Complexity of pattern generation via planar parallel binary fission/fusion grammars
In two previous works [74CGP,86CGP], we defined and studied a scheme for grammar-based generation of planar maps, using deterministic parallel (in the sense of Lindenmayer) derivation rules of tow types — binary splitting of countries (fission) and binary merging (fusion). Countries would be visualized as cells in applications to modeling the development of (planar) biological organisms, for example. We are concerned primarily with intrinsic properties of our model (rather than specific applications), including its power as measured by the complexity of parallel generation of the class of all legal (well-formed) planar maps. Our previous work showed that, with fission added to fusion, the complexity decreases from linear to logarithmic in the size of the map. The present paper is a sequel, giving a more complete treatment of several aspects of these basic results and related questions. In particular, we examine: the effect of merging under grammar constraints; constructions involving Hamiltonian and semi-Hamiltonian paths (tours of countries) and spanning trees (with countries as nodes), which are used as intermediate phases in reaching the general logarithmic complexity result; and neighborhood (crowdedness) control in the generations.
Key wordscell-division complexity fission fusion Hamiltonian linear time logarithmic time pattern planarity graph grammar Lindenmayer system map parallel derivation
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