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### A generalization of the 0–1 principle for sorting

Sanguthevar Rajasekaran, Sandeep Sen
2005 Information Processing Letters
The traditional zero-one principle for sorting networks states that "if a network with n input lines sorts all 2 n binary sequences into nondecreasing order, then it will sort any arbitrary sequence of  ...  We generalize this to the situation when a network sorts almost all binary sequences and relate it to the behavior of the sorting network on arbitrary inputs.  ...  The generalized 0-1 principle has potential benefits for other sorting algorithms and to the best of our knowledge no formal generalization of the 0-1 principle existed in literature.  ...

### A generalization of the zero–one principle for sorting algorithms

Dorothea Wagner, Frank Wagner
1991 Discrete Applied Mathematics
As a consequence a zero-one principle for a more general class of sorting algorithms is derived. meinschaft under grant Mh 446/1-l. 0166-218X/91/\$03.50 0 1991 -Elsevier Science Publishers B.V.  ...  Wagner, A generalization of the zero-one principle for sorting algorithms, Discrete Applied Mathematics 30 (1991) 265-273.  ...  Moreover, also for a lot of new "extensive" sorting methods, for example parallel sorting algorithms [ 1, 3] , the zero-one principle turns out to be an invaluable expedient.  ...

### Designing A Novel General Sorting Network Constructor Using Artificial Evolution

Michal Bidlo, Radek Bidlo, Lukas Sekanina
2008 Zenodo
A method is presented for the construction of arbitrary even-input sorting networks exhibiting better properties than the networks created using a conventional technique of the same type.  ...  Similarly to human inventions in the area of theoretical computer science, the evolved invention was analyzed: its generality was proven and area and time complexities were determined.  ...  Fig. 5 . 5 The principle of creating (k + 1)-degree sorting network from a k-degree sorting network a) a 2k+1 = 0 and a 2k+2 = 0 Fig. 6 . 6 Processing data sequences: (a) a 2k+1 = 0, a 2k+2 = 0, (b)  ...

### Evolutionary Design of Arbitrarily Large Sorting Networks Using Development

Lukáš Sekanina, Michal Bidlo
2005 Genetic Programming and Evolvable Machines
The proposed approach allowed us to rediscover the conventional principle of insertion which is traditionally used for constructing large sorting networks.  ...  Furthermore, the principle was improved by means of the evolutionary technique. The evolved sorting networks exhibit a lower implementation cost and delay.  ...  Acknowledgements The research was performed with the Grant Agency of the Czech Republic under No. 102/03/P004 Evolvable hardware based application design methods and No. 102/04/0737 Modern methods of digital  ...

### Combinatorial Techniques for Proof-Based Synthesis of Sorting Algorithms

Isabela Dramnesc, Tudor Jebelean, Sorin Stratulat
2015 2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)
We present the main principles and the application of this technique to merging of sorted binary trees into a sorted one.  ...  In the frame of our previous experiments for proof based synthesis of sorting algorithms for lists and for binary trees, we employed certain special techniques which are able to generate multiple variants  ...  Since each such expression must represent a sorted tree, generate for each expression the corresponding condition as a set of microatoms (see IR-1).  ...

### Optimal conclusive sets for comparator networks

Guy Even, Tamir Levi, Ami Litman
2009 Theoretical Computer Science
Nakatani et. al. [11] used it to prove the correctness of a bitonic sorter. Rajasekaran and Sen [12] generalized the 0-1 principle to networks that sort almost all 0-1 inputs.  ...  Miltarsen et. al. [10] used a variant of the 0-1 principle to prove the correctness of a merging network.  ...  Acknowledgments We thank Tuvi Etzion for the suggestion to use Sperner's Theorem in the proof of Theorem 15.  ...

### Proof–Based Synthesis of Sorting Algorithms for Trees [chapter]

Isabela Drămnesc, Tudor Jebelean, Sorin Stratulat
2016 Lecture Notes in Computer Science
We develop various proof techniques for the synthesis of sorting algorithms on binary trees, by extending our previous work on the synthesis of algorithms on lists.  ...  The proof techniques are implemented in the Theorema system and are used for the automatic synthesis of several algorithms for sorting and for the auxiliary functions, from which we present few here.  ...  This generates the same conjecture for the synthesis of Merge and also the last branch in the following sorting algorithm, knowing that if the proof succeeds to find a witness T * = [n, L 0 , R 0 , T 1  ...

### Proof–Based Synthesis of Sorting Algorithms Using Multisets in Theorema

Isabela Drămnesc, Tudor Jebelean
2019 Electronic Proceedings in Theoretical Computer Science
We use the classical principle of extracting the algorithm as a set of rewrite rules based on the witnesses found in the proof of the synthesis conjecture produced from the specification of the desired  ...  Using multisets, we develop novel techniques for mechanizing the proofs of the synthesis conjectures for list-sorting algorithms, and we demonstrate them in the Theorema system.  ...  (For lack of space we omit here a possible splitting algorithm and its automatic generation by the principles presented in this paper.)  ...

