Algorithm 19 step-cycle generation

For given integerm>1, each step-cycle corresponds to a set of permutations such that the step-cycles constitute a set of equivalence classes on the set of all permutations onm elements. The algorithm has been used in connection with computations to search for groups consisting of a union of disjoint sets of permutations such that each set of permutations corresponds to a step-cycle, see [2] and [8].     

Parallel permutation generation on linear array

Given n items, a parallel algorithm for generating all the n! permutations is presented. The computational model used is a linear array which consists of n identical processing elements with a simple structure. One permutation is produced at each other time step. The elapsed time to produce a permutation is independent of the integer n. The basic idea used is the iterative method and the exchange of two consecutive components in an existing permutation. The design procedures of this algorithm are considered in detail. The ranking and unranking functions of the required permutations are also discussed.     

A Cost-Optimal Pipeline Algorithm for Permutation Generation in Lexicographic Order

In this paper we solve the open problem of designing a cost-optimal parallel algorithm for generating permutations ofMelements out of the set {0, 1, …,N− 1}, in lexicographic order. Our algorithm runs on the simplest model of parallel computation, i.e., a linear array of sizeM, where each processor PE(i) needs only a constant number of words of length logN, and is responsible for producing with constant delay, theith components in the successive permutations. This algorithm runs in timeO(N!/(N−M)!) on an array of sizeMand is therefore cost-optimal when the time needed to output the permutations is taken into account. Moreover, by doubling the number of cells, it is possible to implement this algorithm on a unidirectional linear array.     

Constant time generation of derangements

This paper presents the first constant time generation algorithm for derangements—permutations with no fixed points. Each derangement is obtained from its predecessor by making either one transposition or one rotation of three elements. It also generalizes this to permutations with a bounded number of fixed points.     

Intersection theorem for finite permutations

We find the maximal number of permutations on a set of n elements such that any pair of permutations has at least t common cycles.     

Common Intervals of Multiple Permutations

Given k permutations of n elements, a k-tuple of intervals of these permutations consisting of the same set of elements is called a common interval. We present an algorithm that finds in a family of k permutations of n elements all z common intervals in optimal O(kn+z) time and O(n) additional space. Additionally, we show how to adapt this algorithm to multichromosomal and circular permutations.     

On the Distinguishability of Distance-Bounded Permutations in Ordered Channels

Ordered channels, such as those provided by Internet protocol and transmission control protocol protocols, rely on sequence numbers to recover from packet reordering due to network dynamics. The existence of covert channels in any ordered channel is a well-known fact: Two parties can reorder the elements (packets) to be sent according to some predefined code. Schemes based on distance-bounded permutations have been proposed for steganographic communication with the aim of keeping and controling the increase of latency due to reordering. In this paper, we demonstrate that distance-bounded permutations are highly anomalous from a metric point of view. Our analysis is based on the study of the distribution of distances between normal permutations generated by the channel, and those produced when embedding hidden information. We provide results for four different distances: Kendall's tau, Spearman's rho, Spearman's footrule, and Levenshtein's distance (which is equivalent to Ulam's distance for permutations). In all cases, it is shown how sequences with hidden information can be separated from the normal ones. As a result, very accurate and efficient distinguishers can be easily constructed. Finally, we study the detection capabilities of the associated detectors through a receiver operating characteristic analysis.     

Don't Be Too Clever: Routing BMMC Permutations on the MasPar MP-2

The authors implemented and measured several methods to perform BMMC permutations on the MasPar MP-2. Each method is coded in C and MPL and uses only the global router to transfer data among processors. Implementation in MPL limits the amount of control the programmer has over the global router. The methods used were a naive method; the PE method, which adapts the block BMMC algorithm for parallel disk systems to treat processor elements as independent devices; and the cluster method, which adapts the block BMMC algorithm to treat clusters of processor elements as independent devices. The authors also implemented variations of the first two methods to compute virtual-processor numbers in Gray-code order. Our results indicate that for random BMMC permutations, the best method overall is the naive method with the Gray-code variation. The PE methods are almost as good, especially when each processor element has several items to route. For some permutations, however, the naive method performs very poorly; the best method in these cases is the PE method with the Gray-code variant. Although one might expect the cluster method to be best, since it minimizes congestion at the global router ports, this method turns out to be the worst overall, and it is always much worse than the PE method. If it had been coded at a lower level than MPL, the cluster method may well have turned out to be the best in many cases. Received January 1996, and in final form May 1997.     

