ALGORITHMS ALGORITHMS ALGORITHMS ALGORITHMS ALGORITHMS ALGORITHMS. Reference: Discrete Math by Rosen

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1 Reference: Discrete Math by Rosen Example: Finding the Maximum Element in a Finite Sequence. procedure max(a 1, a 2,..., a n : integers) max : = a 1 f i : = 2 to n if max < a i then max := a i return max {max is the largest element} 1 2 Introduction There are many general classes of problems that arise in discrete mathematics. F instance: given a sequence of integers, find the largest one; given a set, list all its subsets; given a set of integers, put them in increasing der; given a netwk, find the shtest path between two vertices. 3 When presented with such a problem, the first thing to do is to construct a model that translates the problem into a mathematical context. Setting up the appropriate mathematical model is only part of the solution. To complete the solution, a method is needed that will solve the general problem using the model. Ideally, what is required is a procedure that follows a sequence of steps that leads to the desired answer. 4 An algithm is a finite sequence of precise instructions f perfming a computation f solving a problem. 5 Describe an algithm f finding the maximum (largest) value in a finite sequence of integers. 1. Set the tempary maximum equal to the first integer in the sequence. (The tempary maximum will be the largest integer examined at any stage of the procedure.) 2. Compare the next integer in the sequence to the tempary maximum, and if it is larger than the tempary maximum, set the tempary maximum equal to this integer. 3. Repeat the previous step if there are me integers in the sequence. 4. Stop when there are no integers left in the sequence. The tempary maximum at this point is the largest integer in the sequence. 6

2 PSEUDOCODE An algithm can also be described using a computer language A fm of pseudocode is used instead of using a particular computer language like C, C++, Java, etc to specify algithms Pseudocode provides an intermediate step between an English language description of an algithm and an implementation of this algithm in a programming language. PSEUDOCODE In pseudocode, the instructions used can include any well-defined operations statements. A computer program can be produced d in any computer language using the pseudocode description as a starting point. 7 8 COMMON PSEUDOCODE STATEMENTS Procedure Assignment Comments Conditional Constructions Loop Constructions Return 9 Procedure Statements The pseudocode f an algithm begins with a procedure statement that gives the name of an algithm, lists the input variables, and describes what kind of variable each input is. Example: procedure maximum(l: list of integers) is the first statement in the pseudocode description of the algithm, which we have named maximum, that finds the maximum of a list L of integers. 10 Assignment Statements Conditional Constructions used to assign values to variables. symbol := is used the left-hand side is the name of the variable and the right-hand side is an expression that involves constants, variables that have been assigned values, functions defined by procedures and may contain any of the usual arithmetic operations. variable := expression max := a x := largest integer in the list L 11 The simplest fm of the conditional construction that we will use is if condition then statement if condition then block of statements t t if condition then statement 1 else statement 2 If a > b then a = c else a = d 12

3 Loop Constructions f variable := initial value to final value statement f variable := initial value to final value block of statements f i := 1 to n sum := sum +i Loop Constructions while condition statement while condition block of statements while n > 0 sum := sum +n n := n Comments statements enclosed in curly braces are not executed. Such statements serve as comments reminders that help explain how the procedure wks. F instance, the statement {x is the largest element in L} can be used to remind the reader that at that point in the procedure the variable x equals the largest element in the list L. 15 RETURN We use a return statement to show where a procedure produces output. A return statement of the fm return x produces the current value of x as output. The output x can involve the value of one me functions, including the same function under evaluation, but at a smaller value. F instance, the statement return f (n 1) is used to call the algithm with input of n 1. This means that the algithm is run again with input equal to n Example: Finding the Maximum Element in a Finite Sequence. PROPERTIES OF Input. An algithm has input values from a specified set. procedure max(a 1, a 2,..., a n : integers) max : = a 1 f i : = 2 to n if max < a i then max := a i return max {max is the largest element} 17 Output. From each set of input values an algithm produces output values from a specified set. The output values are the solution to the problem. Definiteness. The steps of an algithm must be defined precisely. 18

4 PROPERTIES OF Crectness. An algithm should produce the crect output values f each set of input values. Finiteness. An algithm should produce the desired output after a finite (but perhaps large) number of steps f any input in the set. PROPERTIES OF Effectiveness. It must be possible to perfm each step of an algithm exactly and in a finite amount of time. Generality. The procedure should be applicable f all problems of the desired fm, not just f a particular set of input values USE OF Searching Algithms Sting Greedy Algithms -determine the best choice SAMPLE SEARCHING The Linear Search Algithm procedure linear search(x: integer, a1, a2,..., an: distinct integers) i := 1 while (i n and x = ai ) i := i + 1 if i n then location := i else location := 0 return location{location is the subscript of the term that equals x, is 0 if x is not found} SAMPLE SEARCHING The Binary Search Algithm procedure binary search (x: integer, a1, a2,..., an: increasing integers) i := 1{i is left endpoint of search interval} j := n {j is right endpoint of search interval} while i < j m := (i + j)/2 if x > a m then i := m + 1 else j := m if x = a i then location := i else location := 0 return location {location is the subscript i of the term a i equal to x, 0 if x is not found} 23 SEATWORK Determine which characteristics of an algithm described in the text the following procedures have and which they lack. a) procedure double(n: positive integer) while n > 0 n := 2n 24

5 SEATWORK b) procedure divide(n: positive integer) while n 0 m := 1/n n := n 1 c) procedure sum(n: positive integer) while i < 10 sum := sum + i d) procedure choose(a, b: integers) x := either a b 25