Main Index Contents 11 Main Index Contents Selection Sort Selection SortSelection Sort Selection Sort (3 slides) Selection Sort Alg. Selection Sort Alg.Selection Sort Alg. Selection Sort Alg. (3 slides) Search Algorithms Search AlgorithmsSearch Algorithms Search Algorithms (6 slides) Illustrating the Binary Search -Successful (3 slides)Successful -Unsuccessful (3 slides)Unsuccessful Binary Search Alg. Binary Search Alg.Binary Search Alg. Binary Search Alg. (3 slides) Big-O Notation Big-O Notation Constant Time Algorithms Constant Time Algorithms Linear Time Algorithms Linear Time Algorithms Exponential Algs. Exponential Algs.Exponential Algs. Exponential Algs. (2 slides) Logarithmic Time Algorithms Logarithmic Time Algorithms Chapter 3 – Introduction to Algorithms Selection Sort Algorithm -Integer VersionInteger Version -String VersionString Version Template Syntax Template SyntaxTemplate Syntax Template Syntax (4 slides) Recursive Def n of the Power Fnc Recursive Def n of the Power Fnc Stopping Conditions for- Stopping Conditions for- Recursive Algorithms Recursive Algorithms Implementing the Recursive- Implementing the Recursive- Power Function Power Function Tower of Hanoi w/ Recursion Tower of Hanoi w/ Recursion (3 slides) Fibonacci Numbers Using- Fibonacci Numbers Using- Iteration Iteration Iteration Iteration (2 slides) Summary Slides Summary SlidesSummary Slides Summary Slides (5 slides)

Main Index Contents 22 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.

Main Index Contents 33 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.

Main Index Contents 44 Main Index Contents Selection Sort - 5 Element Array Pass 2: Locate the smallest element in the sublist 40, 75, and 50.

Main Index Contents 55 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].

Main Index Contents 66 Main Index Contents Selection Sort - 5 Element Array

Main Index Contents 77 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

Main Index Contents 88 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

Main Index Contents 99 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; }

Main Index Contents 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.

Main Index Contents 11 Main Index Contents Search Algorithms

Main Index Contents 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 }

Main Index Contents 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;

Main Index Contents 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 <search sublist arr[first]…arr[mid-1]

Main Index Contents 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)

Main Index Contents 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.

Main Index Contents 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.

Main Index Contents 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.

Main Index Contents 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.

Main Index Contents 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.

Main Index Contents 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.

Main Index Contents 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

Main Index Contents 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

Main Index Contents 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 }

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

Main Index Contents 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).

Main Index Contents 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.

Main Index Contents 28 Main Index Contents Exponential Algorithms Algorithms with running time O(n 2 ) 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(n 3 )are cubic. – efficiency is generally poor; doubling the size of n increases the running time eight-fold.

Main Index Contents 29 Main Index Contents Exponential Algorithms

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

Main Index Contents 31 Main Index Contents Selection Sort Algorithm Integer 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... }

Main Index Contents 32 Main Index Contents Selection Sort Algorithm String 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... } }

Main Index Contents 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

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

Main Index Contents 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

Main Index Contents 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; } } }

Main Index Contents 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 x n-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, x n. We compute all successive powers of x by multiplying the previous value by x.

Main Index Contents 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.

Main Index Contents 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 }

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

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

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

Main Index Contents 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;

Main Index Contents 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; }

Main Index Contents 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.

Main Index Contents 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.

Main Index Contents 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.

Main Index Contents 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.

Main Index Contents 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.