Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider collaborative graph exploration with a set of $k$ agents. All agents start at a common vertex of an initially unknown graph and need to collectively visit all other vertices.  assume agents are deterministic, vertices are distinguishable, moves are simultaneous, and  allow agents to communicate globally. For this setting,  give the first non-trivial lor bounds that bridge the gap beten small ($k \leq \sqrt n$) and large ($k \geq n$) teams of agents. Remarkably, our bounds tightly connect to existing results in both domains. #R##N#First,  significantly extend a lor bound of $\Omega(\log k / \log\log k)$ by Dynia et al. on the competitive ratio of a collaborative tree exploration strategy to the range $k \leq n \log^c n$ for any $c \in \mathbb{N}$. Second,  provide a tight lor bound on the number of agents needed for any competitive exploration algorithm. In particular,  show that any collaborative tree exploration algorithm with $k = Dn^{1+o(1)}$ agents has a competitive ratio of $\omega(1)$, while Dereniowski et al. gave an algorithm with $k = Dn^{1+\varepsilon}$ agents and competitive ratio $O(1)$, for any $\varepsilon > 0$ and with $D$ denoting the diameter of the graph. Lastly,  show that, for any exploration algorithm using $k = n$ agents, there exist trees of arbitrarily large height $D$ that require $\Omega(D^2)$ rounds, and  provide a simple algorithm that matches this bound for all trees.
['Yann Disser', 'Frank Mousset', 'Andreas Noever', 'Nemanja Škorić', 'Angelika Steger']
I am writing a research paper on a idea A general lower bound for collaborative tree exploration
[{"from": "human", "value": "I am writing a research paper on a idea A general lower bound for collaborative tree exploration"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider collaborative graph exploration with a set of $k$ agents. All agents start at a common vertex of an initially unknown graph and need to collectively visit all other vertices.  assume agents are deterministic, vertices are distinguishable, moves are simultaneous, and  allow agents to communicate globally. For this setting,  give the first non-trivial lor bounds that bridge the gap beten small ($k \\leq \\sqrt n$) and large ($k \\geq n$) teams of agents. Remarkably, our bounds tightly connect to existing results in both domains. #R##N#First,  significantly extend a lor bound of $\\Omega(\\log k / \\log\\log k)$ by Dynia et al. on the competitive ratio of a collaborative tree exploration strategy to the range $k \\leq n \\log^c n$ for any $c \\in \\mathbb{N}$. Second,  provide a tight lor bound on the number of agents needed for any competitive exploration algorithm. In particular,  show that any collaborative tree exploration algorithm with $k = Dn^{1+o(1)}$ agents has a competitive ratio of $\\omega(1)$, while Dereniowski et al. gave an algorithm with $k = Dn^{1+\\varepsilon}$ agents and competitive ratio $O(1)$, for any $\\varepsilon > 0$ and with $D$ denoting the diameter of the graph. Lastly,  show that, for any exploration algorithm using $k = n$ agents, there exist trees of arbitrarily large height $D$ that require $\\Omega(D^2)$ rounds, and  provide a simple algorithm that matches this bound for all trees."}]