A heuristic to accelerate in-situ permutation algorithms

In-situ permutation algorithms permute an array of n items without using a second array. Little space is needed when one proceeds permuting along a cycle. Thus, those algorithms first search for one element called the leader on each cycle of the permutation. Then they move the items cycle by cycle. A great fraction of the runtime is spent with testing whether elements are leaders. We present a simple-to-implement heuristic to accelerate the search for leaders and present performance gains for randomly chosen permutations on three algorithms.     

Routing automorphisms of the hypercube

We present an online algorithm for routing the automorphisms (BPC permutations) of the queueless MIMD hypercube. The routing algorithm has the virtue of being executed by each node of the hypercube without knowing the state of the others nodes. The algorithm is also vertex and link-contention free. We show, using the proposed algorithm, that BPC permutations are arbitrarily routable in the considered communication model.     

Recognizing sources of random strings

The identification of a source given a sequence of random strings is discussed. Two modes of random string generation are analyzed. In the first mode, arbitrary strings are generated in which the individual symbols occur exactly once in each random string. The latter case corresponds to the situation in which the sources generate random permutations. In both cases, the best match to the distribution being used by each source can be obtained by maintaining an exponential number of statistics. This being infeasible, a simple parameterization of the distributions is proposed. For arbitrary strings, the simple unigram-based model (U-model) is proposed. For the case of permutations, a new model called the S-model is proposed, and it is used to analyze and/or approximate unknown distributions of permutations. The relevant estimation procedures, together with the applications to source recognition, are presented. The method presents a unique blend of syntactic and statistical pattern recognition     

A genetic algorithm approach to a nurse rerostering problem

The nurse rerostering problem occurs when one or more nurses cannot work in shifts that were previously assigned to her or them. If no pool of reserve nurses exists to replace those absent, then the current roster must be rebuilt. This new roster must comply with the labour rules and institutional constraints. Moreover, it must be as similar as possible to the current one. The present paper describes constructive heuristics, besides several versions of genetic algorithms based on specific encoding and operators for sequencing problems applied to the nurse rerostering problem, defined with hard constraints. In the genetic algorithms described, each individual in the population is associated with a pair of chromosomes, representing permutations of tasks and nurses. Those permutations are used as input to a procedure that generates rosters. The fitness of individuals is given by the similarity between the roster generated from the permutations and the current one. The authors developed several versions of the genetic algorithm, whose difference lay in the encoding of permutations and in the genetic operators used for each encoding. These heuristics were tested with real data from a Lisbon hospital and yielded good quality solutions.     

Sorting with two ordered stacks in series

The permutations that can be sorted by two stacks in series are considered, subject to the condition that each stack remains ordered. A forbidden characterisation of such permutations is obtained and the number of permutations of each length is determined by a generating function.     

超启发式交叉熵算法求解模糊分布式流水线绿色调度问题

针对实际生产过程中普遍存在的加工时间不确定性,采用模糊数表示工件的加工时间,以同时最小化模糊最大完工时间和模糊总能耗为优化目标,建立模糊分布式流水线绿色调度问题(green distributed permutation flow-shop scheduling problem with fuzzy processing time,GDPFSP_FPT)的模型,进而提出一种超启发式交叉熵算法(hyper-heuristic cross-entropy algorithm,HHCE)进行求解.首先,HHCE采用一种新颖的三角模糊数排序准则合理计算个体的目标函数值,可在算法搜索过程中较准确发现优质解区域;其次,HHCE在高层利用基于贡献率的评价方法确定8种特定邻域操作所构成的各排列的优劣,同时采用交叉熵(cross-entropy,CE)方法学习较优排列的信息并生成新排列,进而在低层把高层生成的每个排列作为一种启发式算法,对低层相应个体执行一系列邻域操作,以实现对问题解空间较多不同区域的搜索;然后,HHCE将基于非关键路径的节能策略用于对低层每代种群中的较优个体执行局部搜索,从而进一步提高算法获取低能耗非劣个体或解的能力;最后,仿真实验与算法对比表明,HHCE可有效求解GDPFSP_FPT.     

More on permutation generation methods

We study a class of recursive permutation generation methods which construct a sequence of alln! permutations ofn elements by repeatedly generating all permutations of the elements in the firstn?1 positions and exchanging one of them with the element in then-th position. We give a general principle which enables us to obtain a whole class of permutation generation methods. This class includes the well-known algorithms of Wells and Heap as special cases, but contains also many new simple algorithms. Moreover, we are able to produce permutation generation methods with prescribed properties concerning the change that should be made in order to skip a block ofm! permutations with fixed elements in positionsm+1, …,n. For any method in our class, this change is a single transposition for any oddm>1, and a cyclic shift along a cycle of lengthm for any evenm.     

