Armin Weiß

Armin Weiß
Universität Stuttgart · Institut für Formale Methoden der Informatik

About

56
Publications
4,896
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214
Citations
Citations since 2017
41 Research Items
174 Citations
2017201820192020202120222023051015202530
2017201820192020202120222023051015202530
2017201820192020202120222023051015202530
2017201820192020202120222023051015202530
Introduction
Armin Weiß currently works at the Institut für Formale Methoden der Informatik, Universität Stuttgart. Armin does research in Algorithms, and algorithmic and combinatorial group theory.
Additional affiliations
April 2016 - February 2017
Stevens Institute of Technology
Position
  • PostDoc Position
October 2011 - present
Universität Stuttgart
Position
  • Research Assistant

Publications

Publications (56)
Article
Full-text available
Let Gamma 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 structur...
Conference Paper
Full-text available
The conjugacy problem belongs to algorithmic group theory. It is the following question: 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 complexi...
Article
Baumslag-Solitar groups were introduced in 1962 by Baumslag and Solitar as examples for finitely presented non-Hopfian two-generator groups. Since then, they served as examples for a wide range of purposes. As Baumslag-Solitar groups are HNN extensions, there is a natural generalization in terms of graph of groups. Concerning algorithmic aspects of...
Article
It is well known that Quicksort -- which is commonly considered as one of the fastest in-place sorting algorithms -- suffers in an essential way from branch mispredictions. We present a novel approach to addressing this problem by partially decoupling control from dataflow: in order to perform the partitioning, we split the input into blocks...
Article
Full-text available
We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem and acts over a binary alphabet. Thus, it is optimal in terms of the alphabet size. Our construction directly simulates the c...
Article
We give lower bounds on the complexity of the word problem for a large class of non-solvable infinite groups that we call strongly efficiently non-solvable (SENS) groups. This class includes free groups, Grigorchuk’s group and Thompson’s groups. We prove that these groups have an \(\mathsf {NC}^{1} \) -hard word problem and that for some of them (i...
Chapter
The Baumslag group had been a candidate for a group with an extremely difficult word problem until Myasnikov, Ushakov, and Won succeeded to show that its word problem can be solved in polynomial time. Their result used the newly developed data structure of power circuits allowing for a non-elementary compression of integers. Later this was extended...
Article
Full-text available
The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell in (Inf. Comput. 178 (1), 253–262, 2002) where they showed that this problem is in for nilpotent groups while it is -complete for non-solvable groups. Since then, several results have appeared showing that the problem can b...
Conference Paper
Over twenty years ago, Goldmann and Russell initiated the study of the complexity of the equation satisfiability problem (PolSat and the NUDFA program satisfiability problem (ProgramSat) in finite groups. They showed that these problems are in 𝖯 for nilpotent groups while they are NP-complete for non-solvable groups. In this work we completely char...
Preprint
The Baumslag group had been a candidate for a group with an extremely difficult word problem until Myasnikov, Ushakov, and Won succeeded to show that its word problem can be solved in polynomial time. Their result used the newly developed data structure of power circuits allowing for a non-elementary compression of integers. Later this was extended...
Preprint
Full-text available
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...
Conference Paper
Full-text available
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. Sénizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in PSPACE when the input is given in terms of so-called virtually free presentations. Here we consider t...
Conference Paper
Full-text available
Power circuits have been introduced in 2012 by Myasnikov, Ushakov and Won as a data structure for non-elementarily compressed integers supporting the arithmetic operations addition and (x, y) → x·2 y. The same authors applied power circuits to give a polynomial-time solution to the word problem of the Baumslag group, which has a non-elementary Dehn...
Preprint
Full-text available
The power word problem of a group $G$ asks whether an expression $p_1^{x_1} \dots p_n^{x_n}$, where the $p_i$ are words and the $x_i$ binary encoded integers, is equal to the identity of $G$. We show that the power word problem in a fixed graph product is $\mathsf{AC}^0$-Turing-reducible to the word problem of the free group $F_2$ and the power wor...
Chapter
The power word problem of a group G asks whether an expression \(p_1^{x_1} \dots p_n^{x_n}\), where the \(p_i\) are words and the \(x_i\) binary encoded integers, is equal to the identity of G. We show that the power word problem in a fixed graph product is \({{{\textsf {{AC}}}}}^0\)-Turing-reducible to the word problem of the free group \(F_2\) an...
Preprint
Full-text available
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S\'enizergues and Wei\ss (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in $\mathsf{PSPACE}$ when the input is given in terms of so-called virtually free presentations. Here we conside...
Preprint
Full-text available
