Triangulations Of Convex Polygon Research Articles

Convex Polygon Triangulation Based on Symmetry

A polygon with n nodes can be divided into two subpolygons by an internal diagonal through node n. Splitting the polygon along diagonal δi,n and diagonal δn−i,n, i∈{2,…,⌊n/2⌋} results in mirror images. Obviously, there are ⌊n/2⌋−1 pairs of these reflectively symmetrical images. The influence of the observed symmetry on polygon triangulation is studied. The central result of this research is the construction of an efficient algorithm used for generating convex polygon triangulations in minimal time and without generating repeat triangulations. The proposed algorithm uses the diagonal values of the Catalan triangle to avoid duplicate triangulations with negligible computational costs and provides significant speedups compared to known methods.

Extensions of dynamic programming for multi-stage combinatorial optimization

We propose a dynamic programming framework for an exact multi-stage (lexicographic) combinatorial optimization. Unlike conventional algorithms of dynamic programming that return one optimal solution, two dynamic programming algorithms proposed in this paper are coping with the whole set of optimal solutions or with its essential part. We describe the set of elements for optimization by a labeled directed acyclic graph, which in some sense, is similar to the structure of subproblems of the considered problem. For a given cost function (objective), the first algorithm constructs a subgraph of this graph that describes the whole set of optimal elements or its essential part. This algorithm can be used for multi-stage optimization of elements relative to a sequence of cost functions. The second algorithm counts elements before the optimization and after each optimization step. The considered labeled directed acyclic graph is a kind of circuit. This circuit builds the set of elements for optimization from one-element sets attached to input nodes. It uses the operation of union of sets attached to unifying nodes and functional operations attached to functional nodes. The algorithms for optimization and counting elements are defined for an arbitrary circuit without repetitions in which each element is generated exactly one time. For a considered problem with a known conventional dynamic programming solution algorithm, usually, it is easy to construct a corresponding circuit without repetitions. Once the circuit and cost functions are defined, our framework provides the correctness proofs and detailed time complexity analysis for the proposed algorithms. To make this approach more intuitive, we consider an illustrative example related to the maximum subarray problem. We tested our approach on the following nine combinatorial optimization problems: matrix chain multiplication, global sequence alignment, optimal paths in directed graphs, binary search trees, optimal bitonic tour, segmented least squares, convex polygon triangulation, one-dimensional clustering, and line breaking (text justification). We consider the last three problems in detail: construct a circuit without repetitions, describe at least two cost functions, evaluate a number of operations and time required by the algorithms, discuss an example, and consider the results of computer experiments with randomly generated instances of the problem.

COMPUTING TRIANGULATIONS OF THE CONVEX POLYGON IN PHP/MYSQL ENVIRONMENT

In this paper we implement Block method for convex polygon triangulation in web environment (PHP/MySQL). Our main aim is to show the advantages of usage of web technologies in performing complex algorithm from computer graphics. The basic assumption is that one obtained results we store in database and use it for other calculation. Databases are convenient and structured methods of sharing and retrieving data. We have performed a comparative analysis of developed program with respect to two criteria: CPU time in generating triangulation and CPU time in reading results from database.

Convex polygon triangulation based on planted trivalent binary tree and ballot problem

This paper presents a new technique of generation of convex polygon triangulation based on planted trivalent binary tree and ballot notation. The properties of the Catalan numbers were examined and their decomposition and application in developing the hierarchy and triangulation trees were analyzed. The method of storage and processing of triangulation was constructed on the basis of movements through the polygon. This method was derived from vertices and leaves of the planted trivalent binary tree. The research subject of the paper is analysis and comparison of a constructed method for solving of convex polygon triangulation problem with other methods and generating graphical representation. The application code of the algorithms was done in the Java programming language.

AN ALTERNATIVE DECOMPOSITION OF CATALAN NUMBER

A particular integer sequence derived by the convex polygon triangulation is introduced and investigated. After some underlying results are presented, the forbidden (or improper) integer values relative to the triangulation are concerned. It is understood that the forbidden sequences do not correspond to any triangulation. Some of their properties are presented. These properties are used to count the forbidden values, which is, finally, exploited in stating another decomposition of the Catalan number.

Orbiting triangle method for convex polygon triangulation

Finding all possible triangulations of convex polygon is a highly time and memory space consuming combinatorial problem. Therefore, it is very important to develop algorithms for generating triangulations as efficiently as possible. This paper presents a new method for generating triangulations of a convex polygon, called Orbiting triangle method (OTM). The method is based on using the set of (n-1)-gon triangulations during the set of n-gon triangulations creation. The main feature of the OTM algorithm is the use of the Catalan triangle to identify valid triangulations, so that the algorithm spends almost no time to eliminate duplicates. In this way, the method possesses small complexity and saves the computational time required for detecting and eliminating duplicates.

