ALGORITHMS

Reference: Discrete Math by Rosen

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Count the number of persons in the room

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ALGORITHMS Example: Finding the Maximum Element in a

Finite Sequence. procedure max(a1, a2 , . . . , an : integers) max : = a1 for i : = 2 to n if max < ai then max := ai return max {max is the largest element}

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ALGORITHMS Introduction There are many general classes of problems that

arise in discrete mathematics. For 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 order; given a network, find the shortest path between

two vertices.

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ALGORITHMS • 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.

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ALGORITHMS • An algorithm is a finite sequence of precise

instructions for performing a computation or for solving a problem.

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ALGORITHMS • Describe an algorithm for finding the maximum

(largest) value in a finite sequence of integers. • 1. Set the temporary maximum equal to the first integer in the

sequence. (The temporary maximum will be the largest integer examined at any stage of the procedure.)

• 2. Compare the next integer in the sequence to the temporary maximum, and if it is larger than the temporary maximum, set the temporary maximum equal to this integer.

• 3. Repeat the previous step if there are more integers in the sequence.

• 4. Stop when there are no integers left in the sequence. The temporary maximum at this point is the largest integer in the sequence.

FLOW CHART • diagram that

represents an algorithm, workflow or process, showing the steps as boxes of various kinds, and their order by connecting them with arrows.

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6 BASIC SYMBOLS USED • Terminal

– Indicates the starting or ending

• Input/Output • Process

– Indicates the operation 9

6 BASIC SYMBOLS USED • Decision

– Yes or No

• Connector – Used to connect breaks

• Control Flow

– Show direction 10

FLOW CHART RULES • Top to bottom

• Boxes must connected with arrows

• Start with a Terminal or Process symbol

• Decision symbols have 2 exit points,

YES and NO 11

Create Flow Charts for the following

• Addition of 6 and 4

• Sum of 3 numbers

• Print “Passed” 5 times

• Logging in to Facebook

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PSEUDOCODE • An algorithm can also be described using a

computer language

• A form of pseudocode is used instead of using a particular computer language like C, C++, Java, etc… to specify algorithms

• Pseudocode provides an intermediate step between an English language description of an algorithm and an implementation of this algorithm in a programming language.

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PSEUDOCODE

• In pseudocode, the instructions used can include any well-defined operations or statements.

• A computer program can be produced in any computer language using the pseudocode description as a starting point.

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COMMON PSEUDOCODE STATEMENTS

Procedure Assignment Comments Conditional Constructions Loop Constructions Return

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Procedure Statements The pseudocode for an algorithm begins with a

procedure statement that gives the name of an algorithm, 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 algorithm, which we have named maximum, that finds the maximum of a list L of integers.

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Assignment Statements • 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, or functions defined by procedures and may contain any of the usual arithmetic operations.

variable := expression • Example: max := a x := largest integer in the list L

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Conditional Constructions • The simplest form of the conditional construction

that we will use is • if condition then statement or • if condition then block of statements or • if condition then statement 1 else statement 2 • Example: If a > b then a = c

else a = d

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Loop Constructions • for variable := initial value to final value statement or • for variable := initial value to final value block of

statements

• Example: sum := 0 for i := 1 to n sum := sum +i

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Loop Constructions • while condition statement or • while condition block of statements

• Example:

sum := 0 while n > 0 sum := sum +n n := n − 1

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Comments • statements enclosed in curly braces are not

executed. Such statements serve as comments or reminders that help explain how the procedure works.

• For 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.

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RETURN • We use a return statement to show where a

procedure produces output.

• A return statement of the form return x produces the current value of x as output. The output x can involve the value of one or more functions, including the same function under evaluation, but at a smaller value.

• For instance, the statement return f (n − 1) is used to call the algorithm with input of n − 1. This means that the algorithm is run again with input equal to n − 1.

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ALGORITHMS Example: Finding the Maximum Element in a

Finite Sequence. procedure max(a1, a2 , . . . , an : integers) max : = a1 for i : = 2 to n if max < ai then max := ai return max {max is the largest element}

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PROPERTIES OF ALGORITHMS • Input. An algorithm has input values from a

specified set.

• Output. From each set of input values an algorithm produces output values from a specified set. The output values are the solution to the problem.

• Definiteness. The steps of an algorithm must be defined precisely.

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PROPERTIES OF ALGORITHMS • Correctness. An algorithm should produce

the correct output values for each set of input values.

• Finiteness. An algorithm should produce the desired output after a finite (but perhaps large) number of steps for any input in the set.

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PROPERTIES OF ALGORITHMS • Effectiveness. It must be possible to perform

each step of an algorithm exactly and in a finite amount of time.

• Generality. The procedure should be applicable for all problems of the desired form, not just for a particular set of input values.

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USE OF ALGORITHMS • Searching Algorithms

• Sorting

• Greedy Algorithms

– -determine the best choice

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SAMPLE SEARCHING ALGORITHMS

The Linear Search Algorithm 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, or is 0 if x is not found}

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SAMPLE SEARCHING ALGORITHMS

• The Binary Search Algorithm 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 > am then i := m + 1 else j := m if x = ai then location := i else location := 0 return location {location is the subscript i of the term ai equal to x, or 0 if x is not found}