Ingo Wegener

TCS Journal 2007 Journal Article

A comparison of simulated annealing with a simple evolutionary algorithm on pseudo-boolean functions of unitation

• Thomas Jansen
• Ingo Wegener

The development in the area of randomized search heuristics has shown the importance of a rigorous theoretical analysis of the performance of these heuristics. Unfortunately, the analysis of the expected optimization time of a specific algorithm has in general no implications on the behaviour of other algorithms — even if they differ only in some aspects. Indeed, small differences may imply large differences in the optimization time. Hence, it is an important issue to compare fundamental heuristics and to find out for which problems they behave in such a similar way that results on one heuristic can be transferred to the other one and to describe problems where they behave quite differently. Such an approach is performed here to the simple and well-known (1+1) EA, which is based on elitist selection and a global search operator, and simulated annealing, which is based on nonelitist selection and a local search operator.

TCS Journal 2007 Journal Article

Randomized local search, evolutionary algorithms, and the minimum spanning tree problem

• Frank Neumann
• Ingo Wegener

Randomized search heuristics, among them randomized local search and evolutionary algorithms, are applied to problems whose structure is not well understood, as well as to problems in combinatorial optimization. The analysis of these randomized search heuristics has been started for some well-known problems, and this approach is followed here for the minimum spanning tree problem. After motivating this line of research, it is shown that randomized search heuristics find minimum spanning trees in expected polynomial time without employing the global technique of greedy algorithms.

TCS Journal 2005 Journal Article

On converting CNF to DNF

• Peter Bro Miltersen
• Jaikumar Radhakrishnan
• Ingo Wegener

We study how big the blow-up in size can be when one switches between the CNF and DNF representations of Boolean functions. For a function f: { 0, 1 } n → { 0, 1 }, cnfsize ( f ) denotes the minimum number of clauses in a CNF for f; similarly, dnfsize ( f ) denotes the minimum number of terms in a DNF for f. For 0 ⩽ m ⩽ 2 n - 1, let dnfsize ( m, n ) be the maximum dnfsize ( f ) for a function f: { 0, 1 } n → { 0, 1 } with cnfsize ( f ) ⩽ m. We show that there are constants c 1, c 2 ⩾ 1 and ε > 0, such that for all large n and all m ∈ [ 1 ε n, 2 ε n ], we have 2 n - c 1 ( n / log ( m / n ) ) ⩽ dnfsize ( m, n ) ⩽ 2 n - c 2 ( n / log ( m / n ) ). In particular, when m is the polynomial n c, we get dnfsize ( n c, n ) = 2 n - θ ( c - 1 ( n / log n ) ).

I&C Journal 2005 Journal Article

On the influence of the variable ordering for algorithmic learning using OBDDs

• Matthias Krause
• Petr Savický
• Ingo Wegener

OBDDs with a fixed variable ordering are used successfully as data structure in experiments with learning heuristics based on examples. In this paper, it is shown that, for some functions, it is necessary to develop an algorithm to learn also a good OBDD variable ordering. There are functions with the following properties. They have OBDDs of linear size for optimal variable orderings. But for all but a small fraction of all variable orderings one needs large size to represent a list of randomly chosen examples. These properties are shown for simple functions like the multiplexer and the inner product.

TCS Journal 2005 Journal Article

The one-dimensional Ising model: Mutation versus recombination

• Simon Fischer
• Ingo Wegener

The investigation of genetic and evolutionary algorithms on Ising model problems gives much insight into how these algorithms work as adaptation schemes. The one-dimensional Ising model with periodic boundary conditions has been considered as a typical example with a clear building block structure suited well for two-point crossover. It has been claimed that GAs based on recombination and appropriate diversity-preserving methods by far outperform EAs based on mutation only. Here, a rigorous analysis of the expected optimization time proves that mutation-based EAs are surprisingly effective. The ( 1 + λ ) EA with an appropriate λ -value is almost as efficient as typical GAs. Moreover, it is proved that specialized GAs do even better and this holds for two-point crossover as well as for one-point crossover.

