In this paper we generalize the idea of QuickHeapsort leading to the notion of QuickXsort. Given some external sorting algorithm X, QuickXsort yields an internal sorting algorithm if X satisfies certain natural conditions. We show that up to o(n) terms the average number of comparisons incurred by QuickXsort is equal to the average number of comparisons of X.We also describe a new variant of WeakHeapsort. With QuickWeak-Heapsort and QuickMergesort we present two examples for the QuickXsort construction. Both are efficient algorithms that perform approximately n log n − 1.26n + o(n) comparisons on average. Moreover, we show that this bound also holds for a slight modification which guarantees an n log n + O(n) bound for the worst case number of comparisons.Finally, we describe an implementation of MergeInsertion and analyze its average case behavior. Taking MergeInsertion as a base case for QuickMergesort, we establish an efficient internal sorting algorithm calling for at most n log n − 1.3999n + o(n) comparisons on average. QuickMergesort with constant size base cases shows the best performance on practical inputs and is competitive to STL-Introsort.

No abstract

Let Γ be a connected, locally finite graph of finite tree width and G be a group acting on it with finitely many orbits and finite node stabilizers. We provide an elementary and direct construction of a tree T on which G acts with finitely many orbits and finite vertex stabilizers. Moreover, the tree is defined directly in terms of the structure tree of optimally nested cuts of Γ. Once the tree is constructed, Bass-Serre theory yields that G is virtually free. This approach simplifies the existing proofs for the fundamental result of Muller and Schupp that characterizes context-free groups as f.g. virtually free groups. Our construction avoids the explicit use of Stallings' structure theorem and it is self-contained.We also give a simplified proof for an important consequence of the structure tree theory by Dicks and Dunwoody which has been stated by Thomassen and Woess. It says that a f.g. group is accessible if and only if its Cayley graph is accessible.

No abstract

Abstract. The conjugacy is the following question in algorithmic group theory: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz −1 = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the conjugacy problem for two prominent groups: the Baumslag-Solitar group BS1,2 and the Baumslag(-Gersten) group G1,2. The conjugacy problem in BS1,2 is TC 0 -complete. To the best of our knowledge BS1,2 is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group G1,2 is an HNN extension of BS1,2. We show that the conjugacy problem is decidable (which has been known before); but our results go far beyond decidability. In particular, we are able to show that conjugacy in G1,2 can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs conjugacy in G1,2 can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in G1,2 by reducing the division problem in power circuits to the conjugacy problem in G1,2. The complexity of the division problem in power circuits is an open and interesting problem in integer arithmetic. To date it is believed that this problem has non-elementary time complexity. Another contribution of the paper concerns a general statement about HNN extension of the form G = H, b | bab −1 = ϕ(a), a ∈ A with a finitely generated base group H. We show that the complement of H is strongly generic if and only if A = H = B. This is the situation for G1,2; and yields an important piece of information why it is possible to solve conjugacy for G1,2 in strongly generic polynomial time. Note also that the complement of H is strongly generic if and only if the Schreier graph of G with respect to the subgroup H is non-amenable.