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. 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 the identity problem that has not been considered before.

翻译：长期以来一直在研究群体理论算法最坏的复杂情况。 通用复杂情况,或随机输入的复杂情况,最近才被引入研究。 在本文中,我们处理了若干类群体中单词问题的平均复杂时间,并表明,就输入单词的长度而言,平均复杂情况往往是线性的。 我们认为包括理性矩阵组(特别是多周期组)、某些可溶性组和自由产品组。对于自由产品,我们也处理分组成员问题的平均复杂情况,并表明它往往是线性的。最后,我们讨论了以前未曾审议过的身份问题。