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 Raman, [SIAM J. Comput. 31 (2001), 762–776], who gave O(n)-bit and 2n + o(n)-bit representations, respectively, that supported the above operations in O(1) time on the RAM model of computation. This data structure is a fundamental tool in succinct representations, and has applications in representing suffix trees, ordinal trees, planar graphs and permutations.

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