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.Ĭonsider the set of unlabeled and connected finite planar graphs, where neither loops nor multiple edges are allowed, that admit planar embeddings in which each face has three and only three incident edges. Additionally, we provide an implementation of our proposed data structure. 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. 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 proposed solution differs from previous approaches in that operations in planar triangulations are reduced to operations in particular parentheses sequences encoding PS-trees. 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 main result is a new succinct encoding achieving the information-theory optimal bound of 3.24 bits per vertex, while allowing efficient navigation. ![]() We consider the problem of designing a succinct data structure for representing the connectivity of planar triangulations.
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