|Date: Friday, April 15, 2016
Location: 3725 BBB (10:30 AM to 11:30 AM)
Title: An Exponential Separation Between Randomized and Deterministic Complexity in the LOCAL Model
Abstract: 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. In this paper we prove that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message:
1. Fast Delta-coloring of trees requires random bits: Building on the recent lower bounds of Brandt et al., we prove that the randomized complexity of Delta-coloring a tree with maximum degree Delta>=55 is Theta(log_Delta logn) , whereas its deterministic complexity is Theta(log_Delta n) for any Delta >= 3. This also establishes a large separation between the deterministic complexity of Delta-coloring and (Delta +1) -coloring trees.
2. Randomized lower bounds imply deterministic lower bounds: We prove that any deterministic algorithm for a natural class of problems that runs in O(1)+o(log_Delta n) rounds can be transformed to run in O(log* n - log^*(Delta + 1)) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Omega(log Delta n) time deterministically.
3. Deterministic lower bounds imply randomized lower bounds: We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size sqrt( logn). This shows that a deterministic Omega(log_Delta n) lower bound for any problem implies a randomized Omega(log_Delta log n) lower bound. It also illustrates that the graph shattering technique is absolutely essential to the LOCAL model.
Joint work with Tsvi Kopelowitz and Seth Pettie http://arxiv.org/abs/1602.08166
Speaker: Yi-Jun Chang