Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Minimum Spanning Trees of ighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages beten network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances beten pairs. While known lor bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [4] that there exists a spanning tree with constant average distortion. Yet, the ight of such a tree may be significantly larger than that of the MST. ,  show that any ighted undirected graph admits a  spanning tree  whose ight is at most (1 + ρ) times that of the MST, providing  constant average distortion O (1/ρ 2 ). 1    The constant average distortion bound is implied by a stronger property of  scaling distortion , i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low ight  spanner  with small  prioritized distortion , a property allowing to prioritize the nodes whose associated distortions will be improved.  show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular,  obtain an embedding for arbitrary metrics into Euclidean space with optimal prioritized distortion.
['Yair Bartal', 'Arnold Filtser', 'Ofer Neiman']
I am writing a research paper on a idea On notions of distortion and an almost minimum spanning tree with constant average distortion
[{"from": "human", "value": "I am writing a research paper on a idea On notions of distortion and an almost minimum spanning tree with constant average distortion"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Minimum Spanning Trees of ighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages beten network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances beten pairs. While known lor bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [4] that there exists a spanning tree with constant average distortion. Yet, the ight of such a tree may be significantly larger than that of the MST. ,  show that any ighted undirected graph admits a  spanning tree  whose ight is at most (1 + \u03c1) times that of the MST, providing  constant average distortion O (1/\u03c1 2 ). 1    The constant average distortion bound is implied by a stronger property of  scaling distortion , i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low ight  spanner  with small  prioritized distortion , a property allowing to prioritize the nodes whose associated distortions will be improved.  show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular,  obtain an embedding for arbitrary metrics into Euclidean space with optimal prioritized distortion."}]