Loopless Algorithm Research Articles

On the central levels problem

The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1−ℓ many 1s and at most m+ℓ many 1s, i.e., the vertices in the middle 2ℓ levels, has a Hamilton cycle for any m≥1 and 1≤ℓ≤m+1. This problem was raised independently by Buck and Wiedemann, Savage, Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case ℓ=1, and classical binary Gray codes, namely the case ℓ=m+1. In this paper we present a general constructive solution of the central levels problem. Our results also imply the existence of optimal cycles through any sequence of ℓ consecutive levels in the n-dimensional hypercube for any n≥1 and 1≤ℓ≤n+1. Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the n-dimensional hypercube, n≥2, that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code.

Universality class for loopless invasion percolation models and a percolation avalanche burst model for hydraulic fracturing.

Invasion percolation is a model that was originally proposed to describe growing networks of fractures. Here we describe a loopless algorithm on random lattices, coupled with an avalanche-based model for bursts. The model reproduces the characteristic b-value seismicity and spatial distribution of bursts consistent with earthquakes resulting from hydraulic fracturing ("fracking"). We test models for both site invasion percolation and bond invasion percolation. These have differences on the scale of site and bond lengths l. But since the networks are characterized by their large-scale behavior, l≪L, we find small differences between scaling exponents. Though data may not differentiate between models, our results suggest that both models belong to different universality classes.

A loopless algorithm for generating (k, m)-ary trees in Gray code order

A family of (k, m)-ary trees was firstly introduced by Du and Liu when they studied hook length polynomial for plane trees. Recently, Amani and Nowzari-Dalini presented a generation algorithm to produce (k, m)-ary trees of order n encoding by Z-sequences in reverse lexicographic order. In this paper, we propose a loopless algorithm to generate all such Z-sequences in Gray code order. Hence, the worst-case time complexity of generating one Z-sequence is $${\mathcal {O}}(1)$$ , and the space requirement of our algorithm is $$2n+{\mathcal {O}}(1)$$ . Moreover, based on this ordering, we also provide ranking and unranking algorithms. The ranking algorithm can be run in $${\mathcal {O}}(\max \{kmn,n^2\})$$ time and $${\mathcal {O}}(kmn)$$ space, whereas the unranking algorithm requires $${\mathcal {O}}(kmn^2)$$ time and space.

Faster search by lackadaisical quantum walk

In the typical model, a discrete-time coined quantum walk searching the 2D grid for a marked vertex achieves a success probability of $O(1/\log N)$ in $O(\sqrt{N \log N})$ steps, which with amplitude amplification yields an overall runtime of $O(\sqrt{N} \log N)$. We show that making the quantum walk lackadaisical or lazy by adding a self-loop of weight $4/N$ to each vertex speeds up the search, causing the success probability to reach a constant near $1$ in $O(\sqrt{N \log N})$ steps, thus yielding an $O(\sqrt{\log N})$ improvement over the typical, loopless algorithm. This improved runtime matches the best known quantum algorithms for this search problem. Our results are based on numerical simulations since the algorithm is not an instance of the abstract search algorithm.

An Assessment of SPSS Grid Interfaced SPV System under Abnormal Grid Conditions

ABSTRACTThis paper deals with a single-phase single-stage grid interfaced solar photo-voltaic (SPV) array system. A simple phase lock loop-less (PLL-less) control algorithm is proposed which works under wide variation in the grid voltage. Moreover, the assessment of all the intermediate control signals are described which helps in intuitive understanding of the SPV system. A suitable feed-forward term is included in the control algorithm to take care of fast sudden changes in the grid voltage or SPV insolation level. The proposed system is simulated in MATLAB and simulation results are verified experimentally. The operation of proposed solar energy conversion system (SECS) is demonstrated under abnormal grid conditions. The proposed SECS avoid generation loss in the deep voltage sag and swell at the grid. Both simulation and experimental results are shown for complete range of voltage variations. The grid current and voltage total harmonic distortion are observed well under IEEE-519 standard.

