CSCE350 Algorithms and Data Structure

1. CSCE350 Algorithms and Data Structure Lecture 7 Jianjun Hu Department of Computer Science and Engineering University of South Carolina 2009.9.

2. Outline • Learning how to understand Pseudocodes of Mysterious programs • Brute Force Strategy for Algorithm Design • The art of lazy algorithm is to count/estimate its running time • Is it doable within given timeframe?

3. Sequential Search – Smallest Distance of Two numbers • Find smallest distance between any 2 numbers • What is time efficiency of this algorithm?

4. Sequential Search – Brute Force • Find whether a search key is present in an array • What is time efficiency of this algorithm?

5. Brute-Force String Matching • Find a pattern in the text: Pattern – ‘NOT’, text – ‘NOBODY_NOTICED_HIM’ • Typical Applications – ‘find’ function in the text editor, e.g., MS-Word, Google search • What is the time efficiency of this algorithm?

6. Closest-Pair and Convex Hull Problems by Brute Force • Closest-Pair problem • Given n points in a plane, find the closest pair • How to solve this problem and what is the time efficiency of this algorithm? • Convex-Hull problem • Convex hull is the tightest convex polygon that bounds a set of n points in a plane • Convex polygon – any two points in this polygon results in the inclusion of the segment that links these two points also in this polygon

7. Closest-Pair problem Given n points in a plane, find the closest pair

8. Convex/NonConvex Polygons

9. Convex Hull • Imagine a rubber band around a set of nails • Nails touched by the band  extreme points

10. Solve Convex-Hull Problem • Connect any pair of points by a line segment. • Each line segment partitions the plane to the two half planes • If all the n points are on the same side of this line segment • ax+by-c >0 or <0 • Such a line segment is an edge of the convex-hull polygon • What is the time efficiency of this Brute-Force algorithm? • For each possible pair of points, we need to check whether all the remaining n-2 points are on the same side of the line segment that connects these pair of points. • For Sorting, String Matching, and Convex-Hull problems, we will revisit them by designing more efficient algorithms.

11. Exhaustive Search • A brute-force approach to combinatorial problem • Generate each and every element of the problem’s domain • Then compare and select the desirable element that satisfies the set constraints • Involve combinatorial objects such as permutations, combinations, and subsets of a given set • The time efficiency is usually bad – usually the complexity grows exponentially with the input size • Three examples • Traveling salesman problem • Knapsack problem • Assignment problem

12. Traveling Salesman Problem • Find the shortest tour through a given n cities that visits each city exactly once before returning to the starting city • Using graph model: city  vertex, road  edge, length of the road  edge weight. • TSP  shortest Hamiltonian Circuit – a cycle that passes through all the vertices of the graph exactly once • Exhaustive search: • List all the possible Hamiltonian circuits (starting from any vertex) • Ignore the direction • How many candidate circuits do we have?  (n-1)!/2 • Very high complexity

13. TSP Example

14. Knapsack Problem • Given n items of known weight w1, w2, …wn with values v1, v2, ..vn, • And a knapsack of capacity W • How to select the items to fill the knapsack such that you get max value? • Steal money from bank?