Recursive solution of a class of combinatorial problems

1965 ◽  
Vol 8 (10) ◽  
pp. 617-620 ◽  
Author(s):  
W. C. Lynch
Keyword(s):  
Combinatorial Problems ◽  
Recursive Solution

Exchanges of "Quanta" Among Distinguishable and Indistinguishable Objects

1997 ◽  
Vol 11 (19) ◽  
pp. 2281-2301
Author(s):  
Antonio Bonelli ◽  
Stefano Ruffo
Keyword(s):  
Asymptotic Behavior ◽  
Recursion Relation ◽  
Combinatorial Problems ◽  
Physical Problem ◽  
Explicit Formulas ◽  
Recursive Solution ◽  
Theoretic Problem ◽  
Recursive Formulas ◽  
Bose Einstein ◽  
Positive Integers

Beginning with a physical problem of exchange of n indistinguishable "quanta" of energy in an ensemble of k oscillators we define a new wide class of combinatorial problems, which also contains statistics intermediate between Fermi–Dirac and Bose–Einstein. One such problem is related to the number theoretic problem of computing the partitions of positive integers. After establishing such a connection, we give explicit formulas for the partitions M(n,k) of an integer n into k parts with k ≤ 4. Moreover, we derive a recursion relation for fixed n and varying k which is valid for any k. A formula which relates partitions to the cardinality of the partition set taking order into account is also derived. The leading asymptotic behavior for n large is obtained for any k. A suggestive interpretation of this formulas is proposed in terms of simplicial lattices. Recursive formulas at fixed k and varying n are then written for k ≤ 5 using the concept of factorial triangle, which is amenable for generalizations to larger k's. The problem of distinct partitions is mapped onto the probability problem of ball removal with replacement, for which we give again recursive solution formulas. Finally, the method of generalized Tartaglia triangle allows the derivation of recursive formulas for limited partitions which take order into account. This latter result is related to the problem of finding the number of distinct ways of dividing n indistinguishable objects into k distinguishable groups, for which explicit summations had been previously found.

Preliminaries

2017 ◽  
Author(s):  
David J. Lobina
Keyword(s):  
Data Structures ◽  
Production Systems ◽  
Specific Data ◽  
Recursive Functions ◽  
Recursive Solution ◽  
Partial Recursive Functions ◽  
Recursive Structures ◽  
Recursive Data Structures ◽  
Recursive Data ◽  
The Relationship

Recursion, or the capacity of ‘self-reference’, has played a central role within mathematical approaches to understanding the nature of computation, from the general recursive functions of Alonzo Church to the partial recursive functions of Stephen C. Kleene and the production systems of Emil Post. Recursion has also played a significant role in the analysis and running of certain computational processes within computer science (viz., those with self-calls and deferred operations). Yet the relationship between the mathematical and computer versions of recursion is subtle and intricate. A recursively specified algorithm, for example, may well proceed iteratively if time and space constraints permit; but the nature of specific data structures—viz., recursive data structures—will also return a recursive solution as the most optimal process. In other words, the correspondence between recursive structures and recursive processes is not automatic; it needs to be demonstrated on a case-by-case basis.

scholarly journals Q-Learnheuristics: Towards Data-Driven Balanced Metaheuristics

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1839
Author(s):  
Broderick Crawford ◽  
Ricardo Soto ◽  
José Lemus-Romani ◽  
Marcelo Becerra-Rozas ◽  
José M. Lanza-Gutiérrez ◽  
...  
Keyword(s):  
Combinatorial Problems ◽  
Set Covering ◽  
Covering Problem ◽  
Integration Framework ◽  
Q Learning ◽  
Whale Optimization ◽  
Sine Cosine Algorithm ◽  
Exploration Exploitation ◽  
Selection Of

One of the central issues that must be resolved for a metaheuristic optimization process to work well is the dilemma of the balance between exploration and exploitation. The metaheuristics (MH) that achieved this balance can be called balanced MH, where a Q-Learning (QL) integration framework was proposed for the selection of metaheuristic operators conducive to this balance, particularly the selection of binarization schemes when a continuous metaheuristic solves binary combinatorial problems. In this work the use of this framework is extended to other recent metaheuristics, demonstrating that the integration of QL in the selection of operators improves the exploration-exploitation balance. Specifically, the Whale Optimization Algorithm and the Sine-Cosine Algorithm are tested by solving the Set Covering Problem, showing statistical improvements in this balance and in the quality of the solutions.

