Graph Coding and Connectivity Compression

This paper looks at the theoretic roots of current connectivity compression schemes to establish a visual framework within which the differences and similarities of various scheme become intuitive. We show the intimate connections between the classic work on planar graph coding by Turan and recent schemes, such as Edgebreaker, Face Fixer, and the optimal coding by method of Poulalhon and Schaefer. Furthermore we fit Touma and Gotsman’s valence coder into this classification. This helps to explain what information is hidden in the "split offsets" and suggests a strategy for doing valence coding without using offsets. Other results are an elegant method for reverse decoding of meshes encoded with Poulalhon and Schaefer’s optimal coder, and the insight that the classic Keeler and Westbrook method and the Edgebreaker scheme are really the same algorithm for the case of encoding planar triangulations. Finally, we conjecture that (a) optimal encodings are never streamable and (b) encodings that avoid offsets necessarily result in uncoherent mesh layouts.