A loopless algorithm for generating multiple binary tree sequences simultaneously

Pallo and Wu et al. respectively introduced the left-weight sequences (LW-sequences) and right-weight sequences (RW-sequences) for representing binary trees. In this paper, we introduce two new types of binary tree sequences called the left-child sequences (LC-sequences) and right-child sequences (RC-sequences). Next, we propose a loopless algorithm associated with rotations of binary trees for generating LW-, RW-, LC-, and RC-sequences simultaneously. Moreover, we show that LW- and RW-sequences are generated in Gray-code order, and LC- and RC-sequences are generated so that each sequence can be obtained from its predecessor by changing at most two digits. Our algorithm is shown to be more efficient in both space and time than the existing known algorithms.

The Coolest Way to Generate Binary Strings

Pick a binary string of length n and remove its first bit b. Now insert b after the first remaining 10, or insert $\overline{b}$ at the end if there is no remaining 10. Do it again. And again. Keep going! Eventually, you will cycle through all 2 n of the binary strings of length n. For example, [InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.] are the binary strings of length n=4, where [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. And if you only want strings with weight (number of 1s) between l and u? Just insert b instead of $\overline{b}$ when the result would have too many 1s or too few 1s. For example, [InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.][InlineEquation not available: see fulltext.] are the strings with n=4, l=0 and u=2. This generalizes `cool-lex' order by Ruskey and Williams (The coolest way to generate combinations, Discrete Mathematics) and we present two applications of our `cooler' order. First, we give a loopless algorithm for generating binary strings with any weight range in which successive strings have Levenshtein distance two. Second, we construct de Bruijn sequences for (i) l=0 and any u (maximum specified weight), (ii) any l and u=n (minimum specified weight), and (iii) odd uÂ?l (even size weight range). For example, all binary strings with n=6, l=1, and u=4 appear once (cyclically) in [InlineEquation not available: see fulltext.]. We also investigate the recursive structure of our order and show that it shares certain sublist properties with lexicographic order.

Ranking and Unranking of Non-regular Trees in Gray-Code Order

A non-regular tree T with a prescribed branching sequence (s 1 , s 2 ,..., s n ) is a rooted and ordered tree such that its internal nodes are numbered from 1 to n in preorder and every internal node i in T has s i children. Recently, Wu et al. (2010) introduced a concise representation called RD-sequences to represent all non-regular trees and proposed a loopless algorithm for generating all non-regular trees in a Gray-code order. In this paper, based on such a Gray-code order, we present efficient ranking and unranking algorithms of non-regular trees with n internal nodes. Moreover, we show that the ranking algorithm and the unranking algorithm can be run in O(n 2 ) time and O(n 2 + nS n―1 ) time, respectively, provided a preprocessing takes O(n 2 S n―1 ) time and space in advance, where S n―1 = Σ n―1 i=1 (s i ―1).

A Ranking Algorithm of Non-Regular Trees in Gray-Code Order

A non-regular tree T with a prescribed branching sequence is an ordered tree whose internal nodes are numbered from 1 to n in preorder such that every node in T has a prescribed number of children. Recently, Wu et al. (2010) introduced a concise representation called RD-sequences to represent all non-regular trees and proposed a loopless algorithm for generating all non-regular trees in a Gray-code order. In this paper, based on such a Gray-code order, we show that a ranking algorithm can be done in quadratic time provided a preprocessing in advance.

Amortized efficiency of generating planar paths in convex position

Let S be a set of n ⩾ 3 points arranged in convex position in the plane and suppose that all points of S are labeled from 1 to n in clockwise direction. A planar path P on S is a path whose edges connect all points of S with straight line segments such that no two edges of P cross. Flipping an edge on P means that a new path P ′ is obtained from P by a single edge replacement. In this paper, we provide efficient algorithms to generate all planar paths. With the help of a loopless algorithm to produce a set of 2-way binary-reflected Gray codes, the proposed algorithms generate the next planar path by means of a flip and such that the number of position changes for points in the path has a constant amortized upper bound.

On the Solution of the Towers of Hanoi Problem

In this paper, two versions of an iterative loopless algorithm for the classical towers of Hanoi problem with O(1) storage complexity and O(2) time complexity are presented. Based on this algorithm the number of different moves in each of pegs with its direction is formulated. Keywords—Loopless algorithm, Binary tree, Towers of Hanoi.