Power circuits have been introduced in 2012 by Myasnikov, Ushakov and Won as a data structure for non-elementarily compressed integers supporting the arithmetic operations addition and $(x,y) \mapsto x2^y$. The same authors applied power circuits to give a polynomial-time solution to the word problem of the Baumslag group, which has a non-elementar...
Preprint
Full-text available
The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell (2002) where they showed that this problem is in polynomial time for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in poly...
Article
Full-text available
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...
Conference Paper
Full-text available
Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in P for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of F...
Conference Paper
Full-text available
We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk's group and Thompson's groups, we prove that their word problem is ALOGTIME-hard. For some of these groups (including Grigorchuk's group and Thompson's groups)...
Article
Full-text available
QuickXsort is a highly efficient in-place sequential sorting scheme that mixes Hoare’s Quicksort algorithm with X, where X can be chosen from a wider range of other known sorting algorithms, like Heapsort, Insertionsort and Mergesort. Its major advantage is that QuickXsort can be in-place even if X is not. In this work we provide general transfer t...
Conference Paper
Full-text available
We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem. Our construction directly simulates the computation of a Turing machine in an automaton group and, therefore, seems to be qu...
Preprint
Full-text available
Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in P for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of F...
Preprint
Full-text available
We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk's group and Thompson's groups, we prove that their word problem is $\mathsf{NC}^1$-hard. For some of these groups (including Grigorchuk's group and Thompson's g...
Conference Paper
Full-text available
In this work we introduce a new succinct variant of the word problem in a finitely generated group G, which we call the power word problem: the input word may contain powers p^x , where p is a finite word over generators of G and x is a binary encoded integer. The power word problem is a restriction of the compressed word problem, where the input w...
Conference Paper
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...
Preprint
Full-text available
We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem. Our construction directly simulates the computation of a Turing machine in an automaton group and, therefore, seems to be qu...
Preprint
Full-text available
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...
Article
Full-text available
We show that the conjugacy problem in a wreath product A ≀ B is uniform-TC0-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. If B is torsion free, the power problem in B can be replaced by the slightly weaker cyclic submonoid membership problem in B. Moreover, if A is abelian, the cyclic subgroup membersh...
Preprint
Full-text available
In this work we introduce a new succinct variant of the word problem in a finitely generated group $G$, which we call the power word problem: the input word may contain powers $p^x$, where $p$ is a finite word over generators of $G$ and $x$ is a binary encoded integer. The power word problem is a restriction of the compressed word problem, where th...
Preprint
Full-text available
QuickXsort is a highly efficient in-place sequential sorting scheme that mixes Hoare's Quicksort algorithm with X, where X can be chosen from a wider range of other known sorting algorithms, like Heapsort, Insertionsort and Mergesort. Its major advantage is that QuickXsort can be in-place even if X is not. In this work we provide general transfer t...
Preprint
Full-text available
The two most prominent solutions for the sorting problem are Quicksort and Mergesort. While Quicksort is very fast on average, Mergesort additionally gives worst-case guarantees, but needs extra space for a linear number of elements. Worst-case efficient in-place sorting, however, remains a challenge: the standard solution, Heapsort, suffers from a...
Conference Paper
We present an algorithm for the following problem: given a context-free grammar for the word problem of a virtually free group G, compute a finite graph of groups G with finite vertex groups and fundamental group G. Our algorithm is non-deterministic and runs in doubly exponential time. It follows that the isomorphism problem of context-free groups...
Preprint
Full-text available
We consider the fundamental problem of internally sorting a sequence of $n$ elements. In its best theoretical setting QuickMergesort, a combination Quicksort with Mergesort with a Median-of-$\sqrt{n}$ pivot selection, requires at most $n \log n - 1.3999n + o(n)$ element comparisons on the average. The questions addressed in this paper is how to mak...
Preprint
Full-text available
We present an algorithm for the following problem: given a context-free grammar for the word problem of a virtually free group $G$, compute a finite graph of groups $\mathcal{G}$ with finite vertex groups and fundamental group $G$. Our algorithm is non-deterministic and runs in doubly exponential time. It follows that the isomorphism problem of con...
Chapter
