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Williamson and David B. David Mount on Computational Geometry. Tuesday , Thursday , Friday Room No. Bose Bhavan Library Building. B1 Algorithm Design J.
Introduction to Algorithms
Shmoys Cambridge. W1 Lecture Notes by Prof. Convex Hull: lower bound, Graham's scan algorithm, proof of correctness, etc. Jarvis March for convex hull -- [ W3 ] Asymptotic notation: big-oh, big-omega, theta, small-oh Introduction to sorting: merge sort, insertion sort, quick sort, etc. Recurrences: divide and conquer, linear homogeneous and inhomogeneous recurrence; Problems around recurrences -- B1 , B2 , B3 , W2.
Lower bound on the worst case of sorting; Heapsort; Quicksort worst case -- B3 , B2. Building heap in linear time -- B2 , B3 probabilistic analysis average case of deterministic quick sort -- [ B3 , B2 ] randomized quick sort -- [ B5 , B6 , B2 ]. It is very useful for solving small size instances of a problem, even though it is inefficient. The brute-force approach yields reasonable algorithms of at least some practical value with no limitation on instance size for sorting, searching, and string matching.
The analysis of selection sort is straightforward. Bubble Sort The bubble sorting algorithm is to compare adjacent elements of the list and exchange them if they are out of order. In the worst case of decreasing arrays, it is the same as the number of key comparisons. These problems are very useful in important applied areas like computational geometry and operations research.
Closest-Pair Problem The closest-pair problem finds the two closest points in a set of n points. It is the simplest of a variety of problems in computational geometry that deals with proximity of points in the plane or higher-dimensional spaces. Consider the two-dimensional case of the closest-pair problem.
The points are specified in a standard fashion by their x, y Cartesian coordinates and that the distance between two points pi xi, yi and pj xj, yj is the standard Euclidean distance. All the sets depicted in Figure 2. On the other hand, the sets depicted in Figure 2. Take a rubber band and stretch it to include all the nails, then let it snap into place. The convex hull is the area bounded by the snapped rubber band as shown in Figure 2.
Convex hull The convex hull of a set S of points is the smallest convex set containing S. The smallest convex hull of S must be a subset of any convex set containing S. If S is convex, its convex hull is obviously S itself. If S is a set of two points, its convex hull is the line segment connecting these points.
If S is a set of three points not on the same line, its convex hull is the triangle with the vertices at the three points given; if the three points do lie on the same line, the convex hull is the line segment with its endpoints at the two points that are farthest apart. For an example of the convex hull for a larger set, see Figure 2. If all the points do lie on the same line, the polygon degenerates to a line segment but still with the endpoints at two points of S. The convex-hull problem is the problem of constructing the convex hull for a given set S of n points.
To solve it, we need to find the points that will serve as the vertices of the polygon in question. For example, the extreme points of a triangle are its three vertices, the extreme points of a circle are all the points of its circumference, and the extreme points of the convex hull of the set of eight points in Figure 2.
Application Extreme points have several special properties other points of a convex set do not have. One of them is exploited by the simplex method, This algorithm solves linear programming Problems. We are interested in extreme points because their identification solves the convex-hull problem. Note that this issue can also be addressed by listing the extreme points in a clockwise or a counterclockwise order. We can solve the convex-hull problem by brute-force manner. The convex hull problem is one with no obvious algorithmic solution. Facts A few elementary facts from analytical geometry are needed to implement the above algorithm.
Such exhaustive examination of all possibilities is known as exhaustive search, complete search or direct search.
Exhaustive search is simply a brute force approach to combinatorial problems Minimization or maximization of optimization problems and constraint satisfaction problems. First, unlike some of the other strategies, brute force is applicable to a very wide variety of problems. In fact, it seems to be the only general approach for which it is more difficult to point out problems it cannot tackle.
Second, for some important problems, e.
Design and Analysis of Algorithms Pdf Notes – DAA notes pdf
Third, the expense of designing a more efficient algorithm may be unjustifiable if only a few instances of a problem need to be solved and a brute-force algorithm can solve those instances with acceptable speed. Fourth, even if too inefficient in general, a brute-force algorithm can still be useful for solving small-size instances of a problem. The problem asks to find the shortest tour through a given set of n cities that visits each city exactly once before returning to the city where it started. Then the problem can be stated as the problem of finding the shortest Hamiltonian circuit of the graph.
A Hamiltonian circuit is defined as a cycle that passes through all the vertices of the graph exactly once. All circuits start and end at one particular vertex. Figure 2. Hence, we could cut the number of vertex permutations by half because cycle total lengths in both directions are same. It is useful for very small values of n. The exhaustive-search approach to this problem leads to generating all the subsets of the set of n items given, computing the total weight of each subset in order to identify feasible subsets i.
The information about the optimal selection is in bold. Time efficiency: As given in the example, the solution to the instance of Figure 2. Note: Exhaustive search of both the traveling salesman and knapsack problems leads to extremely inefficient algorithms on every input. In fact, these two problems are the best-known examples of NP-hard problems. No polynomial-time algorithm is known for any NP-hard problem.
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Moreover, most computer scientists believe that such algorithms do not exist. Alternatively, we can use one of many approximation algorithms.www.cantinesanpancrazio.it/components/pyfuwyn/
Design and Analysis of Algorithms Spring 2017
There are n people who need to be assigned to execute n jobs, one person per job. That is, each person is assigned to exactly one job and each job is assigned to exactly one person. The problem is to find an assignment with the minimum total cost. Assignment problem solved by exhaustive search is illustrated with an example as shown in figure 2. A small instance of this problem follows, with the table entries representing the assignment costs C[i, j]. An instance of the assignment problem is completely specified by its cost matrix C.
Similarly we can have! The requirements of the assignment problem imply that there is a one-to-one correspondence between feasible assignments and permutations of the first n integers. Therefore, the exhaustive-search approach to the assignment problem would require generating all the permutations of integers , ,. A few first iterations of applying this algorithm to the instance given above are given below. Since the number of permutations to be considered for the general case of the assignment problem is n! Fortunately, there is a much more efficient algorithm for this problem called the Hungarian method.
Divide-and-conquer algorithms work according to the following general plan: 1. A problem is divided into several subproblems of the same type, ideally of about equal size. The subproblems are solved typically recursively, though sometimes a different algorithm is employed, especially when subproblems become small enough.
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If necessary, the solutions to the subproblems are combined to get a solution to the original problem.