The Cycles of the Multiway Perfect Shuffle Permutation

The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρ _k,n: i → ki \bmod (kn+1), for 1 ≤ i ≤ kn. We uncover the cycle structure of the (k,n)-perfect shuffle permutation by a group-theoretic analysis and show how to compute representative elements from its cycles by an algorithm using O(kn) time and O((\log kn)^2) space. Consequently it is possible to realise the (k,n)-perfect shuffle via an in-place, linear-time algorithm. Algorithms that accomplish this for the 2-way shuffle have already been demonstrated

Parallel Cache-Efficient Algorithms on GPUs

Ph.D

Elmsley's Problem

Ovaj završni rad bavi se ulogom matematike u miješanju igraćih karata, konkretno Elmsleyevim problemom. Taj problem se odnosi na jedno od najtežih načina miješanja karata, popularno savršeno miješanje. Osim njega, detaljno ćemo objasniti i razne druge načine kako promiješati karte. Kod savršenog miješanja, pokušat ćemo odgovoriti na sljedeća pitanja: koliki je najmanji broj istovrsnih miješanja koje dovode karte do početne pozicije? Postoji li niz savršenih miješanja koji kartu s vrha špila dovode na odabranu poziciju u špilu? Za kraj ostavljamo glavno pitanje koje je postavio Elmsley: može li se nizom savršenih miješanja karta s pozicije p prebaciti na poziciju q? U radu ćemo ukratko opisati grupu generiranu savršenim miješanjima, a također ćemo spomenuti i neke od primjena koncepta savršenog miješanja.This bachelor’s thesis considers the role of mathematics in shuffling playing cards, specifically with the Elmsley’s problem. This problem is concerned with one of the most difficult shuffle methods, the popular perfect shuffling. Besides this problem, we will explain in details various types of card shuffling. Concerning perfect shuffle, we will try to give an answer to the following questions: what is the smallest number of perfect shuffles of the same kind required to bring deck of cards in the initial position? Is there a sequence of perfect shuffles which bring top card to the given position in a deck? The last question is the famous Elmsley’s problem: is there a sequence of perfect shuffles that bring card at the position p to the position q? In this thesis we will also shortly describe group generated by the sequence of perfect shuffles and mention some applications of the concept of perfect shuffling

Elmsley's Problem

Ovaj završni rad bavi se ulogom matematike u miješanju igraćih karata, konkretno Elmsleyevim problemom. Taj problem se odnosi na jedno od najtežih načina miješanja karata, popularno savršeno miješanje. Osim njega, detaljno ćemo objasniti i razne druge načine kako promiješati karte. Kod savršenog miješanja, pokušat ćemo odgovoriti na sljedeća pitanja: koliki je najmanji broj istovrsnih miješanja koje dovode karte do početne pozicije? Postoji li niz savršenih miješanja koji kartu s vrha špila dovode na odabranu poziciju u špilu? Za kraj ostavljamo glavno pitanje koje je postavio Elmsley: može li se nizom savršenih miješanja karta s pozicije p prebaciti na poziciju q? U radu ćemo ukratko opisati grupu generiranu savršenim miješanjima, a također ćemo spomenuti i neke od primjena koncepta savršenog miješanja.This bachelor’s thesis considers the role of mathematics in shuffling playing cards, specifically with the Elmsley’s problem. This problem is concerned with one of the most difficult shuffle methods, the popular perfect shuffling. Besides this problem, we will explain in details various types of card shuffling. Concerning perfect shuffle, we will try to give an answer to the following questions: what is the smallest number of perfect shuffles of the same kind required to bring deck of cards in the initial position? Is there a sequence of perfect shuffles which bring top card to the given position in a deck? The last question is the famous Elmsley’s problem: is there a sequence of perfect shuffles that bring card at the position p to the position q? In this thesis we will also shortly describe group generated by the sequence of perfect shuffles and mention some applications of the concept of perfect shuffling

