Divide and Conquer For Convex Hull

JOURNAL OF COMPUTING, VOLUME 5, ISSUE 3, MARCH 2013, ISSN (Online) 2151-9617 https://sites.google.com/site/journalofcomputing WWW.JOURNALOFCOMPUTING.ORG

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Divide and Conquer For Convex Hull Ngonidzashe Zanamwe Abstract— The divide and concqur approach has many applications in computer science. With this approach a large problem is broken down into small and manageable subproblems and each is solved then the solutions are combined to give the grand solution to the problem. The problem of finding the convex hull of a polygon can be viewed as one problem that can be solved using the divide and conqurer approach. This paper therefore presents a discussion on the divide and conquer for the Convex Hull. The paper looks at generic divide and conquer approach before using the approach in finding the convex hull of a polygon. The paper presents a step by step procedure for finding the convex hull using the divide and conqurer approach. Also, the paper discusses running times of the Convex Hull divide-and-conquer algorithm, advantages and disadvantages of the divide and conquers approach for convex hull. The paper concludes by giving other algorithms for solving the convex hull problem apart from the divide and conquer. Index Terms—divide and conqurer, convex hull, polygon, divide and conqurer for convex hull.

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1 INTRODUCTION

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HIS paper presents a discussion on the divide and conquer for Convex Hull (hereinafter referred to as CH). The paper begins by looking at the divide and conquer in general before focusing on its applications in computational geometry, notably to the CH problem. This paper looks at two distinct ways of computing the convex hull using the divide and conquer approach. Also, the paper discusses running times of Convex Hull divideand-conquer algorithm, advantages and disadvantages of the divide and conquers approach for convex hull. The paper concludes by giving other algorithms for solving the convex hull problem apart from the divide and conquer approach.

1.1 Divide and Conquer algorithms Naher and Schmitt [1] note that divide and conquer algorithms solve problems by dividing them into instances, solving each instance recursively and merging the corresponding results to a complete solution. Further, [1] asserts that all instances have exactly the same structure as the original problem and can be solved independently from each other, and so can easily be distributed over a number of parallel processes or threads. The divide and conquer approach is presented in Fig. 1 below.

Figure 1: The divide and conquer approach

The above diagram shows that if one is faced with a problem of size n, one begins by dividing it into two subproblems of size n/2 each and then finds a solution to each sub-problem then merges the solutions to subproblems. Similar assertions were made by [5], he presents the following divide and conquer algorithm: if trivial (small), solve it “brute force” else { divide into a number of sub-problems; solve each sub-problem recursively; combine solutions to sub-problems; }

1.2 Polygon

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 Ngonidzashe Zanamwe is with the University of Zimbabwe, Zimbabwe

A polygon is defined as a closed shape with more than two sides or line segments. A simple polygon has no two consecutive edges that are intersecting.

JOURNAL OF COMPUTING, VOLUME 5, ISSUE 3, MARCH 2013, ISSN (Online) 2151-9617 https://sites.google.com/site/journalofcomputing WWW.JOURNALOFCOMPUTING.ORG

1.3 Convexity A convex is a polygon in which any line joining two points within the polygon lies within the polygon. Alexander Kolesnikov [7] gives similar definitions of convexity, the first is that, a subset set S of the plane is called convex if and only if for any pair of points P, QS the line segment PQ is completely contained in S. The second goes as, a set S is convex if it is exactly equal to the intersection of all the half planes containing it. The figures below show convex and non-convex shapes.

Figure 2: The convex and non-convex shapes

1.3. Convex Hull Mohammed Nadeem Ahmed and Raghavendra Kyatha [2] establish that the Convex Hull of a set Q of points is the smallest convex polygon P, for which each point in Q is either on the boundary of P or in its interior. In line with the above, [4] asserts that the convex hull of a set S of points, denoted hull(S) is the smallest polygon P for which each point of S is either on the boundary or in the interior of P. Similarly, [7] establishes that the convex hull CH(P) of a finite point set P is the smallest convex polygon that contains P. Figure 3 below shows a convex hull P.

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1.6 Finding the convex hull using divide and conquer Souvaine [3]establishes that in order to find the convex hull using a divide-and-conquer approach, one has to follow these steps: sort points (p1, p2, . . . , pn) by their x-coordinate recursively find the convex hull of p1 through pn/2 recursively find the convex hull of pn/2 +1 through pn merge the two convex hulls A more detailed algorithm for finding a Convex Hull is presented in [6]. The algorithm is as follows: Hull(S) : If |S|