AN EXPONENTIAL SEPARATION BETWEEN RANDOMIZED AND DETERMINISTIC COMPLEXITY IN THE LOCAL MODEL

作     者：Chang, Yi-Jun Kopelowitz, Tsvi Pettie, Seth 

作者机构：Univ Michigan EECS Ann Arbor MI 48105 USA Bar Ilan Univ Comp Sci IL-5290002 Ramat Gan Israel 

出 版 物：《SIAM JOURNAL ON COMPUTING》 (工业与应用数学会计算杂志)

年 卷 期：2019年第48卷第1期

页      面：122-143页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：NSF [CCF-1217338  CNS-1318294  CCF-1514383] 

主　　题：coloring distributed algorithm local model symmetry breaking 

摘      要：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 numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message: Fast Delta-coloring of trees requires random bits. Building on a recent randomized lower bound of Brandt et al. [A lower bound for the distributed Lovasz local lemma, in Proceedings of the 48th ACM Symposium on Theory of Computing (STOC), ACM, New York, 2016, pp. 479-488], we prove that the randomized complexity of Delta-coloring a tree with maximum degree Delta is O(log(Delta) log n + log* n) for any Delta = 55, whereas its deterministic complexity is Omega(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. There is a gap in the deterministic complexity hierarchy. We show 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. This gives an alternate proof that deterministically Delta-coloring a tree with small Delta takes Omega(log(Delta) n) rounds. Graph shattering is necessary. 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 root logn. This shows that any randomized O(1)+o(log(Delta) log n)-round algorithm can be derandomized to run in determinist