A New Quasi-Human Algorithm for Solving the Packing Problem of Unit Equilateral Triangles

The packing problem of unit equilateral triangles not only has the theoretical significance but also offers broad prospects in material processing and network resource optimization. Because this problem is nondeterministic polynomial (NP) hard and has the feature of continuity, it is necessary to limit the placements of unit equilateral triangles before optimizing and obtaining approximate solution (e.g., the unit equilateral triangles are not allowed to be rotated). This paper adopts a new quasi-human strategy to study the packing problem of unit equilateral triangles. Some new concepts are put forward such as side-clinging action, and an approximation algorithm for solving the addressed problem is designed. Time complexity analysis and the calculation results indicate that the proposed method is a polynomial time algorithm, which provides the possibility to solve the packing problem of arbitrary triangles.

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A Heuristic Algorithm for Solving Triangle Packing Problem

Discrete Dynamics in Nature and Society ◽

10.1155/2013/686845 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Ruimin Wang ◽

Yuqiang Luo ◽

Jianqiang Dong ◽

Shuai Liu ◽

Xiaozhuo Qi

Keyword(s):

Approximate Solution ◽

Time Complexity ◽

Complexity Analysis ◽

Packing Problem ◽

Optimization Method ◽

Resource Optimization ◽

Network Resource ◽

Analogue Experiment ◽

Processing Network ◽

Application Prospects

The research on the triangle packing problem has important theoretic significance, which has broad application prospects in material processing, network resource optimization, and so forth. Generally speaking, the orientation of the triangle should be limited in advance, since the triangle packing problem is NP-hard and has continuous properties. For example, the polygon is not allowed to rotate; then, the approximate solution can be obtained by optimization method. This paper studies the triangle packing problem by a new kind of method. Such concepts as angle region, corner-occupying action, corner-occupying strategy, and edge-conjoining strategy are presented in this paper. In addition, an edge-conjoining and corner-occupying algorithm is designed, which is to obtain an approximate solution. It is demonstrated that the proposed algorithm is highly efficient, and by the time complexity analysis and the analogue experiment result is found.

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Evaluation of Parallel Algorithms of Traffic Volume Statistics per Cell in Cellular Network Based on MapReduce Programming Model

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.401-403.1859 ◽

2013 ◽

Vol 401-403 ◽

pp. 1859-1863

Author(s):

Qing Yang ◽

Jun Liu ◽

Huan Wang ◽

Wen Li Zhou ◽

Hua Yu

Keyword(s):

Big Data ◽

Parallel Algorithms ◽

Network Design ◽

Time Complexity ◽

Cellular Network ◽

Complexity Analysis ◽

Programming Model ◽

Resource Optimization ◽

Network Resource ◽

Data Network

Understanding traffic per unit time in cell dimension in cellular data network can be of great help for mobile operators to improve the performance of the cellular data network. It is important for network design and resource optimization. In this paper, we describe three methods to count the traffic per unit time per cell. Moreover, we compare the results of the three methods by the deviation distribution of the traffic and time complexity analysis. Our work is distinguished from other related work by using big data which contains around 1.4 billion records and 20 thousands cells. Generally, we expect this paper could deliver important insights into cellar data network resource optimization.

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On Counting the Number of Consistent Genotype Assignments for Pedigrees

BRICS Report Series ◽

10.7146/brics.v12i28.21895 ◽

2005 ◽

Vol 12 (28) ◽

Author(s):

Jirí Srba

Keyword(s):

Polynomial Time ◽

Complexity Analysis ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Consistency Checking ◽

Counting Problem ◽

Number Of Children ◽

Genotype Information ◽

Two Generations ◽

Hard Problems

Consistency checking of genotype information in pedigrees plays an important role in genetic analysis and for complex pedigrees the computational complexity is critical. We present here a detailed complexity analysis for the problem of counting the number of complete consistent genotype assignments. Our main result is a polynomial time algorithm for counting the number of complete consistent assignments for non-looping pedigrees. We further classify pedigrees according to a number of natural parameters like the number of generations, the number of children per individual and the cardinality of the set of alleles. We show that even if we assume all these parameters as bounded by reasonably small constants, the counting problem becomes computationally hard (#P-complete) for looping pedigrees. The border line for counting problems computable in polynomial time (i.e. belonging to the class FP) and #P-hard problems is completed by showing that even for general pedigrees with unlimited number of generations and alleles but with at most one child per individual and for pedigrees with at most two generations and two children per individual the counting problem is in FP.

