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A connected undirected graph is called \emph{geodetic} if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the only geodetic Cayley graphs are odd cycles and complete graphs. In this article we present a series of theoretical results which contribute to a computer search verifying this conjecture for all groups of size up to 1024. The conjecture is also verified for several infinite families of groups including dihedral and some families of nilpotent groups. Two key results which enable the computer search to reach as far as it does are: if the center of a group has even order, then the conjecture holds (this eliminates all $2$-groups from our computer search); if a Cayley graph is geodetic then there are bounds relating the size of the group, generating set and center (which {significantly} cuts down the number of generating sets which must be searched)., Comment: 27 pages, 4 tables, 3 figures. Small edits and improvements to exposition

In 1962 Ore initiated the study of geodetic graphs. A graph is called geodetic if the shortest path between every pair of vertices is unique. In the subsequent years a wide range of papers appeared investigating their peculiar properties. Yet, a complete classification of geodetic graphs is out of reach. In this work we present a program enumerating all geodetic graphs of a given size. Using our program, we succeed to find all geodetic graphs with up to 25 vertices and all regular geodetic graphs with up to 32 vertices. This leads to the discovery of two new infinite families of geodetic graphs.

It is a long-standing open question to determine the minimum number of comparisons $S(n)$ that suffice to sort an array of $n$ elements. Indeed, before this work $S(n)$ has been known only for $n\leq 22$ with the exception for $n=16$, $17$, and $18$. In this work, we fill that gap by proving that sorting $n=16$, $17$, and $18$ elements requires $46$, $50$, and $54$ comparisons respectively. This fully determines $S(n)$ for these values and disproves a conjecture by Knuth that $S(16) = 45$. Moreover, we show that for sorting $28$ elements at least 99 comparisons are needed. We obtain our result via an exhaustive computer search which extends previous work by Wells (1965) and Peczarski (2002, 2004, 2007, 2012). Our progress is both based on advances in hardware and on novel algorithmic ideas such as applying a bidirectional search to this problem.

The power word problem for a group $G$ asks whether an expression $u_1^{x_1} \cdots u_n^{x_n}$, 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 $NC^1$-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 $\mathcal{C}$ without order two elements, the uniform power word problem in a graph product can be solved in $\mathsf{AC^0(C_=L^{UPowWP(\mathcal{C})})}$, where $UPowWP(\mathcal{C})$ denotes the uniform power word problem for groups from the class $\mathcal{C}$. 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. In [StoberW22] and previous iterations of this paper 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 previously is true, our proof relies on this additional assumption., Comment: Version 3 fixes a mistake in the previous versions. There our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. Version 3 also includes some content from arXiv:1904.08343

We characterize those strings whose suffix arrays are based on arithmetic progressions, in particular, arithmetically progressed permutations where all pairs of successive entries of the permutation have the same difference modulo the respective string length. We show that an arithmetically progressed permutation $P$ coincides with the suffix array of a unary, binary, or ternary string. We further analyze the conditions of a given $P$ under which we can find a uniquely defined string over either a binary or ternary alphabet having $P$ as its suffix array. For the binary case, we show its connection to lower Christoffel words, balanced words, and Fibonacci words. In addition to solving the arithmetically progressed suffix array problem, we give the shape of the Burrows-Wheeler transform of those strings solving this problem. These results give rise to numerous future research directions.

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 $n \log n - 1.4005n + o(n)$ 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 the recent combination with (1,2)-Insertionsort by Iwama and Teruyama.