Main Index

1. Chapter 3 – Introduction to Algorithms 1 Main Index Contents • Selection Sort (3 slides) • Selection Sort Alg. (3 slides) • Search Algorithms (6 slides) • Illustrating the Binary Search • -Successful (3 slides) • -Unsuccessful (3 slides) • Binary Search Alg. (3 slides) • Big-O Notation • Constant Time Algorithms • Linear Time Algorithms • Exponential Algs. (2 slides) • Logarithmic Time Algorithms • Selection Sort Algorithm • -Integer Version • -String Version • Template Syntax (4 slides) • Recursive Defn of the Power Fnc • Stopping Conditions for- • Recursive Algorithms • Implementing the Recursive- • Power Function • Tower of Hanoi w/ Recursion • (3 slides) • Fibonacci Numbers Using- • Iteration (2 slides) • Summary Slides (5 slides)

2. 2 Main Index Contents Selection Sort- 5 Element Array • Pass 0: • Scan the entire list from arr[0] to arr[4] and identify 20 at index 1 as the smallest element. • Exchange 20 with arr[0] = 50, the first element in the list.

3. 3 Main Index Contents Selection Sort- 5 Element Array • Pass 1: • Scan the sublist 50, 40, 75, and 35. • Exchange the smallest element 35 at index 4 with arr[1] = 50.

4. 4 Main Index Contents Selection Sort- 5 Element Array • Pass 2: • Locate the smallest element in the sublist 40, 75, and 50.

5. 5 Main Index Contents Selection Sort- 5 Element Array • Pass 3: • Two elements remain to be sorted. • Scan the sublist 75, 50 and exchange the smaller element with arr[3]. • The exchange places 50 at index 4 in arr[3].

6. 6 Main Index Contents Selection Sort- 5 Element Array

7. 7 Main Index Contents Selection Sort Algorithm void selectionSort(int arr[], int n) { int smallIndex;// index of smallest // element in the sublist int pass, j; int temp; // pass has the range 0 to n-2

8. 8 Main Index Contents Selection Sort Algorithm for (pass = 0; pass < n-1; pass++) { // scan the sublist starting at index // pass smallIndex = pass; // j traverses the sublist arr[pass+1] // to arr[n-1] for (j = pass+1; j < n; j++) // if smaller element found, assign // smallIndex to that position

9. 9 Main Index Contents Selection Sort Algorithm if (arr[j] < arr[smallIndex]) smallIndex = j; // if smallIndex and pass are not the // same location,exchange the // smallest item in the sublist with // arr[pass] if (smallIndex != pass) { temp = arr[pass]; arr[pass] = arr[smallIndex]; arr[smallIndex] = temp; } } }

10. Search Algorithms • Search algorithms start with a target value and employ some strategy to visit the elements looking for a match. • If target is found, the index of the matching element becomes the return value.

11. 11 Main Index Contents Search Algorithms

12. 12 Main Index Contents Search Algorithms- Sequential Search Algorithm int seqSearch(const int arr[], int first, int last, int target) { int i = first; // scan indices in the range first <= I < last; // test for a match or index out of range. while(i != last && arr[i] != target) i++; return i; // i is index of match or i = last if no match }

13. Search Algorithms Case 1. A match occurs. The search is complete and mid is the index that locates the target. if (midValue == target) // found match return mid;

14. Search Algorithms Case 2. The value of target is less than midvalue and the search must continue in the lower sublist. Reposition the index last to the end of the sublist (last = mid). // search the lower sublist if (target < midvalue)<reposition last to mid><search sublist arr[first]…arr[mid-1]

15. Search Algorithms Case 3. The value of target is greater than midvalue and the search must continue in the upper sublist . Reposition the index first to the front of the sublist (first = mid+1). // search upper sublist if (target > midvalue) <reposition first to mid+1> <search sublist arr[mid+1]…arr[last-1]>

16. 16 Main Index Contents Illustrating the Binary Search- Successful Search 1. Search for target = 23 Step 1: Indices first = 0, last = 9, mid = (0+9)/2 = 4. Since target = 23 > midvalue = 12, step 2 searches the upper sublist with first = 5 and last = 9.

