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ALGORITHMS 2.1 Write pseudo code for an algorithm for finding the square root of a number. guess <-- number / (1 + count of digits in the number) while(absoluteValue((guess * guess) - number) > .01 ){ guess = (guess + number/guess) / 2 } return guess (from: http://www.homeschoolmath.net/teaching/square-root-algorithm.php) 2.2 Write pseudo code for finding the mean of a set of numbers. mean( list_of_numbers ) length <-- length of list_of_numbers index <-- 1 sum <-- 0 while index <= length { sum <-- sum + list_of_numbers[index] index <-- index + 1 } return sum / length 2.3 Count the primitive operations in your algorithm to find the mean. What is the order of growth of your mean algorithm setup 4 loop 5 * length (length = n, the count of numbers to be averaged) return 2 ( n ) 2.4 Write pseudo code for finding the median of a set of numbers. median( list_of_numbers ) length <-- length of list_of_numbers merge_sort( list_of_numbers ) if length is odd // return middle number return list_of_numbers[(length + 1) / 2] else // find subscripts of 2 middle values m1 <-- length / 2 m2 <-- m1 + 1 // and return the average of the middle two return (list_of_numbers[m1]+list_of_numbers[m2]) / 2 end 2.5 What is the order of growth of your algorithm to find the median ( n( lg n ) ), based on merge_sort
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2.6 Suppose that your algorithm to find the mean is (n), and that your algorithm to find the median is (n lg n), what will be the execution speed ratio between your algorithm for the mean and your algorithm for the median when the number of values is 1,000,000 mean : median mean : median mean : median It will be 20 :: 1,000,000 : 1,000,000 * lg 1,000,000 :: 1 : lg 1,000,000 :: 1 : 20 times faster to find the mean than the median.
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2.7 A sort routine which is easy to program is the bubble-sort. The program simply scans all of the elements to be sorted repeatedly. On each pass, the program compares each element with the one next to it, and reorders the two, if they are in inverse order. For instance, to sort the following list: 67314 Bubble-sort starts by comparing 6 and 7. They are in the correct order, so it then compares 7 and 3. They are in inverse order, so bubble-sort exchanges 7 and 3, and then compares 7 and 1. The numbers 7 & 1 are in reverse order, so bubble-sort swaps them, and then compares 7 & 4. Once again, the order is incorrect, so it swaps 7 & 4. End of scan 1: 63147 Scanning left to right again results in: 31467 Scanning left to right again results in a correct ordering: 13467 Write pseudo code for the bubble-sort. bubbleSort( list ) length <-- length of list // look through the whole list to find // mis-ordered pairs, and continue until // we get through the whole list without // swapping any pairs do { // start at the left end of the list: index = 1 swapped_pair <-- false index <-- 1 while index <= length - 1 { // if this pair is mis-ordered, swap them if list[index] > list[index + 1] { swap( list[index], list[index + 1] ) swapped_pair = true // increment the index to look at the // next pair up index <-- index + 1 } } // quit only when we have gone through the whole // list and found it unnecessary to swap any pair } while( swapped = true ) end 2.8 What is the bubble-sort ( n2 ) 2.9 How will the bubble sort compare for speed with the merge sort when the task is to sort 1,000,000 social security numbers which initially are in random order
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