TCS Journal 2004 Journal Article

Real royal road functions for constant population size

• Tobias Storch
• Ingo Wegener

Evolutionary and genetic algorithms (EAs and GAs) are quite successful randomized function optimizers. This success is mainly based on the interaction of different operators like selection, mutation, and crossover. Since this interaction is still not well understood, one is interested in the analysis of the single operators. Jansen and Wegener [Proceedings of GECCO’2001, 2001, pp. 375–382] have described so-called real royal road functions where simple steady-state GAs have a polynomial expected optimization time while the success probability of mutation-based EAs is exponentially small even after an exponential number of steps. This success of the GA is based on the crossover operator and a population whose size is moderately increasing with the dimension of the search space. Here new real royal road functions are presented where crossover leads to a small optimization time, although the GA works with the smallest possible population size—namely 2.

MFCS Conference 2003 Conference Paper

On Converting CNF to DNF

• Peter Bro Miltersen
• Jaikumar Radhakrishnan
• Ingo Wegener

Abstract We study how big the blow-up in size can be when one switches between the CNF and DNF representations of boolean functions. For a function f: {0, 1} n →{0, 1}, \({\mathsf{cnfsize}}\left(f\right)\) denotes the minimum number of clauses in a CNF for f; similarly, \({\mathsf{dnfsize}}\left(f\right)\) denotes the minimum number of terms in a DNF for f. For 0≤ m ≤ 2 n − 1, let \({\mathsf{dnfsize}}\left(m, n\right)\) be the maximum \({\mathsf{dnfsize}}\left(f\right)\) for a function f: {0, 1} n →{0, 1} with \({\mathsf{cnfsize}}\left(f\right) \leq m\). We show that there are constants c 1, c 2 ≥ 1 and ε > 0, such that for all large n and all \(m \in [ \frac{1}{\epsilon}n, 2^{\epsilon{n}}]\), we have $$ 2^{n -- c_1\frac{n}{\log(m/n)}}~ \leq~ {\mathsf{dnfsize}}(m, n) ~\leq~ 2^{n-c_2 \frac{n}{\log(m/n)}}. $$ In particular, when m is the polynomial n c, we get \({\mathsf{dnfsize}} (n^c, n) = 2^{n -\theta(c^{-1}\frac{n}{\log n})}\).

MFCS Conference 2003 Invited Paper

Towards a Theory of Randomized Search Heuristics

• Ingo Wegener

Abstract There is a well-developed theory about the algorithmic complexity of optimization problems. Complexity theory provides negative results which typically are based on assumptions like NP≠P or NP≠RP. Positive results are obtained by the design and analysis of clever algorithms. These algorithms are well-tuned for their specific domain. Practitioners, however, prefer simple algorithms which are easy to implement and which can be used without many changes for different types of problems. They report surprisingly good results when applying randomized search heuristics like randomized local search, tabu search, simulated annealing, and evolutionary algorithms. Here a framework for a theory of randomized search heuristics is presented. It is discussed how randomized search heuristics can be delimited from other types of algorithms. This leads to the theory of black-box optimization. Lower bounds in this scenario can be proved without any complexity-theoretical assumption. Moreover, methods how to analyze randomized search heuristics, in particular, randomized local search and evolutionary algorithms are presented.

TCS Journal 2002 Journal Article

How to analyse evolutionary algorithms

• Hans-Georg Beyer
• Hans-Paul Schwefel
• Ingo Wegener

Many variants of evolutionary algorithms have been designed and applied. The experimental knowledge is immense. The rigorous analysis of evolutionary algorithms is difficult, but such a theory can help to understand, design, and teach evolutionary algorithms. In this survey, first the history of attempts to analyse evolutionary algorithms is described and then new methods for continuous as well as discrete search spaces are presented and discussed.

TCS Journal 2002 Journal Article

On the analysis of the (1+1) evolutionary algorithm

• Stefan Droste
• Thomas Jansen
• Ingo Wegener

Many experimental results are reported on all types of Evolutionary Algorithms but only few results have been proved. A step towards a theory on Evolutionary Algorithms, in particular, the so-called (1+1) Evolutionary Algorithm, is performed. Linear functions are proved to be optimized in expected time O(nlnn) but only mutation rates of size Θ(1/n) can ensure this behavior. For some polynomial of degree 2 the optimization needs exponential time. The same is proved for a unimodal function. Both results were not expected by several other authors. Finally, a hierarchy result is proved. Moreover, methods are presented to analyze the behavior of the (1+1) Evolutionary Algorithm.