A loopless implementation of a gray code for signed permutations

Conway, Sloane and Wilks [1989] proved the existence of a Gray code for the reflection group Bn. The elements of this group are the signed permutations of the set 1, 2, . . . ,n. Here we give a loopless algorithm which generates a specific Gray code for Bn.

Loopless Generation of Non-regular Trees with a Prescribed Branching Sequence

An ordered tree is called a non-regular tree with a prescribed branching sequence (or non-regular tree for short) if its internal nodes have a prespecified degree sequence in preorder list. We define a concise representation, called right distance sequences to describe such trees. A coding tree helps us to systematically investigate the structural representation of non-regular trees. Consequently, we present a loopless algorithm to generate Gray-codes of non-regular trees using right distance sequences.

Gray Codes, Loopless Algorithm and Partitions

The generation of efficient Gray codes and combinatorial algorithms that list all the members of a combinatorial object has received a lot of attention in the last few years. Knuth gave a code for the set of all partitions of [n] = {1,2,...,n}. Ruskey presented a modified version of Knuth’s algorithm with distance 2. Ehrlich introduced a looplees algorithm for the set of the partitions of [n]; Ruskey and Savage generalized Ehrlich’s results and introduced two Gray codes for the set of partitions of [n]. In this paper, we give another combinatorial Gray code for the set of the partitions of [n] which differs from the aforementioned Gray codes. Also, we construct a different loopless algorithm for generating the set of all partitions of [n] which gives a constant time between successive partitions in the construction process.

FUSING LOOPLESS ALGORITHMS FOR COMBINATORIAL GENERATION

Some combinatorial generation problems can be broken into subproblems for which loopless algorithms already exist. This article discusses means by which loopless algorithms can be fused to produce a new loopless algorithm that solves the original problem. It demonstrates this method with two new loopless algorithms. The first generates well-formed parenthesis strings containing two different types of parentheses. The second generates multiset permutations in linear space using only arrays; it is simpler and more efficient than the recent algorithm of Korsh & LaFollette.

A loopless Gray code for rooted trees

Beyer and Hedetniemi [1980] gave the first constant average-time algorithm for the generation of all rooted trees with n nodes. This article presents the first combinatorial Gray code for these trees and a loopless algorithm for its generation.

Low-complexity bit-allocation algorithm for MPEG AAC audio coders

This letter presents a method for estimating the quantization noise introduced by a nonuniform quantizer, like those used in the family of MPEG-2/4 AAC audio coders. The method is generalized for the case of estimating the mean squared or the maximum value of the quantization noise. Its use will allow the bit-allocation algorithm to be adapted to different coding scenarios, depending on the available number of bits. Using the proposed method, it is possible to implement a loopless bit-allocation algorithm, without the need for using any kind of iteration loops. This helps to dramatically reduce the computational complexity of the bit-allocation algorithm, making it easier to implement in real-time applications where computational power is limited.

Loopless Array Generation of Multiset Permutations

This paper presents a loopless algorithm to generate all permutations of a multiset. Each is obtained from its predecessor by making one transposition. It differs from previous such algorithms by using an array for the permutations but requiring storage only linear in its length.

A loopless algorithm for generating the permutations of a multiset

Many combinatorial structures can be constructed from simpler components. For example, a permutation can be constructed from cycles, or a Motzkin word from a Dyck word and a combination. In this paper we present a constructor for combinatorial structures, called shuffle on trajectories (defined previously in a non-combinatorial context), and we show how this constructor enables us to obtain a new loopless generating algorithm for multiset permutations from similar results for simpler objects.

Loopless Generation of Linear Extensions of a Poset

Canfield and Williamson gave the first loopless algorithm for generating all linear extensions of a poset. It elegantly generates all signed extensions, resulting in each extension appearing somewhere with each sign, but retains only every other one independent of its sign. It uses an array for the extension. In this paper we give another loopless algorithm for generating all the linear extensions. It generates each extension only once and uses a list for the extensions.