Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm.  prove that these exponential gaps are necessary and establish numerous connections beten the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message: 1) Building on the recent randomized lor bounds of Brandt et al. [1],  prove that the randomized complexity of Δ-coloring a tree with maximum degree Δ is O(log Δ log n + log*n), for any Δ > = 55, whereas its deterministic complexity is Ω(log Δ n) for any Δ > = 3. This also establishes a large separation beten the deterministic complexity of Δ-coloring and (Δ+1)-coloring trees. 2)  prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(log Δ n) rounds can be transformed to run in O(log*n - log*Δ + 1) rounds. If the transformed algorithm violates a lor bound (even allowing randomization), then one can conclude that the problem requires Ω(log Δ n) time deterministically. This gives an alternate proof that deterministically Δ-coloring a tree with small Δ takes Ω(log Δ n) rounds. 3)  prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √log n. This shows that a deterministic Ω(log Δ n) lor bound for any problem (Δ-coloring a tree, for example) implies a randomized Ω(log Δ log n) lor bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2O(√log log n) term in the complexities of the best MIS and (Δ+1)-coloring algorithms without also improving the 2O(√log n)-round Panconesi-Srinivasan algorithm.
['Yi-Jun Chang', 'Tsvi Kopelowitz', 'Seth Pettie']
I am writing a research paper on a idea An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model
[{"from": "human", "value": "I am writing a research paper on a idea An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm.  prove that these exponential gaps are necessary and establish numerous connections beten the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message: 1) Building on the recent randomized lor bounds of Brandt et al. [1],  prove that the randomized complexity of \u0394-coloring a tree with maximum degree \u0394 is O(log \u0394 log n + log*n), for any \u0394 > = 55, whereas its deterministic complexity is \u03a9(log \u0394 n) for any \u0394 > = 3. This also establishes a large separation beten the deterministic complexity of \u0394-coloring and (\u0394+1)-coloring trees. 2)  prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(log \u0394 n) rounds can be transformed to run in O(log*n - log*\u0394 + 1) rounds. If the transformed algorithm violates a lor bound (even allowing randomization), then one can conclude that the problem requires \u03a9(log \u0394 n) time deterministically. This gives an alternate proof that deterministically \u0394-coloring a tree with small \u0394 takes \u03a9(log \u0394 n) rounds. 3)  prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size \u221alog n. This shows that a deterministic \u03a9(log \u0394 n) lor bound for any problem (\u0394-coloring a tree, for example) implies a randomized \u03a9(log \u0394 log n) lor bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2O(\u221alog log n) term in the complexities of the best MIS and (\u0394+1)-coloring algorithms without also improving the 2O(\u221alog n)-round Panconesi-Srinivasan algorithm."}]