I&C Journal 2002 Journal Article

On the Nonapproximability of Boolean Functions by OBDDs and Read-k-Times Branching Programs

• Beate Bollig
• Martin Sauerhoff
• Ingo Wegener

Branching programs are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e. g. , verification), exact representations are required. For learning boolean functions f on the basis of classified examples, it is sufficient to produce the representation of a function g approximating f. This motivates the investigation of the size of the smallest branching program approximating f. Although several nonapproximability results are contained in the papers on randomized branching programs, these results often do not work for the uniform distribution (which is the most important one in applications). Here, the following nonapproximability results are presented. (1) It is proven that two simple and well-known functions from the branching program literature require exponential size to be approximated with respect to the uniform distribution by OBDDs, which are the most important type of branching programs in applications. (2) The first truly exponential lower bound on the size of approximating syntactic read-k-times branching programs with respect to the uniform distribution and error probability 1/2−2−Ω(n), n the input size, is shown. In order to improve upon the best previous results for error probabilities smaller than 1/3, a strong combinatorial lemma from a paper of Ajtai on linear-length branching programs is exploited.

TCS Journal 2002 Journal Article

Optimization with randomized search heuristics—the (A)NFL theorem, realistic scenarios, and difficult functions

• Stefan Droste
• Thomas Jansen
• Ingo Wegener

The No Free Lunch (NFL) theorem due to Wolpert and Macready (IEEE Trans. Evol. Comput. 1(1) (1997) 67) has led to controversial discussions on the usefulness of randomized search heuristics, in particular, evolutionary algorithms. Here a short and simple proof of the NFL theorem is given to show its elementary character. Moreover, the proof method leads to a generalization of the NFL theorem. Afterwards, realistic complexity theoretical-based scenarios for black box optimization are presented and it is argued why NFL theorems are not possible in such situations. However, an Almost No Free Lunch (ANFL) theorem shows that for each function which can be optimized efficiently by a search heuristic there can be constructed many related functions where the same heuristic is bad. As a consequence, search heuristics use some idea how to look for good points and can be successful only for functions “giving the right hints”. The consequences of these theoretical considerations for some well-known classes of functions are discussed.

TCS Journal 2000 Journal Article

Identification of partial disjunction, parity, and threshold functions

• Ryuhei Uehara
• Kensei Tsuchida
• Ingo Wegener

Let F be a class of functions obtained by replacing some inputs of a Boolean function of a fixed type with some constants. The problem considered in this paper, which is called attribute efficient learning, is to identify “efficiently” a Boolean function g out of F by asking for the value of g at chosen inputs, where “efficiency” is measured in terms of the number of essential variables. We study the query complexity of attribute-efficient learning for three function classes that are, respectively, obtained from disjunction, parity, and threshold functions. In many cases, we obtain almost optimal upper and lower bound on the number of queries.

I&C Journal 1998 Journal Article

Completeness and Non-Completeness Results with Respect to Read-Once Projections

• Beate Bollig
• Ingo Wegener

Several models of restricted branching programs or binary decision diagrams have a lot of applications in hardware verification. These models are investigated here from a complexity theoretical viewpoint. Because of depth restrictions projections are not suitable as reduction type and have to be restricted to read-once projections. Several types of polynomial-size binary decision diagrams have complete problems with respect to polynomial read-once projections. On the contrary it can be proved that the classes of polynomial-size decision trees and polynomial-size read-once branching programs or free binary decision diagrams do not have such complete problems.

TCS Journal 1998 Journal Article

Hierarchy theorems for kOBDDs and kIBDDs

• Beate Bollig
• Martin Sauerhoff
• Detlef Sieling
• Ingo Wegener

Almost the same types of restricted branching programs (or binary decision diagrams BDDs) are considered in complexity theory and in applications like hardware verification. These models are read-once branching programs (free BDDs) and certain types of oblivious branching programs (ordered and indexed BDDs with k layers). The complexity of the satisfiability problem for these restricted branching programs is investigated and tight hierarchy results are proved for the classes of functions representable by k layers of ordered or indexed BDDs of polynomial size.

