties of the input parameters. Charming and GIM penguins allow to reconcile the theoretical predictions with the data. Because of these large effects, we conclude, however, that it is not possible, with the p...

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 Knu...

Succinct Representation of Balanced Parentheses, Static Trees and Planar Graphs

We consider the implementation of abstract data types for the static objects: binary tree, rooted ordered tree and balanced parenthesis expression. Our representations use an amount of space within a lower order term of the information theoretic minimum and support, in constant time, a richer set of navigational operations than has previously been considered in similar work. In the case of bina...