An Efficient CGM-Based Parallel Algorithm Solving the Matrix Chain Ordering Problem

This study focuses on the parallel resolution of the matrix chain ordering problem and the optimal convex polygon triangulation problem on the Coarse grain multicomputer model (CGM for short). There has been intensive work on the parallelization of these dynamic programming problems in PRAM, including the use of systolic arrays, but a BSP/CGM solution is necessary for ease of implementation and portability. Our CGM algorithm is based on Yao's sequential solution running in O(n2) time and O(n2) space. This CGM algorithm uses p processors, each with O(n/p) local memory. It requires at most O(S/p×n2) running time with S communication rounds and with S/p<1. Our algorithm performs better than the algorithm proposed in 2012 by Dilson and Marco when S is less than n/p. We offer several ways of partitioning the problem to solve and study the impact of each partitioning algorithm performance. A CGM solution exists based on Yao's algorithm, but the subdivision of tasks is defined according to the BSP cost model. In this paper, we propose a solution based only on the CGM model specifications. Note that S is the number of super-steps of the CGM algorithm.

Object-Oriented Analysis and Design for one Algorithm of Computational Geometry: Forward, Reverse and Round-Trip Engineering

Triangulation of the polygon is a fundamental algorithm in computational geometry. This paper considers techniques of object-oriented analysis and design as a new tool for solving and analyzing convex polygon triangulation. The triangulation is analyzed from three aspects: forward, reverse and round-trip engineering. We give a suggestion for improving the obtained software solution of the polygon triangulation algorithm using technique that combines UML modeling and Java programming.

Lobb's Generalization of Catalan's Parenthesization Problem

where n > 0 [2, 9]. Such numbers are now called Catalan numbers. However, Swiss mathematician Leonhard Euler (1707-1783) had previously encountered them around 1751 while investigating triangulations of convex polygons. In fact, Chinese mathematician Antu Ming (16927-1763?) discovered them even earlier, about 1730, through his geometric models. To our delight and at same time surprise, Catalan numbers occur in numerous seemingly unrelated places and situations, including combinatorics, abstract algebra, linear algebra, algebraic geometry, and sports [9]. They have same delightful propensity for popping up unexpectedly, particularly in combinatorial problems, Mar tin Gardner wrote in 1976 in his popular column Mathematical games in Scientific American [5]. Indeed, he adds, the Catalan sequence is probably most fre quently encountered sequence that is still obscure enough to cause mathematicians lacking access to N. J. A. Sloane's A Handbook of Integer Sequences to expend inor dinate amounts of energy re-discovering formulas that were worked out long ago. For example, Stanley lists over 70 occurrences of Catalan numbers in his book [11] and another 70 on his website Catalan Addendum.

Properties of Random Graphs via Boltzmann Samplers

This work is devoted to the understanding of properties of random graphs from graph classes with structural constraints. We propose a method that is based on the analysis of the behaviour of Boltzmann sampler algorithms, and may be used to obtain precise estimates for the maximum degree and maximum size of a biconnected block of a "typical'' member of the class in question. We illustrate how our method works on several graph classes, namely dissections and triangulations of convex polygons, embedded trees, and block and cactus graphs.

Parity and Primality of Catalan Numbers

Catalan numbers, like Fibonacci and Lucas numbers, appear in a variety of situ ations, including the enumeration of triangulations of convex polygons, well-formed sequences of parentheses, binary trees, and the ballot problem [l]-[5]. Like the other families, Catalan numbers are a great source of pleasure, and are excellent candidates for exploration, experimentation, and conjecturing. They are named after the Belgian mathematician Eugene Catalan (1814-1894), who discovered them in his study of well-formed sequences of parentheses. However, Leon hard Euler (1707-1783) had found them fifty years earlier while counting the number of triangulations of convex polygons [3]. But the credit for the earliest known discov ery goes to the Chinese mathematician Antu Ming (ca. 1692-1763), who was aware of them as early as 1730 [6]. In 1759 the German mathematician and physicist Johann Andreas von Segner (1707-1777), a contemporary of Euler, found that the number Cn of triangulations of a convex polygon satisfies the recursive formula

A non-recursive algorithm for polygon triangulation

In this paper an algorithm for the convex polygon triangulation based on the reverse Polish notation is proposed. The formal grammar method is used as the starting point in the investigation. This idea is "translated" to the arithmetic expression field enabling application of the reverse Polish notation method. .

Analyzing Glauber dynamics by comparison of Markov chains

A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or “Glauber dynamics.” Typically these single-site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single-point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed. This allows us to give polynomial time bounds for single-point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons. We also survey several other comparison techniques, along with specific applications, which have been used in the context of estimating mixing rates of Markov chains.

Counting thin and bushy triangulations of convex polygons

Triangulation of a simple polygon is a classical problem of immense interest in computational geometry and related fields. Recently, some specialized kinds of triangulations, namely thin and bushy triangulations have received attention owing to their use in pattern recognition and in finding geodesic properties. As triangulation of a given polygon is essentially non-unique, counting the number of ways a polygon can be triangulated is an interesting problem. In this paper, we solve two such counting problems, to find δ 1( n) ( δ b( n)) the number of thin (bushy) triangulations of an n-sided convex polygon and propose many challenging ones. These numbers provide exact upper bounds for the corresponding triangulation and may be used in analyzing average case performance of different triangulation algorithms.