MFCS Conference 1997 Conference Paper

Complexity Theoretical Results on Partitioned (Nondeterministic) Binary Decision Diagrams

• Beate Bollig
• Ingo Wegener

Abstract Ordered binary decision diagrams (OBDDs) have found a lot of applications in the verification of combinational and sequential circuits, protocols, and the synthesis and analysis of systems. The applications are limited, since the expressive power of polynomial-size OBDDs is too restricted. Therefore, several more general BDD models are used. Partitioned OBDDs are an OBDD model allowing a restricted use of nondeterminism and different variable orderings. They are restricted enough such that the essential operations can be performed efficiently and they allow polynomial-size representations for many more functions than OBDDs. Here the expressive power of polynomial-size partitioned OBDDs is investigated. A tight hierarchy with respect to the number of parts in the partition is proved and partitioned OBDDs are compared with other BDD models.

MFCS Conference 1997 Conference Paper

On O versus NP \cap co-NP for Decision Trees and Read-Once Branching Programs

• Stasys Jukna
• Alexander A. Razborov
• Petr Savický
• Ingo Wegener

Abstract It is known that if a Boolean function f in n variables has a DNF and a CNF of size ≤ N then f also has a (deterministic) decision tree of size exp( O (log n log 2 N )). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp ( Ω (log 2 N )) where N is the total number of monomials in minimal DNFs for f and - f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen-Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Whereas other examples have the additional property that f is in AC°.

TCS Journal 1995 Journal Article

Graph driven BDDs — a new data structure for Boolean functions

• Detlef Sieling
• Ingo Wegener

(Ordered) binary decision diagrams (OBDDs) are used as a data structure for Boolean functions in the logical synthesis process, for verification and test pattern generation, and as part of CAD tools. For several important functions like arithmetical and logical units with quite different functions, the indirect storage access function or the hidden weighted bit function OBDDs have exponential size for any ordering of the variables. Since an ordering of the variables may be stored as a list, OBDDs may also be called list driven BDDs. Two new generalized models of graph driven BDDs are presented. The above mentioned and many other functions can be represented in small polynomial size in this model and the usual operations on OBDDs can be performed efficiently also for graph driven BDDs.

TCS Journal 1993 Journal Article

BOTTOM-UP-HEAPSORT, a new variant of HEAPSORT beating, on an average, QUICKSORT (if n is not very small)

• Ingo Wegener

A variant of HEAPSORT, called BOTTOM-UP-HEAPSORT, is presented. It is based on a new reheap procedure. This sequential sorting algorithm is easy to implement and beats, on an average, QUICKSORT if n⩾400 and a clever version of QUICKSORT (where the split object is the median of 3 randomly chosen objects) if n⩾16000. The worst-case number of comparisons is bounded by 1. 5n log n+O(n). Moreover, the new reheap procedure improves the delete procedure for the heap data structure for all n.

I&C Journal 1992 Journal Article

The worst case complexity of McDiarmid and Reed's variant of BOTTOM-UP HEAPSORT is less than n log n + 1.1n

• Ingo Wegener

BOTTOM-UP HEAPSORT is a variant of HEAPSORT which beats on average even the clever variants of QUICKSORT, if n is not very small. Up to now, the worst case complexity of BOTTOM-UP HEAPSORT has been able to be estimated only by 1. 5n log n. McDiarmid and Reed (1989) have presented a variant of BOTTOM-UP HEAPSORT which needs extra storage for n bits. The worst case number of comparisons of this (almost internal) sorting algorithm is estimated by n log n + 1. 1 n. It is discussed how many comparisons can be saved on average.

TCS Journal 1991 Journal Article

The complexity of the parity function in unbounded fan-in, unbounded depth circuits

• Ingo Wegener

Almost everything is known on the complexity of the parity function in fan-in 2 circuits over various bases. Also the minimal depth of polynomial-size, unbounded fan-in {∧, ∨, ⌝\dl; } circuits for the parity function has been studied. Here the complexity without any depth restriction is considered. For the basis {∧, ∨, ⌝\dl; } almost optimal bounds, and for the basis of NOR gates and the basis of all threshold functions optimal bounds on the number of gates are obtained. For the basis {∧, ∨, ⌝\dl; } the minimal number of wires is determined. For threshold circuits an exponential gap between synchronous and asynchronous circuits is proved. The results not only answer open questions in complexity theory but also have implications for the real-life circuit design.

