Succinct Encoding of Binary Strings Representing Triangulations

We consider the problem of designing a succinct data structure for representing the connectivity of planar triangulations. The main result is a new succinct encoding achieving the information-theory optimal bound of 3.24 bits per vertex, while allowing efficient navigation. Our representation is based on the bijection of Poulalhon and Schaeffer (Algorithmica, 46(3):505–527, 2006) that defines a mapping between planar triangulations and a special class of spanning trees, called PS-trees . The proposed solution differs from previous approaches in that operations in planar triangulations are reduced to operations in particular parentheses sequences encoding PS-trees. Existing methods to handle balanced parentheses sequences have to be combined and extended to operate on such specific sequences, essentially for retrieving matching elements. The new encoding supports extracting the d neighbors of a query vertex in O ( d ) time and testing adjacency between two vertices in O (1) time. Additionally, we provide an implementation of our proposed data structure. In the experimental evaluation, our representation reaches up to 7.35 bits per vertex, improving the space usage of state-of-the-art implementations for planar embeddings.