Routing Permutations on Graphs via Factors

We deal with the permutation routing problem on graphs modeling interconnection networks. In our model, calledrouting via factors, at each routing step, the communication pattern is a directed 1-factor in a symmetric digraph. This adds a new feature, that of continuous packet movement, to preciously studied routing types, where the routing of a permutation is reduced to a sequence of permutations from a given class. We especially focus on bipartite graphs and we give sufficient conditions for a graph to be rearrangeable in our model. We propose a general technic for routing via factors that we apply to the 2D mesh and the hypercube.     

Towards understanding optimal MIMD queueless routing of arbitrary permutations on hypercubes

This paper addresses the problem of the offline routing of arbitrary permutations on hypercubes under the MIMD queueless communication constraints which imposes that only one message can be located in each node of the hypercube right through the routing. According to e-cube routing, this kind of communication may require in the worst cases at least n exchanges steps on an n-dimensional hypercube. It has been conjectured that in the general case n steps suffice. The conjecture has been proved either by enumeration or by program for the particular cases of n??3. We revisit the problem through the k-partitioning paradigm based on maximum matching of bipartite graphs concept to take advantage of the recursive structure of the hypercube topology. The paradigm consists in looking for each message a transition node distant of at most say k<n hops from its current node such that all the messages can be routed to their transition nodes in k hops then finally routed to their final destination nodes in n?k hops on two disjoints hypercubes. In others words, the paradigm consists to determine an upstream permutation routable in k steps and which leads to two independent downstream permutations routable in n?k steps on two disjoint hypercubes. With this purpose, we give a characterization of the non-1-partitionable permutations from which the proof of the conjecture comes straightforwardly for n??2 and the non-1-partitionable permutations can be built whatever n may be. For n=3, we thus confirm the existence of exactly three classes of non-1-partitionable permutations and prove that there are two classes of upstream permutations to avoid when 2-partitioning any non-1-partitionable permutation. The process to avoid such upstream permutations is presented, and leads to a formal proof of the conjecture for n=3. For n>3, experiences gained in routing permutations on 4D-hypercubes allow conjecturing that in these cases, too, any non-1-partitionable permutation is 2-partitionable. Indeed, the analysis carried out for the 3D-hypercubes is repeatable to identify, certainly more laboriously given their combinatory, the upstream permutations to avoid when 2-partitioning a non-1-partitionable permutation on a 4D-hypercube. The proof that resulting downstream permutations on a 3D-hypercube can be routed in 2 steps is a consequence of the fact that for n=2 and 3 any permutation on a nD-hypercube can be routed in n steps.     

Semiparametric analysis of randomized response data with missing covariates in logistic regression

In this article, two semiparametric approaches are developed for analyzing randomized response data with missing covariates in logistic regression model. One of the two proposed estimators is an extension of the validation likelihood estimator of Breslow and Cain [Breslow, N.E., and Cain, K.C. 1988. Logistic regression for two-stage case-control data. Biometrika. 75, 11-20]. The other is a joint conditional likelihood estimator based on both validation and non-validation data sets. We present a large sample theory for the proposed estimators. Simulation results show that the joint conditional likelihood estimator is more efficient than the validation likelihood estimator, weighted estimator, complete-case estimator and partial likelihood estimator. We also illustrate the methods using data from a cable TV study.     

Performing BMMC Permutations Efficiently on Distributed-Memory Multiprocessors with MPI

This paper presents an architecture-independent method for performing BMMC permutations on multiprocessors with distributed memory. All interprocessor communication uses the MPI function MPI_Sendrecv_replace(). The number of elements and number of processors must be powers of 2, with at least one element per processor, and there is no inherent upper bound on the ratio of elements per processor. Our method transmits only data without transmitting any source or target indices, which conserves network bandwidth. When data is transmitted, the source and target processors implicitly agree on each other's identity and the indices of the elements being transmitted. A C-callable implementation of our method is available from Netlib. The implementation allows preprocessing (which incurs a modest cost) to be factored out for multiple runs of the same permutation, even if on different data. Data may be laid out in any one of several ways: processor-major, processor-minor, or anything in between. Experimental results indicate that our method works well compared with several other candidate methods on three different platforms. In particular, the slower the interconnection network, the greater the relative advantage of our method. Received June 1, 1997; revised March 10, 1998.     

Translation sets of permutations: Enumeration of selfcomplementary step-cycles

For given integern>2, there corresponds to each step-cycle a set of permutations, called a translation set. All the translation sets of permutations on the setS={1, 2, ...,n?1} form a set of equivalence classes on the set of all permutations onS. The selfcomplementary step-cycles are described and enumerated. A more general introduction to translation sets may be found in [4] and [3].