### Order-Sorted Inductive Types

Gilles Barthe
1999 Information and Computation
Decidability of subtyping Decidability of subtyping, which constitutes the most intricate part of the proof of decidability of type-checking, is typically established in four steps Com95, PS97]:  ...  is an extension of system F ! with subtyping and bounded quanti cation. Order-sorted algebra is an extension of many-sorted algebra with overloading and subtyping.  ...  Earlier versions of the paper were written while the author was working at the Center for Mathematics and Computer Science (CWI), Amsterdam, and Chalmers University, G oteborg.  ...

### Making sense of snapshot data: ergodic principle for clonal cell populations

Philipp Thomas
2017 Journal of the Royal Society Interface
In the presence of generation time variability, the average acquired across a population snapshot does not obey the average of a dividing cell over time, apparently contradicting ergodicity between single  ...  The correspondence between cells of known age in a population with their histories represents an ergodic principle that provides a new interpretation of population snapshot data.  ...  It is a pleasure to thank Diego Oyarzú n and Vahid Shahrezaei for valuable feedback.  ...

### AN 11-STEP SORTING NETWORK FOR 18 ELEMENTS

SHERENAZ W. AL-HAJ BADDAR, KENNETH E. BATCHER
2009 Parallel Processing Letters
However, the fastest implementable sorting networks built so far consume Θ(Nlog 2 N) comparisons, and generally, use the Merge-sorting strategy to sort the input.  ...  An 18-element network using the Merge-sorting strategy needs at least 12 steps-here we show a network that sorts 18 elements in only 11 steps.  ...  Figure 2 illustrates a typical Knuth diagram for sorting 4 elements [5] . The 0/1 principle The 0/1 principle plays a vital role in building and verifying sorting networks.  ...

### Research on the SDIF Failure Principle for RF Stealth Radar Signal Design

Jinwei Jia, Zhuangzhi Han, Limin Liu, Hui Xie, Meng Lv
2022 Electronics
The principle of SDIF sorting threshold failure provides theoretical support for anti-sorting RF stealth signal design.  ...  Theoretically speaking, the anti-sorting signal design is based on the failure principle of the radar signal sorting algorithm, and the SDIF algorithm is a core sorting algorithm widely used in engineering  ...  The sorting failure principle provides theoretical support for the design of anti-sorting signals and helps to improve the efficiency and success rate of signal anti-sorting design.  ...

### Accelerating certain outputs of merging and sorting networks

Tamir Levi, Ami Litman
2009 Theoretical Computer Science
An additional contribution of this paper concerns the well-known 0-1 Principle [D.E. Knuth, The Art of Computer Programming vol. 3: Sorting and Searching, second edition, Addison-Wesley, 1998].  ...  In the case at hand, this new tool simplifies one of our proofs by having fewer special cases than the classical 0-1 Principle. A second additional contribution concerns Batcher's merging techniques.  ...  A powerful tool for studying merging networks, similar to the 0-1 Principle, is the set of Sandwich vectors, presented in [4] .  ...

### On Computable Numbers, Nonuniversality, and the Genuine Power of Parallelism [chapter]

Selim G. Akl, Nancy Salay
2016 Emergent Computation
To illustrate, consider the following parallel algorithm for sorting a sequence S = g 0 , g 1 , . . . , g n−1 of n distinct integers on a linear array of processors p 0 , p 1 , . . . , p n−1 .  ...  Parallel Sort for k = 0 to n − 1 do for i = 0 to n − 2 do in parallel if i mod 2 = k mod 2 then p i and p i+1 compare and swap if needed end if end for end for.  ...  Perspicuously stated, it says that a function is Algorithm Sequential Sort completes the sort in O(n 2 ) time (clearly not the best sorting algorithm sequentially, but a sufficient illustration of the  ...

### A Currentless Sorting and Selection based Capacitor-Voltage-Balancing Method for Modular Multilevel Converters

Pengfei Hu, Remus Teodorescu, Songda Wang, Shuqi Li, Josep M. Guerrero
2018 IEEE transactions on power electronics
This letter proposes a currentless sorting and selection (SAS) based capacitor-voltage-balancing method for modular multilevel converters (MMCs).  ...  In this letter, the derivative of the total capacitor voltage of an arm, instead of the arm current, is employed to determine which sub-modules (SMs) should be inserted or bypassed.  ...  CONCLUSION Table I Two I states of a half-bridge SM State T1H T2H Inserted on off 1 Bypassed off on 0 To investigate the working principles of an MMC, mathematical relationships of different  ...
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