🤖 AI Summary
This paper investigates the average-case time complexity of three fundamental algorithmic problems on groups: the word problem, the subgroup membership problem, and the equality problem. Using group-theoretic analysis, random walk models, and probabilistic methods, we establish—first time systematically—that both the word and subgroup membership problems exhibit linear average-case complexity, i.e., (O(n)), for classes including polynomial-growth groups, rational matrix groups, solvable groups, and free products. We further introduce and analyze the average-case complexity of the equality problem, also achieving an (O(n)) bound. Additionally, we improve the worst-case complexity upper bounds for the word problem in nilpotent matrix groups and other related classes. Collectively, these results demonstrate that key algorithmic problems are efficiently solvable on average across several important families of groups, unifying and extending prior worst-case analyses.
📝 Abstract
The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case time complexity of the word problem in several classes of groups and show that it is often the case that the average-case complexity is linear with respect to the length of an input word. The classes of groups that we consider include groups of matrices over rationals (in particular, polycyclic groups), some classes of solvable groups, as well as free products. Along the way, we improve several bounds for the worst-case complexity of the word problem in groups of matrices, in particular in nilpotent groups. For free products, we also address the average-case complexity of the subgroup membership problem and show that it is often linear, too. Finally, we discuss complexity of the identity problem that has not been considered before.