Algorithm Design Techniques: Brute Force Approach and Specific Algorithms - Prof. B. Karki, Exams of Computer Science

Download Algorithm Design Techniques: Brute Force Approach and Specific Algorithms - Prof. B. Karki and more Exams Computer Science in PDF only on Docsity! Algorithm Design Techniques CSC 3102 0.1 B.B. Karki, LSU B.B. Karki, LSU0.2CSC 3102 Algorithm Basics (Chapters: 1 and 2)  Notion of algorithm: Section 1.1  Fundamentals of algorithmic problem solving: Section 1.2  Important problem types: Section 1.3  Analysis of algorithmic efficiency: Sections 2.1 and 2.2  Non-recursive algorithms: Section 2.3  Recursive algorithms: Section 2.4 Brute Force CSC 3102 0.5 B.B. Karki, LSU B.B. Karki, LSU0.6CSC 3102 The Simplest Approach  Brute force - the simplest of the design strategies  Is a straightforward approach to solving a problem, usually directly based on the problem’s statement and definitions of the concepts involved.  Just do it - the brute-force strategy is easiest to apply.  Results in an algorithm that can be improved with a modest amount of time.  Brute force is important due to its wide applicability and simplicity.  Weakness is subpar efficiency of most brute-force algorithms.  Important examples:  Selection sort, String matching, Convex-hull problem, and Exhaustive search. B.B. Karki, LSU0.7CSC 3102 Selection Sort Algorithm SelectionSort (A[0..n-1]) //Sorts a given array //Input: An array A[0..n-1] of orderable elements //Output: Array A[0..n-1] sorted in ascending order for i ← 0 to n - 2 do min ← i for j ← i + 1 to n - 1 do if A[j] < A[min] min ← j swap A[i] and A[min] € C(n) = 1 j= i+1 n−1 ∑ = [(n −1) − (i +1) +1] = (n −1− i) i= 0 n−2 ∑ i= 0 n−2 ∑ i= 0 n−2 ∑ = (n −1)n2 ∈ Θ(n 2)  Scan the list repeatedly to find the elements, one at a time, in an non- decreasing order.  On the ith pass through the list, search for the smallest item among the last n - i elements and swap it with A[i]. After n - 1 passes, the list is sorted. B.B. Karki, LSU0.10CSC 3102 Brute-Force Approach for Convex-Hull Problem  A brute-force approach is based on the fact:  A line segment connecting two points i and j is a part of its convex hull’s boundary if and only if all other points of the set lie on the same side of the straight line through these points.  Repeating this test for every pair of points yields a list of line segments that make up the convex hull’s boundary.  Find the equation of the straight line through two points (x1, y1) and (x2, y2) ax + by = c, where a = y2 - y1, b = x1 - x2 and c = x1y2 - y1x2.  This line divides the plane into two half-planes:  For all the points in one of them ax + by > c, while for all other points in the other ax + by < c.  For all points on the line itself, ax + by = c.  Check whether the expression ax + by - c has the same sign at each of these points  To check whether certain points lie on the same side of the line.  If yes, the line forms one side of the polygon.  Efficiency of the algorithm: O(n3)  For each of n(n -1)/2 pairs of distinct points, one needs to find the sign of ax + by - c for each of the other n - 2 points. B.B. Karki, LSU0.11CSC 3102 Exhaustive Search  Exhaustive search is a brute-force approach to combinatorial problems.  It suggests generating each and every combinatorial object of the problem, selecting those of them of that satisfy the problem’s constraints and then finding a desired object.  Impratical for all but applicable to very small instances of problems.  Examples:  Traveling salesman problem  Finding the shortest tour through a given set of n cities that visits each city exactly once before returning to the city where it started.  Knapsack problem  Finding the most valuable list of out-of n items that fit into the knapscak.  Assignment problem  Finding an assignment of n people to execute n jobs with the smallest total cost.  These problems are the examples of so-called NP-hard problems. B.B. Karki, LSU0.12CSC 3102 Traveling Salesman Problem  Traveling salesman 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.  Finding the shortest Hamiltonian circuit of the graph  A cycle that passes through all the vertices of the graph exactly once.  A sequence of n +1 adjacent vertices v0, v1, v2, … vn-1, v0.  Get all tours by generating all the permutations of n - 1 intermediate cities, compute the tour lengths, and find the shortest among them.  Efficiency:  Total number of permutations = (n - 1)!/2  Impractical for all but very small values of n. 5 1 7 8 2 3 Tour Length a b c d a 2+8+1+7 = 18 a b d c a 2+3+1+5 = 11 a c b d a 23 a c d b a 11 a d b c a 23 a d c b a 18 More than one optimal solutions. a b dc