scholarly journals Inversion-free geometric mapping construction: A survey

2021 ◽  
Vol 7 (3) ◽  
pp. 289-318
Author(s):  
Xiao-Ming Fu ◽  
Jian-Ping Su ◽  
Zheng-Yu Zhao ◽  
Qing Fang ◽  
Chunyang Ye ◽  
...  
Keyword(s):  
Mesh Generation ◽  
Optimization Problem ◽  
Texture Mapping ◽  
Deformation Texture ◽  
Research Field ◽  
Combinatorial Problems ◽  
Open Problems ◽  
Nonlinear Constrained Optimization ◽  
Construction Methods ◽  
Geometric Mapping

AbstractA geometric mapping establishes a correspondence between two domains. Since no real object has zero or negative volume, such a mapping is required to be inversion-free. Computing inversion-free mappings is a fundamental task in numerous computer graphics and geometric processing applications, such as deformation, texture mapping, mesh generation, and others. This task is usually formulated as a non-convex, nonlinear, constrained optimization problem. Various methods have been developed to solve this optimization problem. As well as being inversion-free, different applications have various further requirements. We expand the discussion in two directions to (i) problems imposing specific constraints and (ii) combinatorial problems. This report provides a systematic overview of inversion-free mapping construction, a detailed discussion of the construction methods, including their strengths and weaknesses, and a description of open problems in this research field.

scholarly journals Feature Selection and Parameter Optimization of Support Vector Machines Based on Modified Artificial Fish Swarm Algorithms

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Kuan-Cheng Lin ◽  
Sih-Yang Chen ◽  
Jason C. Hung
Keyword(s):  
Feature Selection ◽  
Parameter Optimization ◽  
Combinatorial Problem ◽  
Combinatorial Problems ◽  
Support Vector ◽  
Local Optimum ◽  
Artificial Fish Swarm Algorithm ◽  
Swarm Algorithms ◽  
Artificial Fish Swarm ◽  
Information And Communication

Rapid advances in information and communication technology have made ubiquitous computing and the Internet of Things popular and practicable. These applications create enormous volumes of data, which are available for analysis and classification as an aid to decision-making. Among the classification methods used to deal with big data, feature selection has proven particularly effective. One common approach involves searching through a subset of the features that are the most relevant to the topic or represent the most accurate description of the dataset. Unfortunately, searching through this kind of subset is a combinatorial problem that can be very time consuming. Meaheuristic algorithms are commonly used to facilitate the selection of features. The artificial fish swarm algorithm (AFSA) employs the intelligence underlying fish swarming behavior as a means to overcome optimization of combinatorial problems. AFSA has proven highly successful in a diversity of applications; however, there remain shortcomings, such as the likelihood of falling into a local optimum and a lack of multiplicity. This study proposes a modified AFSA (MAFSA) to improve feature selection and parameter optimization for support vector machine classifiers. Experiment results demonstrate the superiority of MAFSA in classification accuracy using subsets with fewer features for given UCI datasets, compared to the original FASA.

Lower bounds for combinatorial problems on graphs

1985 ◽  
Vol 6 (3) ◽  
pp. 393-399 ◽  
Author(s):  
Hiroyuki Nakayama ◽  
Takao Nishizeki ◽  
Nobuji Saito
Keyword(s):  
Lower Bounds ◽  
Combinatorial Problems

scholarly journals On the Number of Tetrahedra with Minimum, Unit, and Distinct Volumes in Three-Space

2008 ◽  
Vol 17 (2) ◽  
pp. 203-224 ◽  
Author(s):  
ADRIAN DUMITRESCU ◽  
CSABA D. TÓTH
Keyword(s):  
Unit Volume ◽  
Time Algorithm ◽  
Combinatorial Problems ◽  
List Type ◽  
Type Number ◽  
Point Sets ◽  
Minimum Number ◽  
Three Space

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined bynpoints in 3-space, and in general inddimensions.(i)The number of tetrahedra of minimum (non-zero) volume spanned bynpoints in$\mathbb{R}$3is at most$\frac{2}{3}n^3-O(n^2)$, and there are point sets for which this number is$\frac{3}{16}n^3-O(n^2)$. We also present anO(n3) time algorithm for reporting all tetrahedra of minimum non-zero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke and Seidel. In general, for every$k,d\in \mathbb{N}, 1\leq k \leq d$, the maximum number ofk-dimensional simplices of minimum (non-zero) volume spanned bynpoints in$\mathbb{R}$dis Θ(nk).(ii)The number of unit volume tetrahedra determined bynpoints in$\mathbb{R}$3isO(n7/2), and there are point sets for which this number is Ω(n3log logn).(iii)For every$d\in \mathbb{N}$, the minimum number of distinct volumes of all full-dimensional simplices determined bynpoints in$\mathbb{R}$d, not all on a hyperplane, is Θ(n).

scholarly journals Some polyhedra related to combinatorial problems

1969 ◽  
Vol 2 (4) ◽  
pp. 451-558 ◽  
Author(s):  
Ralph E. Gomory
Keyword(s):  
Combinatorial Problems
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