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There is no standard pseudocode form, and many computer scientists develop a personal style of pseudocode that suits them and their tasks. We will use the following pseudocode style to represent the GCD algorithm: GCD ( a, b ) While b ! = 0 { r <-- a modulo b a <-- b b <-- r } return a CHARACTERIZING ALGORITHMS To illustrate how different algorithms can have different performance characteristics, we will discuss a variety of algorithms that computer scientists have developed to solve common problems in computing. Sequential search Suppose one is provided with a list of people in the class, and one is asked to look up the name Debbie Drawe. A sequential search is a brute force algorithm that one can use. With a sequential search, the algorithm simply compares each name in the list to the name for which we are searching. The search ends when the algorithm finds a matching name, or when the algorithm has inspected all names in the list. Here is pseudocode for the sequential search. The double forward slash // indicates a comment. Note, too, the way we use the variable index to refer to a particular element in list_of_names. For instance, list_of_names is the third name in the list. Sequential_Search(list_of_names, name) length <-- length of list_of_names match_found <-- false index <-- 1 // While we have not found a match AND // we have not looked at every person in the list, // (The symbol <= means "less than or equal to.") // continue ... // Once we find a match or get to the end of the list, // we are finished while match_found = false AND index <= length { // // // if The index keeps track of which name in the list we are comparing with the test name. If we find a match, set match_found to true list_of_names[index] = name then match_found <-- true index <-- index + 1 } // match_found will be true if we found a match, and // false if we looked at every name and found no match return match_found end Function name and arguments ! = means not equal indentation shows what to do while b ! = 0 set r = a modulo b ( = remainder a/b) set a = original b set b = r (i.e., the remainder) border of the while repetition when b = 0, return value of a as the GCD
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ANALYZING ALGORITHMS If we know how long each statement takes to execute, and we know how many names are in the list, we can calculate the time required for the algorithm to execute. However, the important thing to know about an algorithm is usually not how long it will take to solve any particular problem. The important thing to know is how the time taken to solve the problem will vary as the size of the problem changes. The sequential search algorithm will take longer as the number of comparisons becomes greater. The real work of the algorithm is in comparing each name to the search name. Most other statements in the algorithm get executed only once, but as long as the while condition remains true, the comparisons occur again and again. If the name we are searching for is in the list, on average the algorithm will have to look at half the names on the list before finding a match. If the name we are searching for is not on the list, the algorithm will have to look at all the names on the list. If the list is twice as long, approximately twice as many comparisons will be necessary. If the list is a million times as long, approximately a million times as many comparisons will be necessary. In that case, the time devoted to the statements executed only once will become insignificant with respect to the execution time overall. The running time of the sequential search algorithm grows in proportion to the size of the list being searched. We say that the order of growth of the sequential search algorithm is n. The notation for this is T(n). We also say that an algorithm whose order of growth is within some constant factor of T(n) has a theta of NL say. The sequential search has a theta of n. The size of the problem is n, the length of the list being searched. Since for large problems the one-time-only or a-few-times-only statements make little difference, we ignore those constant or nearly constant times and simply focus on the fact that the running time will grow in proportion to the length of the list being searched. Of course, for any particular search, the time required will depend on where in the list the match occurs. If the first name is a match, then it doesn t matter how long the list is. If the name does not occur in the list, the search will always require comparing the search name with all the names in the list. We say the sequential search algorithm is (n) because in the average case, and the worst case, its performance slows in proportion to n, the length of the list. Sometimes algorithms are characterized for best-case performance, but usually average performance, and particularly worst-case performance are reported. The average case is usually better for setting expectations, and the worst case provides a boundary upon which one can rely. Insertion sort An example of order of growth n2 Q(n2) Programmers have designed many algorithms for sorting numbers, because one needs this functionality frequently. One sorting algorithm is called the insertion sort, and it works in a manner similar to a card player organizing his hand. Each time the algorithm reads a number (card), it places the number in its sorted position among the numbers (cards) it has already sorted. On the next page we show the pseudocode for the insertion sort. In this case, we use two variables, number_index and sorted_index, to keep track of two positions in the list of numbers. We consider the list as two sets of numbers. We start with only one set of numbers the numbers we want to sort. However, immediately the algorithm considers the list to be comprised of two sets of numbers; the first set consists of the first number in the original list, and the second set consists of all the rest of the numbers. The first set is the set of sorted numbers (like the cards already sorted in your hand), and the second set is the remaining set of unsorted numbers. The sorted set of numbers starts out containing only a single number, but as the algorithm proceeds, more and more of the unsorted numbers will be moved to their proper position in the sorted set. The variable number_index keeps track of where we are in the list of unsorted numbers; it starts at 2, the first number which is unsorted. The variable sorted_index keeps track of where we are among the sorted numbers; it starts at 1, since the first element of the original list starts the set of sorted numbers. The algorithm compares the next number to be inserted into the sorted set against the largest of the sorted numbers. If the new number is smaller, then the algorithm shifts all the numbers up one position in the list. This repeats, until eventually the algorithm will find that the new number is greater than the next sorted number, and the algorithm will put the new number in the proper position next to the smaller number. It s also possible that the new number is smaller than all of the numbers in the sorted set. The algorithm will know that has happened when sorted_index becomes 0. In that case, the algorithm inserts the new number as the first element in the sorted set.
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