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The power word problem for a group G asks whether an expression u 1 x 1 ⋯ u n x n , where the u i are words over a finite set of generators of G and the x i binary encoded integers, is equal to the identity of G . It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G ). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group G is uNC 1 -many-one reducible to the power word problem for a finite-index subgroup of G . For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is AC 0 -Turing-reducible to the word problem for the free group F 2 and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups C without order two elements, the uniform power word problem in a graph product can be solved in AC 0 [ C = L UPowWP (C) ] , where UPowWP (C) denotes the uniform power word problem for groups from the class C . As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is NP -complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated in Stober and Weiß (2022a) is true, our proof relies on this additional assumption. [ABSTRACT FROM AUTHOR]

MergeInsertion, also known as the Ford-Johnson algorithm, is a sorting algorithm which, up to today, for many input sizes achieves the best known upper bound on the number of comparisons. Indeed, it gets extremely close to the information-theoretic lower bound. While the worst-case behavior is well understood, only little is known about the average case. This work takes a closer look at the average case behavior. In particular, we establish an upper bound of n log n − 1.4005 n + o (n) comparisons. We also give an exact description of the probability distribution of the length of the chain a given element is inserted into and use it to approximate the average number of comparisons numerically. Moreover, we compute the exact average number of comparisons for n up to 148. Furthermore, we experimentally explore the impact of different decision trees for binary insertion. To conclude, we conduct experiments showing that a slightly different insertion order leads to a better average case and we compare the algorithm to Manacher's combination of merging and MergeInsertion as well as to the recent combined algorithm with (1,2)-Insertionsort by Iwama and Teruyama. [ABSTRACT FROM AUTHOR]