Adaptation of multiway-merge sorting algorithm to MIMD architectures with an experimental study

Ankara : The Department of Computer Engineering and the Institute of Engineering and Science of Bilkent University, 2002.Thesis (Master's) -- Bilkent University, 2002.Includes bibliographical references leaves 73-78.Sorting is perhaps one of the most widely studied problems of computing. Numerous asymptotically optimal sequential algorithms have been discovered. Asymptotically optimal algorithms have been presented for varying parallel models as well. Parallel sorting algorithms have already been proposed for a variety of multiple instruction, multiple data streams (MIMD) architectures. In this thesis, we adapt the multiwaymerge sorting algorithm that is originally designed for product networks, to MIMD architectures. It has good load balancing properties, modest communication needs and well performance. The multiway-merge sort algorithm requires only two all-to-all personalized communication (AAPC) and two one-to-one communications independent from the input size. In addition to evenly distributed load balancing, the algorithm requires only size of 2N/P local memory for each processor in the worst case, where N is the number of items to be sorted and P is the number of processors. We have implemented the algorithm on the PC Cluster that is established at Computer Engineering Department of Bilkent University. To compare the results we have implemented a sample sort algorithm (PSRS Parallel Sorting by Regular Sampling) by X. Liu et all and a parallel quicksort algorithm (HyperQuickSort) on the same cluster. In the experimental studies we have used three different benchmarks namely Uniformly, Gaussian, and Zero distributed inputs. Although the multiwaymerge algorithm did not achieve better results than the other two, which are theoretically cost optimal algorithms, there are some cases that the multiway-merge algorithm outperforms the other two like in Zero distributed input. The results of the experiments are reported in detail. The multiway-merge sort algorithm is not necessarily the best parallel sorting algorithm, but it is expected to achieve acceptable performance on a wide spectrum of MIMD architectures.Cantürk, LeventM.S

The Design, modeling and simulation of switching fabrics: For an ATM network switch

The requirements of today\u27s telecommunication systems to support high bandwidth and added flexibility brought about the expansion of (Asynchronous Transfer Mode) ATM as a new method of high-speed data transmission. Various analytical and simulation methods may be used to estimate the performance of ATM switches. Analytical methods considerably limit the range of parameters to be evaluated due to extensive formulae used and time consuming iterations. They are not as effective for large networks because of excessive computations that do not scale linearly with network size. One the other hand, simulation-based methods allow determining a bigger range of performance parameters in a shorter amount of time even for large networks. A simulation model, however, is more elaborate in terms of implementation. Instead of using formulae to obtain results, it has to operate software or hardware modules requiring a certain amount of effort to create. In this work simulation is accomplished by utilizing the ATM library - an object oriented software tool, which uses software chips for building ATM switches. The distinguishing feature of this approach is cut-through routing realized on the bit level abstraction treating ATM protocol data units, called cells, as groups of 424 bits. The arrival events of cells to the system are not instantaneous contrary to commonly used methods of simulation that consider cells as instant messages. The simulation was run for basic multistage interconnection network types with varying source arrival rate and buffer sizes producing a set of graphs of cell delays, throughput, cell loss probability, and queue sizes. The techniques of rearranging and sorting were considered in the simulation. The results indicate that better performance is always achieved by bringing additional stages of elements to the switching system

Scalable String and Suffix Sorting: Algorithms, Techniques, and Tools

This dissertation focuses on two fundamental sorting problems: string sorting and suffix sorting. The first part considers parallel string sorting on shared-memory multi-core machines, the second part external memory suffix sorting using the induced sorting principle, and the third part distributed external memory suffix sorting with a new distributed algorithmic big data framework named Thrill.Comment: 396 pages, dissertation, Karlsruher Instituts f\"ur Technologie (2018). arXiv admin note: text overlap with arXiv:1101.3448 by other author

Sorting networks using k-comparators

Bibliography: leaves 160-167