TCS Journal 1991 Journal Article

The conjunctive complexity of quadratic Boolean functions

• Katja Lenz
• Ingo Wegener

The minimal number, of conjuctions in monotone circuits for quadratic Boolean functions, i. e. disjunctions of quadratic monomials x i x j, is investigated. Single level circuits which have only one level of conjuctions are compared with arbitrary monotone circuits. The computation of the single level complexity is shown to be NP complete. For almost all quadratic functions, almost optimal circuits can be computed in polynomial time. The single level conjecture is disproved, i. e. a quadratic function is defined whose single level complexity is larger than its conjuctive complexity

MFCS Conference 1990 Conference Paper

Bottom-Up-Heap Sort, a New Variant of Heap Sort Beating on Average Quick Sort (if n is not very small)

• Ingo Wegener

Abstract A new variant of HEAP SORT, called BOTTOM-UP-HEAP SORT, is presented. This new sequential sorting algorithm is easy to implement and beats QUICK SORT on average, if n ≥ 400, and the best-off-three version of QUICK SORT, if n ≥ 16000. The worst case number of comparisons can be bounded by 1. 5 n log n + O ( n ).

MFCS Conference 1990 Conference Paper

Symmetric Functions in AC 0 A Can Be Computed in Constant Depth With Very Small Size

• Ingo Wegener
• Norbert Wurm
• Sang-Zin Yi

Abstract It is well-known which symmetric Boolean functions can be computed by constant depth, polynomial size, unbounded fan-in circuits, i. e. which are contained in the complexity class AC 0. This result is sharpened. Symmetric Boolean functions in AC 0 can be computed by unbounded fan-in circuits with small constant depth, an almost linear number of n log O (1) n wires and a subpolynomial (but superpolylogarithmic) number of 2 O (logδ n ) gates for some δ<1.

CSL Conference 1989 Conference Paper

A Remark on Minimal Polynomials of Boolean Functions

• Bernd Voigt
• Ingo Wegener

Abstract We consider two problems in connection with minimal polynomials of Boolean functions. We first discuss the Disjoint Computation Scheme Hypothesis. Then we show how to compute efficiently minimal polynomials for symmetric Boolean functions.

I&C Journal 1989 Journal Article

Minimal polynomials for the conjunction of functions on disjoint variables can be very simple

• Bernd Voigt
• Ingo Wegener

Let h(x, y) = f(x) ∧ g(y) be the conjunction of Boolean functions f(x) and g(y), where f and g depend on x- (resp. , y)-variables only. It is shown that a minimal polynomial for h may be remarkably small. The Disjoint Computation Scheme Hypothesis is discussed.

CSL Conference 1988 Conference Paper

The Conjunctive Complexity of Quadratic Boolean Functions

• Katja Lenz
• Ingo Wegener

Abstract The minimal number of conjunctions in monotone circuits for quadratic Boolean functions, i. e. disjunctions of quadratic monomials x i x j, is investigated. Single level circuits which have only one level of conjunctions are compared with arbitrary monotone circuits. The computation of the single level complexity is shown to be NP complete. For almost all quadratic functions almost optimal circuits can be computed in polynomial time. The single level conjecture is disproved, i. e. some quadratic function is defined whose single level complexity is larger than its conjunctive complexity.

TCS Journal 1986 Journal Article

More on the complexity of slice functions

• Ingo Wegener

One of the hardest problems in complexity theory seems to be to find a proof that the circuit complexity of some explicitly defined Boolean function is nonlinear in the number of inputs and outputs. An interesting approach for such a bound is the investigation of slice functions since for these functions circuits over a complete basis, monotone circuits, and the so-called set circuits are nearly equivalent. In this paper the theory of slice functions is extended. A monotone representation of each Boolean function whose monotone complexity is at most a factor n larger than its circuit complexity is presented. For some slice functions and some other functions new pseudocomplements with small monotone complexity are presented. These results lead to new and intuitive problems whose solution would lead to nonlinear lower bounds on circuit complexity. Some connections between the complexity of n-output and 1-output slices and between the complexity of the k-slice and the (k + 1)-slice of the same function are discussed.

