21,796 research outputs found
Graham Higman's PORC theorem
Graham Higman published two important papers in 1960. In the first of these
papers he proved that for any positive integer the number of groups of
order is bounded by a polynomial in , and he formulated his famous
PORC conjecture about the form of the function giving the number of
groups of order . In the second of these two papers he proved that the
function giving the number of -class two groups of order is PORC. He
established this result as a corollary to a very general result about vector
spaces acted on by the general linear group. This theorem takes over a page to
state, and is so general that it is hard to see what is going on. Higman's
proof of this general theorem contains several new ideas and is quite hard to
follow. However in the last few years several authors have developed and
implemented algorithms for computing Higman's PORC formulae in special cases of
his general theorem. These algorithms give perspective on what are the key
points in Higman's proof, and also simplify parts of the proof.
In this note I give a proof of Higman's general theorem written in the light
of these recent developments
An Efficient Algorithm for Enumerating Chordless Cycles and Chordless Paths
A chordless cycle (induced cycle) of a graph is a cycle without any
chord, meaning that there is no edge outside the cycle connecting two vertices
of the cycle. A chordless path is defined similarly. In this paper, we consider
the problems of enumerating chordless cycles/paths of a given graph
and propose algorithms taking time for each chordless cycle/path. In
the existing studies, the problems had not been deeply studied in the
theoretical computer science area, and no output polynomial time algorithm has
been proposed. Our experiments showed that the computation time of our
algorithms is constant per chordless cycle/path for non-dense random graphs and
real-world graphs. They also show that the number of chordless cycles is much
smaller than the number of cycles. We applied the algorithm to prediction of
NMR (Nuclear Magnetic Resonance) spectra, and increased the accuracy of the
prediction
Enumerating Cyclic Orientations of a Graph
Acyclic and cyclic orientations of an undirected graph have been widely
studied for their importance: an orientation is acyclic if it assigns a
direction to each edge so as to obtain a directed acyclic graph (DAG) with the
same vertex set; it is cyclic otherwise. As far as we know, only the
enumeration of acyclic orientations has been addressed in the literature. In
this paper, we pose the problem of efficiently enumerating all the
\emph{cyclic} orientations of an undirected connected graph with vertices
and edges, observing that it cannot be solved using algorithmic techniques
previously employed for enumerating acyclic orientations.We show that the
problem is of independent interest from both combinatorial and algorithmic
points of view, and that each cyclic orientation can be listed with
delay time. Space usage is with an additional setup cost
of time before the enumeration begins, or with a setup cost of
time
Real Rational Curves in Grassmannians
Fulton asked how many solutions to a problem of enumerative geometry can be
real, when that problem is one of counting geometric figures of some kind
having specified position with respect to some general fixed figures. For the
problem of plane conics tangent to five general conics, the (surprising) answer
is that all 3264 may be real. Similarly, given any problem of enumerating
p-planes incident on some general fixed subspaces, there are real fixed
subspaces such that each of the (finitely many) incident p-planes are real. We
show that the problem of enumerating parameterized rational curves in a
Grassmannian satisfying simple (codimension 1) conditions may have all of its
solutions be real.Comment: 9 pages, 1 eps figure, uses epsf.sty. Below the LaTeX source is a
MAPLE V.5 file which computes an example in the paper, and its outpu
Complete enumeration of two-Level orthogonal arrays of strength with constraints
Enumerating nonisomorphic orthogonal arrays is an important, yet very
difficult, problem. Although orthogonal arrays with a specified set of
parameters have been enumerated in a number of cases, general results are
extremely rare. In this paper, we provide a complete solution to enumerating
nonisomorphic two-level orthogonal arrays of strength with
constraints for any and any run size . Our results not only
give the number of nonisomorphic orthogonal arrays for given and , but
also provide a systematic way of explicitly constructing these arrays. Our
approach to the problem is to make use of the recently developed theory of
-characteristics for fractional factorial designs. Besides the general
theoretical results, the paper presents some results from applications of the
theory to orthogonal arrays of strength two, three and four.Comment: Published at http://dx.doi.org/10.1214/009053606000001325 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Shortest Distances as Enumeration Problem
We investigate the single source shortest distance (SSSD) and all pairs
shortest distance (APSD) problems as enumeration problems (on unweighted and
integer weighted graphs), meaning that the elements -- where
and are vertices with shortest distance -- are produced and
listed one by one without repetition. The performance is measured in the RAM
model of computation with respect to preprocessing time and delay, i.e., the
maximum time that elapses between two consecutive outputs. This point of view
reveals that specific types of output (e.g., excluding the non-reachable pairs
, or excluding the self-distances ) and the order of
enumeration (e.g., sorted by distance, sorted row-wise with respect to the
distance matrix) have a huge impact on the complexity of APSD while they appear
to have no effect on SSSD.
In particular, we show for APSD that enumeration without output restrictions
is possible with delay in the order of the average degree. Excluding
non-reachable pairs, or requesting the output to be sorted by distance,
increases this delay to the order of the maximum degree. Further, for weighted
graphs, a delay in the order of the average degree is also not possible without
preprocessing or considering self-distances as output. In contrast, for SSSD we
find that a delay in the order of the maximum degree without preprocessing is
attainable and unavoidable for any of these requirements.Comment: Updated version adds the study of space complexit
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