265,272 research outputs found
Cycles of free words in several independent random permutations with restricted cycle lengths
In this text, we consider random permutations which can be written as free
words in several independent random permutations: firstly, we fix a non trivial
word in letters , secondly, for all , we
introduce a -tuple of independent random permutations
of , and the random permutation we are going to
consider is the one obtained by replacing each letter in by .
For example, for , . Moreover, we restrict the set of possible lengths of
the cycles of the 's: we fix sets of positive integers
and suppose that for all , for all , is uniformly distributed on
the set of permutations of which have all their cycle lengths in
. For all positive integer , we are going to give asymptotics, as
goes to infinity, on the number of cycles of length of
. We shall also consider the joint distribution of the random vectors
. We first prove that the order of in a
certain quotient of the free group with generators determines
the rate of growth of the random variables as goes to
infinity. We also prove that in many cases, the distribution of
converges to a Poisson law with parameter and that the random variables
are asymptotically independent. We notice
the surprising fact that from this point of view, many things happen as if
were uniformly distributed on the -th symmetric group.Comment: 28 page
A complete proof of the safety of Nöcker's strictness analysis
This paper proves correctness of Nöcker's method of strictness analysis, implemented in the Clean compiler, which is an effective way for strictness analysis in lazy functional languages based on their operational semantics. We improve upon the work of Clark, Hankin and Hunt did on the correctness of the abstract reduction rules. Our method fully considers the cycle detection rules, which are the main strength of Nöcker's strictness analysis. Our algorithm SAL is a reformulation of Nöcker's strictness analysis algorithm in a higher-order call-by-need lambda-calculus with case, constructors, letrec, and seq, extended by set constants like Top or Inf, denoting sets of expressions. It is also possible to define new set constants by recursive equations with a greatest fixpoint semantics. The operational semantics is a small-step semantics. Equality of expressions is defined by a contextual semantics that observes termination of expressions. Basically, SAL is a non-termination checker. The proof of its correctness and hence of Nöcker's strictness analysis is based mainly on an exact analysis of the lengths of normal order reduction sequences. The main measure being the number of 'essential' reductions in a normal order reduction sequence. Our tools and results provide new insights into call-by-need lambda-calculi, the role of sharing in functional programming languages, and into strictness analysis in general. The correctness result provides a foundation for Nöcker's strictness analysis in Clean, and also for its use in Haskell
Minimum-weight Cycle Covers and Their Approximability
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L.
We investigate how well L-cycle covers of minimum weight can be approximated.
For undirected graphs, we devise a polynomial-time approximation algorithm that
achieves a constant approximation ratio for all sets L. On the other hand, we
prove that the problem cannot be approximated within a factor of 2-eps for
certain sets L.
For directed graphs, we present a polynomial-time approximation algorithm
that achieves an approximation ratio of O(n), where is the number of
vertices. This is asymptotically optimal: We show that the problem cannot be
approximated within a factor of o(n).
To contrast the results for cycle covers of minimum weight, we show that the
problem of computing L-cycle covers of maximum weight can, at least in
principle, be approximated arbitrarily well.Comment: To appear in the Proceedings of the 33rd Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2007). Minor change
Long path and cycle decompositions of even hypercubes
We consider edge decompositions of the -dimensional hypercube into
isomorphic copies of a given graph . While a number of results are known
about decomposing into graphs from various classes, the simplest cases of
paths and cycles of a given length are far from being understood. A conjecture
of Erde asserts that if is even, and divides the number
of edges of , then the path of length decomposes . Tapadia et
al.\ proved that any path of length , where , satisfying these
conditions decomposes . Here, we make progress toward resolving Erde's
conjecture by showing that cycles of certain lengths up to
decompose . As a consequence, we show that can be decomposed into
copies of any path of length at most dividing the number of edges of
, thereby settling Erde's conjecture up to a linear factor
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