Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the parameterized complexity of Geometric Graph Isomorphism (Known as the Point Set Congruence problem in computational geometry): given two sets of n points A and B with rational coordinates in k-dimensional euclidean space, with k as the fixed parameter, the problem is to decide if there is a bijection $$\pi :A \rightarrow B$$ź:AźB such that for all $$x,y \in A$$x,yźA, $$\Vert x-y\Vert = \Vert \pi (x)-\pi (y)\Vert $$źx-yź=źź(x)-ź(y)ź, where $$\Vert \cdot \Vert $$ź·ź is the euclidean norm. Our main result is the following:  give a $$O^*(k^{O(k)})$$Oź(kO(k)) time (The $$O^*(\cdot )$$Oź(·) notation here, as usual, suppresses polynomial factors) FPT algorithm for Geometric Isomorphism. This is substantially faster than the previous best time bound of $$O^*(2^{O(k^4)})$$Oź(2O(k4)) for the problem (Evdokimov and Ponomarenko in Pure Appl Algebra 117---118:253---276, 1997). In fact,  show the stronger result that even canonical forms for finite point sets with rational coordinates can also be computed in $$O^*(k^{O(k)})$$Oź(kO(k)) time.  also briefly discuss the isomorphism problem for other $$l_p$$lp metrics. Specifically,  describe a deterministic polynomial-time algorithm for finite point sets in $$\mathbb {Q}^2$$Q2.
['Vikraman Arvind', 'Gaurav Rattan']
I am writing a research paper on a idea The Parameterized Complexity of Geometric Graph Isomorphism
[{"from": "human", "value": "I am writing a research paper on a idea The Parameterized Complexity of Geometric Graph Isomorphism"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the parameterized complexity of Geometric Graph Isomorphism (Known as the Point Set Congruence problem in computational geometry): given two sets of n points A and B with rational coordinates in k-dimensional euclidean space, with k as the fixed parameter, the problem is to decide if there is a bijection $$\\pi :A \\rightarrow B$$\u017a:A\u017aB such that for all $$x,y \\in A$$x,y\u017aA, $$\\Vert x-y\\Vert = \\Vert \\pi (x)-\\pi (y)\\Vert $$\u017ax-y\u017a=\u017a\u017a(x)-\u017a(y)\u017a, where $$\\Vert \\cdot \\Vert $$\u017a\u00b7\u017a is the euclidean norm. Our main result is the following:  give a $$O^*(k^{O(k)})$$O\u017a(kO(k)) time (The $$O^*(\\cdot )$$O\u017a(\u00b7) notation here, as usual, suppresses polynomial factors) FPT algorithm for Geometric Isomorphism. This is substantially faster than the previous best time bound of $$O^*(2^{O(k^4)})$$O\u017a(2O(k4)) for the problem (Evdokimov and Ponomarenko in Pure Appl Algebra 117---118:253---276, 1997). In fact,  show the stronger result that even canonical forms for finite point sets with rational coordinates can also be computed in $$O^*(k^{O(k)})$$O\u017a(kO(k)) time.  also briefly discuss the isomorphism problem for other $$l_p$$lp metrics. Specifically,  describe a deterministic polynomial-time algorithm for finite point sets in $$\\mathbb {Q}^2$$Q2."}]