TCS Journal 1986 Journal Article

Properties of complexity measures for PRAMs and WRAMs

• Siegfried Bublitz
• Ute Schürfeld
• Ingo Wegener
• Bernd Voigt

The computation of Boolean functions by parallel computers with shared memory (PRAMs and WRAMs) is considered. In particular, complexity measures for parallel computers like critical and sensitive complexity are compared with other complexity measures for Boolean functions like branching program depth and length of prime implicants and clauses. The relations between these complexity measures and their asymptotic behaviour are investigated for the classes of Boolean functions, monotone functions and symmetric functions.

MFCS Conference 1986 Conference Paper

Properties of Complexity Measures for PRAMs and WRAMs

• Siegfried Bublitz
• Ute Schürfeld
• Bernd Voigt
• Ingo Wegener

Abstract The computation of Boolean functions by parallel computers with shared memory (PRAMs and WRAMs) is considered. In particular complexity measures for parallel computers like critical and sensitive complexity are compared with other complexity measures for Boolean functions like branching program depth and length of prime implicants and clauses. The relations between these complexity measures and their asymptotic behavior are investigated for the classes of all Boolean functions, monotone functions and symmetric functions.

TCS Journal 1985 Journal Article

On the complexity of slice functions

• Ingo Wegener

By a result of Berkowitz (1982), the monotone circuit complexity of slice functions cannot be much larger than the circuit (combinational) complexity of these functions for arbitrary complete bases. This result strengthens the importance of the theory of monotone circuits. We show in this paper that monotone circuits for slice functions can be understood as special circuits called set circuits. Here, disjunction and conjunction are replaced by set union and set intersection. All the main methods known for proving lower bounds on the monotone complexity of Boolean functions fail to work in their present form for slice functions. Furthermore, we show that the canonical slice functions of the Boolean convolution, the Nechiporuk Boolean sums, and the clique function can be computed with a linear number of gates.

MFCS Conference 1984 Conference Paper

On the Complexity of Slice Functions

• Ingo Wegener

Abstract By a result of Berkowitz the monotone circuit complexity of slice functions cannot be much larger than the circuit (combinational) complexity of these functions for arbitrary complete bases. This result strengthens the importance of the theory of monotone circuits. We show in this paper that monotone circuits for slice functions can be understood as special circuits called set circuits. Here disjunction and conjunction are replaced by set union and set intersection. All known methods for proving lower bounds on the monotone complexity of Boolean functions do not work in their present form for slice functions. Furthermore we show that the canonical slice functions of the Boolean convolution, the Nechiporuk Boolean sums and the clique function can be computed with linear many gates.

TCS Journal 1982 Journal Article

Boolean functions whose monotone complexity is of size n2log n

• Ingo Wegener

We construct a sequence of monotone Boolean functions h n: {0, 1} n →{0, 1} n, such that the monotone complexity of h n is of order n2 log n. This result includes the largest known lower bound of this kind. Previously there were an Ω(n 3 2 ) bound for the Boolean matrix product, an Ω(n 5 3 ) bound for Boolean sums and an Ω( n2 log2n ) bound by the author for the same functions h n. This new lower bound is proved by new methods which probably will turn out to be useful also for other problems.

TCS Journal 1979 Journal Article

A counterexample to a conjecture of Schnorr referring to monotone networks

• Ingo Wegener

Schnorr[1] proved a lower bound on the number of additions in monotone computations of rational polynomials. He conjectured a similar lower bound on the number of ∨-gates in monotone networks computing monotone Boolean functions. We disprove this conjecture.

TCS Journal 1979 Journal Article

Switching functions whose monotone complexity is nearly quadratic

• Ingo Wegener

A sequence of monotone switching functions h n: {0, 1}n→ {0, 1}n is constructed, such that the monotone complexity of h n grows faster than Ω(n 2 log−2 n). Previously the best lower bounds of this nature were several Ω(n 3 2 bounds due to Pratt, Paterson, Mehlhorn and Galil and Savage.

STOC Conference 1978 Conference Paper

Switching Functions Whose Monotone Complexity Is Nearly Quadratic

• Ingo Wegener