Lecture Slides on Brute Force - Analysis Of Algorithms | CS 3343, Study notes of Algorithms and Programming

Download Lecture Slides on Brute Force - Analysis Of Algorithms | CS 3343 and more Study notes Algorithms and Programming in PDF only on Docsity! Chapter 3: Brute Force Adequacy is sufficient. (Adam Osborne) Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Bubble Sort and Selection Sort 3 Bubble Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Correctness of Bubble Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Selection Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Efficiency of Selection Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Brute-Force String Matching 7 Brute-Force String Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Analysis of Brute-Force String Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Closest-Pair and Convex-Hull 9 Closest Pair Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Closest-Pair Brute-Force Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 The Convex Hull Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Examples of Convex Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Example of Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Idea for Solving Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Development of Idea for Convex Hull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Exhaustive Search 16 Exhaustive Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 TSP Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 TSP Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Knapsack Problem Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1 Knapsack Problem Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Introduction Brute force is a straightforward approach to solving a problem without regard for efficiency. Example: an O(n) algorithm for an: algorithm Power(a, n) // Input: A real number a and an integer n ≥ 0 // Output: an result ← 1 for i ← 1 to n do result ← result ∗ a return result CS 3343 Analysis of Algorithms Chapter 3: Slide – 2 Bubble Sort and Selection Sort 3 Bubble Sort My bubble sort varies from the book. algorithm BubbleSort(A[0..n− 1]) // Sorts a given array by bubble sort // Input: An array A of orderable elements // Output: Array A[0..n− 1] in ascending order sorted ← false while ¬sorted do sorted ← true for j ← 0 to n− 2 do if A[j] > A[j + 1] then swap A[j] and A[j + 1] sorted ← false CS 3343 Analysis of Algorithms Chapter 3: Slide – 3 3 Correctness of Bubble Sort  If A is not sorted, sorted is set to false, and loop continues.  Once a pair of elements are swapped, they won’t be swapped again.  n elements have n(n− 1)/2 different pairs, so at most n(n− 1)/2 swaps, so loop must eventually exit.  The number of comparisons is Θ(n2). See book. CS 3343 Analysis of Algorithms Chapter 3: Slide – 4 Selection Sort algorithm SelectionSort(A[0..n− 1]) // Sorts a given array by selection sort // Input: An array A of orderable elements // Output: Array A[0..n− 1] 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] then min ← i swap A[i] and A[min] CS 3343 Analysis of Algorithms Chapter 3: Slide – 5 Efficiency of Selection Sort  Correct because A[i] is minimum of A[i..n− 1].  The i = 0 pass (outer loop iteration) performs n− 1 comparisons.  The i = 1 pass performs n− 2 comparisons.  The last i = n− 2 pass performs 1 comparison.  The number of comparisons C(n) is Θ(n2). C(n) = n−2 ∑ i=0 (n− 1− i) = n−1 ∑ k=1 k = (n− 1)n 2 ≈ n2 2 CS 3343 Analysis of Algorithms Chapter 3: Slide – 6 4