We show a transfer result from individual groups to wreath products. Namely, we prove that the conjugacy problem in the wreath product AB of two groups A and B is log-space decidable, provided the factor groups A and B both have log-space decidable conjugacy problem and B has log-space computable power problem. If, additionally, A and B have bounde...
Chapter
The word problem of a finitely generated group is the set of words over the generators that are equal to the identity in the group. The word problem is therefore a formal language. If this language happens to be context-free, then the group is called context-free. Finitely generated virtually free groups are context-free. In the seminal paper Mulle...
Conference Paper
We show that the conjugacy problem in a wreath product \(A \wr B\) is uniform-\({{\mathsf {T}}}{{\mathsf {C}}}^0\)-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. Moreover, if B is torsion free, the power problem for B can be replaced by the slightly weaker cyclic submonoid membership problem for B, whic...
Preprint
Full-text available
Recently, MacDonald et. al. showed that many algorithmic problems for nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of presentations of subgroups can be done in Logspace. Here we follow their approach and show that all these problems are actually complete for the unif...
Article
Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices $UT_N(Z)$ for some $N$. In 2006, Nickel proposed an algorithm to calculate such embeddings. In this work, we show that if $UT_n(Z)$ is embedded into $UT_N(Z)$ using Nickel's algorithm, then...
Book
This volume presents the lecture notes from the authors’ three summer courses offered during the program “Automorphisms of Free Groups: Geometry, Topology, and Dynamics” held at the Centre de Recerca Matemàtica (CRM) in Bellaterra, Spain. The first two chapters present the basic tools needed, from formal language theory (regular and context-free l...
Preprint
Full-text available
We show that the conjugacy problem in a wreath product $A \wr B$ is uniform-$\mathsf{TC}^0$-Turing-reducible to the conjugacy problem in the factors $A$ and $B$ and the power problem in $B$. Moreover, if $B$ is torsion free, the power problem for $B$ can be replaced by the slightly weaker cyclic submonoid membership problem for $B$, which itself tu...
Article
Full-text available
The conjugacy problem asks whether two words over generators of a fixed group G are conjugated, i.e., it is the problem to decide on input words x, y whether there exists z such that 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 gr...
Article
In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. Thus, the question arises “how many” such elements there are. This question can be formalized using the notion of strongly generic sets and lower bounds can...
Article
Full-text available
QuickHeapsort is a combination of Quicksort and Heapsort. We show that the expected number of comparisons for QuickHeapsort is always better than for Quicksort if a usual median-of-constant strategy is used for choosing pivot elements. In order to obtain the result we present a new analysis for QuickHeapsort splitting it into the analysis of the pa...
Preprint
Full-text available
Since the work of Kaligosi and Sanders (2006), it is well-known that Quicksort -- which is commonly considered as one of the fastest in-place sorting algorithms -- suffers in an essential way from branch mispredictions. We present a novel approach to address this problem by partially decoupling control from data flow: in order to perform the partit...
Article
Full-text available
In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. This observation asks for a bound on how many such elements there are. Such bounds can be derived using the theory of amenable graphs: In this work we exami...
Thesis
Full-text available
This thesis deals with the conjugacy problem in classes of groups which can be written as HNN extension or amalgamated product. The conjugacy problem is one of the fundamental problems in algorithmic group theory which were introduced by Max Dehn in 1911. It poses the question whether two group elements given as words over a fixed set of generators...
Conference Paper
Full-text available
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 compar...
Preprint
Full-text available
The word problem of a finitely generated group is the formal language of words over the generators which are equal to the identity in the group. If this language happens to be context-free, then the group is called context-free. Finitely generated virtually free groups are context-free. In a seminal paper Muller and Schupp showed the converse: A co...
Preprint
Full-text available
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. With QuickWeakHeapsort and QuickMergesort we present two examples for the QuickXsort-construction. Both are efficient algorithm...
Conference Paper
Full-text available
A weak heap is a variant of a binary heap where, for each node, the heap ordering is enforced only for one of its two children. In 1993, Dutton showed that this data structure yields a simple worst-case-efficient sorting algorithm. In this paper we review the refinements proposed to the basic data structure that improve the efficiency even further....
Conference Paper
Full-text available
We present a new analysis for QuickHeapsort splitting it into the analysis of the partition-phases and the analysis of the heap-phases. This enables us to consider samples of non-constant size for the pivot selection and leads to better theoretical bounds for the algorithm. Furthermore we introduce some modifications of QuickHeapsort, both in-place...

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