17. 17 Main Index Contents Illustrating the Binary Search- Successful Search Step 2: Indices first = 5, last = 9, mid = (5+9)/2 = 7. Since target = 23 < midvalue = 33, step 3 searches the lower sublist with first = 5 and last = 7.

18. 18 Main Index Contents Illustrating the Binary Search- Successful Search Step 3: Indices first = 5, last = 7, mid = (5+7)/2 = 6. Since target = midvalue = 23, a match is found at index mid = 6.

19. Illustrating the Binary Search- Unsuccessful Search Search for target = 4. Step 1: Indices first = 0, last = 9, mid = (0+9)/2 = 4. Since target = 4 < midvalue = 12, step 2 searches the lower sublist with first = 0 and last = 4.

20. Illustrating the Binary Search- Unsuccessful Search Step 2: Indices first = 0, last = 4, mid = (0+4)/2 = 2. Since target = 4 < midvalue = 5, step 3 searches the lower sublist with first = 0 and last 2.

21. Illustrating the Binary Search- Unsuccessful Search Step 3: Indices first = 0, last = 2, mid = (0+2)/2 = 1. Since target = 4 > midvalue = 3, step 4 should search the upper sublist with first = 2 and last =2. However, since first >= last, the target is not in the list and we return index last = 9.

22. 22 Main Index Contents Binary Search Algorithm Int binSearch(const int arr[], int first, int last, int target) { int mid; // index of the midpoint int midvalue; // object that is // assigned arr[mid] int origLast = last; // save original value of last

23. 23 Main Index Contents Binary Search Algorithm while (first < last) // test for nonempty sublist { mid = (first+last)/2; midvalue = arr[mid]; if (target == midvalue) return mid; // have a match // determine which sublist to // search

24. 24 Main Index Contents Binary Search Algorithm else if (target < midvalue) last = mid; // search lower sublist. reset last else first = mid+1; // search upper sublist. Reset first } return origLast; // target not found }

25. Entire expression is called the "Big-O" measure for the algorithm. Big-O notation ** Big-O notation provides a machine independent means for determining the efficiency of an Algorithm. For the selection sort, the number of comparisons is T(n) = n2/2 - n/2. n = 100: T(100) = 1002/2 -100/2 = 10000/2 - 100/2 = 5,000 - 50 = 4,950

26. 26 Main Index Contents Constant Time Algorithms An algorithm is O(1) when its running time is independent of the number of data items. The algorithm runs in constant time. The storing of the element involves a simple assignment statement and thus has efficiency O(1).

27. Linear Time Algorithms An algorithm is O(n) when its running time is proportional to the size of the list. When the number of elements doubles, the number of operations doubles.

28. 28 Main Index Contents Exponential Algorithms • Algorithms with running time O(n2) are quadratic. • practical only for relatively small values of n. • Whenever n doubles, the running time of the algorithm increases by a factor of 4. • Algorithms with running time O(n3)are cubic. • efficiency is generally poor; doubling the size of n increases the running time eight-fold.

29. 29 Main Index Contents Exponential Algorithms

30. Logarithmic Time Algorithms The logarithm of n, base 2, is commonly used when analyzing computer algorithms. Ex. log2(2) = 1 log2(75) = 6.2288 When compared to the functions n and n2, the function log2 n grows very slowly.

31. 31 Main Index Contents Selection Sort AlgorithmInteger Version void selectionSort(int arr[], int n) { . . . int temp; // int temp used for the exchange for (pass = 0; pass < n-1; pass++) {. . . if (arr[j] < arr[smallIndex]) // compare integer elements . . . } }

32. 32 Main Index Contents Selection Sort AlgorithmString Version void selectionSort(string arr[], int n) { . . . string temp; // double temp used for the exchange for (pass = 0; pass < n-1; pass++) { . . . if (arr[j] < arr[smallIndex]) // compare string element . . . } }

