Transcript Slide 1
Discrete Structures
Chapter 4: Elementary Number Theory and Methods of Proof 4.8 Application: Algorithms
Begin at the beginning…and go on till you come to the end: then stop. – Lewis Carroll, 1832 – 1898
Alice’s Adventures in Wonderland, 1865
4.8 Application: Algorithms 1
Definitions
•
Variable
In higher-level computer languages, the term
variable
is used to refer to a specific storage location in a computer’s memory.
•
Data Type
The
data type
takes its values.
of a variable indicates the set in which the variables •
Assignment Statement
An
assignment statement
x
: =
e
where
x
gives a value to the variable in the form is the variable and
e
is the expression.
4.8 Application: Algorithms 2
Conditional Statements
Ordinarily, algorithm statements are executed one after another in the order in which they are written. Conditional statements allow this natural order to be overridden by using the current values of program variables to determine which algorithm statement will be executed next.
4.8 Application: Algorithms 3
Conditional Statements
Conditional statements are denoted in one of two ways.
1. If
(
condition
)
2. If
then s 1 else s 2 (
condition
) then s 1 Notice that we use indentation to indicate that the statements belong together. We can also bind statements by
do
and ending with
end do
.
4.8 Application: Algorithms 4
Execution of if-then-else statements
1. The
condition
is evaluated by substituting the current values of all algorithm values appearing in it and evaluating the truth or falsity of the resulting statement.
2.
If the
condition
is true, then
s
1 is executed and execution moves to the next algorithm statement following the
if-then else
statement.
3.
If the
condition
is false, then
s
2 is executed and execution moves to the next algorithm statement following the
if-then else
statement.
Note: Sometimes
condition
before
s
1 and
s
2 is called
guard
because it is stationed and restricts access to them. 4.8 Application: Algorithms 5
Examples – pg. 225 # 1 & 2
• Find the value of
z
when each of the algorithm segments is executed.
1.
i
:= 2
if
(
i
> 3 or
i
0) then z := 1 else z := 0 2.
i
if
:= 3 (
i
3 or
i
> 6) then z := 2 else z := 0 4.8 Application: Algorithms 6
Iterative Statements
Iterative statements
are used when a sequence of algorithm statements is to be executed over and over again. We use two types of iterative statements:
while
loops and
for-next
loops.
4.8 Application: Algorithms 7
While Loop
A
while
loop has the form
while
(
condition
) [
statements that make up the body of the loop
]
end while
4.8 Application: Algorithms 8
Execution of while Loop
1. The
condition
is evaluated by substituting the current values of all algorithm variables and evaluating the truth or falsity of the resulting statement.
2. If the
condition
is true, all statements in the body of the loop are executed in order. Then execution moves back to the beginning of the loop and the process repeats.
3. If the
condition
is false, execution passes to the next algorithm statement following the loop.
Note: Each execution of the body of the loop is called an iteration of the loop. 4.8 Application: Algorithms 9
Example - While Loop
i
:= 10
while
(
i
> 0) display
i i
:=
i
– 1
end while
display “Blast Off” 4.8 Application: Algorithms 10
for-next Loop
A
for-next
loop has the form for variable := initial expression to final expression [
statements that make up the body of the loop
]
next
(
same
)
variable
4.8 Application: Algorithms 11
Execution of for-next Loop
1. The
for-next
loop
variable initial expression
.
is set equal to the value of the 2.
A check is made to determine whether the value of variables is less than or equal to the value of the
final expression
.
3.
If the value of the
variable
is less than or equal to the value of the
final
expression, then the statements in the body of the loop are executed in order,
variable
is increased by 1, and execution returns back to step 2.
4.
If the value of the
variable
is greater than the value of the
final
expression, then execution passes to the next algorithm statement following the loop.
4.8 Application: Algorithms 12
Example – pg 225 # 4
Find the values of
a
after execution of the loop.
a
:= 2 for i := 1
to
2
a
:=
a
/2 + 1/
a
next i 4.8 Application: Algorithms 13
Algorithm Notation
We generally include the following information when describing algorithms.
1. The name of the algorithm, together with a list of input and output variables.
2. A brief description of how the algorithm works.
3. The input variable names, labeled by data type (integer, real number, etc.).
4. The statements that make up the body of the algorithm, possibly with explanatory comments.
5. The output variable names, labeled by data type.
4.8 Application: Algorithms 14
Algorithm 4.8.1 – Division Algorithm
4.8 Application: Algorithms 15
Example – pg 225 # 6
Make a trace table to trace the action of Algorithm 4.8.1 for the given input values.
6.
a
= 26
d
= 7 4.8 Application: Algorithms 16
Definition – Greatest Common Divisor
Let
a
and
b
be integers that are not both zero. The
greatest common divisor
of
a
and
b
, denoted
gcd(a, b)
, is that integer
d
with the following properties: 1.
d
is a common divisor of both
a
and
b
. In other words,
d
|
a
and
d
|
b
.
2. For all integers
c
, if
c
is a common divisor of both
a
and
b
, then
c
is less than or equal to
d
. In other words, for all integers
c
, if
c
|
a
and
c
|
b
, then
c
d
.
4.8 Application: Algorithms 17
Lemmas
•
Lemma 4.8.1
If
r
is a positive integer, then gcd(
r
, 0) =
r
.
•
Lemma 4.8.2
If
a
and
b
are any integers not both zero, and if
q
and
r
are any integers s.t.
a
=
bq
+
r
, then gcd(
a
,
b
) = gcd(
b
,
r
).
4.8 Application: Algorithms 18
Algorithm 4.8.2 – Euclidean Algorithm
4.8 Application: Algorithms 19
Example – pg 225 # 17
Make a trace table to trace the action of Algorithm 4.8.2 for the given input values.
17. 1,001 and 871 4.8 Application: Algorithms 20