Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Graph Isomorphism is the prime example of a computational problem with a wide#R##N#difference beten the best known lor and upper bounds on its complexity. There#R##N#is a significant gap beten extant lor and upper bounds for planar graphs as ll.#R##N# bridge the gap for this natural and important special case by presenting an upper#R##N#bound that matches the known log-space hardness [JKMT03]. In fact,  show the#R##N#formally stronger result that planar graph canonization is in log-space. This improves the#R##N#previously known upper bound of AC1 [MR91].#R##N#Our algorithm first constructs the biconnected component tree of a connected planar#R##N#graph and then refines each biconnected component into a triconnected component#R##N#tree. The next step is to log-space reduce the biconnected planar graph isomorphism and#R##N#canonization problems to those for 3-connected planar graphs, which are known to be in#R##N#log-space by [DLN08]. This is achieved by using the above decomposition, and by making#R##N#significant modifications to Lindell’s algorithm for tree canonization, along with changes#R##N#in the space complexity analysis.#R##N#The reduction from the connected case to the biconnected case requires further new#R##N#ideas including a non-trivial case analysis and a group theoretic lemma to bound the#R##N#number of automorphisms of a colored 3-connected planar graph.
['Samir Datta', 'Nutan Limaye', 'Prajakta Nimbhorkar', 'Thomas Thierauf', 'Fabian Wagner']
I am writing a research paper on a idea Planar Graph Isomorphism is in Log-Space
[{"from": "human", "value": "I am writing a research paper on a idea Planar Graph Isomorphism is in Log-Space"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Graph Isomorphism is the prime example of a computational problem with a wide#R##N#difference beten the best known lor and upper bounds on its complexity. There#R##N#is a significant gap beten extant lor and upper bounds for planar graphs as ll.#R##N# bridge the gap for this natural and important special case by presenting an upper#R##N#bound that matches the known log-space hardness [JKMT03]. In fact,  show the#R##N#formally stronger result that planar graph canonization is in log-space. This improves the#R##N#previously known upper bound of AC1 [MR91].#R##N#Our algorithm first constructs the biconnected component tree of a connected planar#R##N#graph and then refines each biconnected component into a triconnected component#R##N#tree. The next step is to log-space reduce the biconnected planar graph isomorphism and#R##N#canonization problems to those for 3-connected planar graphs, which are known to be in#R##N#log-space by [DLN08]. This is achieved by using the above decomposition, and by making#R##N#significant modifications to Lindell\u2019s algorithm for tree canonization, along with changes#R##N#in the space complexity analysis.#R##N#The reduction from the connected case to the biconnected case requires further new#R##N#ideas including a non-trivial case analysis and a group theoretic lemma to bound the#R##N#number of automorphisms of a colored 3-connected planar graph."}]