33. Template Syntax • template function syntax includes the keyword template followed by a non-empty list of formal types enclosed in angle brackets. • In the argument list, each type is preceded by the keyword typename, and types are separated by commas. // argument list with a multiple template // types template <typename T, typename U, typename V, ...>

34. Template Syntax Example template <typename T> void selectionSort(T arr[], int n) { int smallIndex; // index of smallest element in the // sublist int pass, j; T temp;

35. Template Syntax Example // pass has the range 0 to n-2 for (pass = 0; pass < n-1; pass++) { // scan the sublist starting at // index pass smallIndex = pass; // j traverses the sublist // a[pass+1] to a[n-1] for (j = pass+1; j < n; j++) // update if smaller element found

36. Template Syntax Example if (arr[j] < arr[smallIndex]) smallIndex = j; // if smallIndex and pass are not // the same location, exchange the // smallest item in the sublist with // arr[pass] if (smallIndex != pass) { temp = arr[pass]; arr[pass] = arr[smallIndex]; arr[smallIndex] = temp; } } }

37. 37 Main Index Contents Recursive Definition of the Power Function • A recursive definition distinguishes between the exponent n = 0 (starting point) and n  1 which assumes we already know the value xn-1. • After determining a starting point, each step uses a known power of 2 and doubles it to compute the next result. • Using this process gives us a new definition for the power function, xn. • We compute all successive powers of x by multiplying the previous value by x.

38. Stopping Conditions for Recursive Algorithms • Use a recursive function to implement a recursive algorithm. • The design of a recursive function consists of 1. One or more stopping conditions that can be directly evaluated for certain arguments. 2. One or more recursive steps in which a current value of the function can be computed by repeated calling of the function with arguments that will eventually arrive at a stopping condition.

39. 39 Main Index Contents Implementing the Recursive Power Function Recursive power(): double power(double x, int n) // n is a non-negative integer { if (n == 0) return 1.0; // stopping condition else return x * power(x,n-1); // recursive step }

40. 40 Main Index Contents Solving the Tower of Hanoi Puzzle using Recursion

41. 41 Main Index Contents Solving the Tower of Hanoi Puzzle using Recursion

42. 42 Main Index Contents Solving the Tower of Hanoi Puzzle using Recursion

43. Fibonacci Numbers using Iteration int fibiter(int n) { // integers to store previous two // Fibonacci value int oneback = 1, twoback = 1, current; int i; // return is immediate for first two numbers if (n == 1 || n == 2) return 1;

44. Fibonacci Numbers using Iteration else // compute successive terms beginning at 3 for (i = 3; i <= n; i++) { current = oneback + twoback; twoback = oneback; // update for next calculation oneback = current; } return current; }

45. 45 Main Index Contents Summary Slide 1 §- The simplest form of searching is the sequential search. §-It compares the target with every element in a list until matching the target or reaching the end of the list. §- If the list is in sorted order, the binary search algorithm is more efficient. §-It exploits the structure of an ordered list to produce very fast search times.

46. 46 Main Index Contents Summary Slide 2 §- Big-O notation measures the efficiency of an algorithm by estimating the number of certain operations that the algorithm must perform. - For searching and sorting algorithms, the operation is data comparison. - Big-O measure is very useful for selecting among competing algorithms.

47. 47 Main Index Contents Summary Slide 3 §- The running time of the sequential search is O(n) for the worst and the average cases. §- The worst and average case for the binary search is O(log2n). §- Timing data obtained from a program provides experimental evidence to support the greater efficiency of the binary search.

48. 48 Main Index Contents Summary Slide 4 §- C++ provides a template mechanism that allows a programmer to write a single version of a function with general type arguments. - If a main program wants to call the function several times with different runtime arguments, the compiler looks at the types of the runtime arguments and creates different versions of the function that matches the types.

49. 49 Main Index Contents Summary Slide 5 §- An algorithm is recursive if it calls itself for smaller problems of its own type. §-Eventually, these problems must lead to one or more stopping conditions. - The solution at a stopping condition leads to the solution of previous problems. - In the implementation of recursion by a C++ function, the function calls itself.