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Traps to the BGJT-algorithm for discrete logarithms

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157014000242 ◽

2014 ◽

Vol 17 (A) ◽

pp. 218-229 ◽

Cited By ~ 5

Author(s):

Qi Cheng ◽

Daqing Wan ◽

Jincheng Zhuang

Keyword(s):

Finite Fields ◽

Polynomial Time ◽

Time Complexity ◽

Complexity Analysis ◽

Discrete Logarithm ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

New Approach ◽

Discrete Logarithms ◽

Polynomial Time Complexity

AbstractIn the recent breakthrough paper by Barbulescu, Gaudry, Joux and Thomé, a quasi-polynomial time algorithm is proposed for the discrete logarithm problem over finite fields of small characteristic. The time complexity analysis of the algorithm is based on several heuristics presented in their paper. We show that some of the heuristics are problematic in their original forms, in particular when the field is not a Kummer extension. We propose a fix to the algorithm in non-Kummer cases, without altering the heuristic quasi-polynomial time complexity. Further study is required in order to fully understand the effectiveness of the new approach.

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A POLYNOMIAL-TIME APPROXIMATION ALGORITHM FOR A GEOMETRIC DISPERSION PROBLEM

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195909002952 ◽

2009 ◽

Vol 19 (03) ◽

pp. 267-288 ◽

Cited By ~ 5

Author(s):

MARC BENKERT ◽

JOACHIM GUDMUNDSSON ◽

CHRISTIAN KNAUER ◽

RENÉ VAN OOSTRUM ◽

ALEXANDER WOLFF

Keyword(s):

Approximation Algorithm ◽

Polynomial Time ◽

Packing Problem ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Time Approximation ◽

Polynomial Time Approximation Algorithm ◽

Dispersion Problem ◽

Unit Disks ◽

Polynomial Time Approximation

We consider the following packing problem. Let α be a fixed real in (0, 1]. We are given a bounding rectangle ρ and a set [Formula: see text] of n possibly intersecting unit disks whose centers lie in ρ. The task is to pack a set [Formula: see text] of m disjoint disks of radius α into ρ such that no disk in B intersects a disk in [Formula: see text], where m is the maximum number of unit disks that can be packed. In this paper we present a polynomial-time algorithm for α = 2/3. So far only the case of packing squares has been considered. For that case, Baur and Fekete have given a polynomial-time algorithm for α = 2/3 and have shown that the problem cannot be solved in polynomial time for any α > 13/14 unless [Formula: see text].

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A Polynomial Time Algorithm for 3SAT

ACM Transactions on Computation Theory ◽

10.1145/3460950 ◽

2021 ◽

Vol 13 (3) ◽

pp. 1-13

Author(s):

Lizhi Du

Keyword(s):

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

New Concepts ◽

Unit Layer

By creating new concepts and methods—the checking tree, long unit path, direct contradiction unit pair, indirect contradiction unit pair, additional contradiction unit pair, two-unit layer and three-unit layer—we successfully transform solving a 3SAT problem to solving 2SAT problems in polynomial time. Each time, we add only one layer of the three-unit layers to the two-unit layers to calculate 2SAT paths, respectively. The key is as follows: in each 2SAT path, any two units cannot be a direct contradiction unit pair and cannot be an indirect contradiction unit pair and additional contradiction unit pair. This guarantees that all of the 2SAT paths we got, respectively, can shape at least one long path without contradictions. Thus, we proved that NP = P.

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Learning Importance of Preferences

10.29007/v68w ◽

2018 ◽

Author(s):

Ying Zhu ◽

Mirek Truszczynski

Keyword(s):

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Scoring Rules ◽

Np Hard ◽

Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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Metric Dimension Parameterized By Treewidth

Algorithmica ◽

10.1007/s00453-021-00808-9 ◽

2021 ◽

Author(s):

Édouard Bonnet ◽

Nidhi Purohit

Keyword(s):

Computable Function ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Minimum Size ◽

Metric Dimension ◽

Resolving Set ◽

Input Graph ◽

Outerplanar Graphs ◽

Bounded Treewidth ◽

Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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