Introduction to Algorithms
What is Computer Science?
Computer Science is the study of computers (??)
• This leaves aside the theoretical work in CS, which does not make use
of real computers, but of formal models of computers
• A lot of work in CS is done with pen and paper!
• Actually, the early work in CS took place before the development of
the first computer
• Computer Science is no more about computers than astronomy is
about telescopes, biology is about microscopes, or chemistry is about
test tubes. Science is not about tools. It is about how we use them, and
what we find out we can do.
What is Computer Science?
Computer Science is the study of how to write computer
programs (programming) (??)
•
Programming is a big part of CS.. ..but it is not the most important part.
Computer Science is the study of the uses and applications
of computers and software (??)
• Learning to use software packages is no more a part of CS than
driver’s education is part of automotive engineering.
• CS is responsible for building and designing software.
What is computer science?
• The study of algorithms:
– their formal properties
• correctness, limits
• efficiency/cost
– their hardware realizations
• computer design
– their linguistic realizations
• programming languages
– their applications
• network design, ocean modeling, bioinformatics, ...
What is an algorithm?
… a well-defined procedure that allows an agent to solve a
problem.
Algorithms must:
1. Be well-ordered and unambigous
2. Be executable (understandable)
3. Solve the problem, and
4. Terminate.
Note: often the agent is a computer or a robot…
Example Algorithms
•
•
•
•
•
Cooking a dish
Making a peanut-butter jelly sandwich
Shampooing hair
Programming a VCR
Making a pie
Examples
Is this an algorithm?
•
•
•
•
Step 1: Wet hair
Step 2: Lather
Step 3: Rinse
Step 4: Repeat
Example
• Problem: Adding two n-digit numbers
7597831 +
1287525
------------------8885356
How would you write an algorithm to solve this
problem? Assume the basic operation is adding
one-digit numbers.
Types of Operations
• Basic operations
– Wet hair
– Rinse
– Turn on VCR
• Conditional operations
– If batter is too dry add water
• Repeat/looping operations
– Repeat step 1 and 2 three times
– Repeat steps 2,3,4,…10 until batter becomes soft.
Expressing algorithms
• Is natural language good?
– For daily life, yes…but for CS it lacks structure and
would be hard to follow
– Too rich, ambiguous, depends on context
• How about a programming language?
– Good, but not when we try to solve a problem..we
want to think at an abstract level
– It shifts the emphasis from how to solve the problem
to tedious details of syntax and grammar.
Pseudocode
• Pseudocode = English but looks like programming
• Good compromise
– Simple, readable, no rules, don’t worry about
punctuation. Lets you think at an abstract level about
the problem.
– Contains only instructions that have a well-defined
structure and resemble programming languages
Pseudocode elements
• Basic operations
– Read the input from user
– Print the output to the user
– Carry out basic arithmetical computations
• Conditional operations
– Execute an operation if a condition is true
• Repeat operations
– Execute a block of operation multiple times until a
certain condition is met
Basic operations
– Read the input from user
• Get x
• Get a, b, c
– Print the output to the user
• Print x
• Print “Your mileage is ” x
– Cary out basic arithmetical computations
• Set x to 10
• Set y to x*x/3
Conditional statements
Specify a statement that may or may not be done:
if <condition> then
<statement to be done>
else <statement to be done otherwise>
Example
if the value of carry is 0 then set the value of
a to
0
else set the vale of a to a+1
Loop statements
specify a group of statements that may be done several
times (repeated):
repeat until <condition>
< statements to be repeated >
• How does this work?
– Condition is evaluated
– If it is true than the loop terminates and the next instruction to be
executed will be the instruction immediately following the loop
– If it is false, then the algorithm executes the <statements to be
repeated> in order, one by one
Example
Variables: count, square
Step 1: set count to 1
Step 2: repeat step 3 to step 5 until count is > 10
Step 3: set square to count *count
Step 4: print value of square and value of count
Step 5: add 1 to count
Step 6: end
• What does this algorithm do?
• Note: indentation
– Not necessary, but makes reading/understanding algorithms
easier
Pseudocode examples
Equivalent:
– Set the value of a to 1
– Set a to 1
– a=1
Equivalent
–
–
–
–
Add 1 to count
Set count to count + 1
Increment the value of count by 1
count = count + 1
Writing in pseudocode gives you the freedom to choose
any of these!
Pseudocode Examples
• Incorrect
– Set 1 to a
– Add a + b (what do you do with the result?)
– Set a+b +3
• Note: not the same
– set a to b
– set b to a
• Example: what is the output of the following algorithms?
set a to 2
set a to 2
set b to 4
set b to 4
set a to b
set b to a
print a, b
print a, b
Problem
• Adding two n-digit numbers
7597831 +
1287525
------------------8885356
How would you write an algorithm to solve this problem?
Assume the basic operation is adding one-digit numbers.
List variables
• How to represent (in pseudocode) inputs of arbitrary size?
• Suppose that we need to read 100 numbers from the user, or
1000, or..
– we could give each variable a different name…tedious!!
• Use a list variable:
– Variable: list a of size n
– This means that a is a list of n elements: a1, a2, a3,…,an
– To read the list from the user use
• Get n and a1, a2, a3,…,an
– To print the list use
• Print a1, a2, a3,…,an
– We can treat each element in the list as a variable
• Set a3 to 5
• Set a4 to a3 +2
Algorithm for adding two m-digit numbers (Fig 1.2)
Given: m ≥ 1 and two positive numbers a and b, each containing m digits,
compute the sum c = a + b.
Variables: m, list a0 ..am-1, list b0 …. bm-1 , list c0 …cm-1 cm, carry, i
0
Get values for m, am-1 … a0 and bm-1 … b0
1
Set the value of carry to 0.
2
Set the value of i to 0.
3
Repeat steps 4-6 until i > m-1
4
Set the value of ci to ai + bi + carry
5
if ci ≥ 10 then
subtract 10 from ci and set the value of carry to 1
else set the value of carry to 0
6
Add 1 to i
7
Set the value of cm to carry
8
Print value of c = cm cm-1 cm-2 … c0
So, how does this work?
For example, the input is m = 4, a = 3276, and b = 7345.
After step 0, the variables m, a, and b have those values:
m
4
3
2
a3 a2
7
a1 a0
6
7
b3 b2
3
4
5
b1 b0
After steps 1 and 2, the variables i and carry are initialized.
i
0
carry
0
Next, steps 4-6 are repeated until the value of i > 3. Each repetition computes a
single digit of c.
c4
c3
c2
c1
c0
A model for visualizing an algorithm’s behavior
Computer
Algorithm
Input
(keyboard)
Output
(screen)
Variables
E.g., Visualizing Fig
Computer
0 Get values for …
…
8 Print value of …
Input
(keyboard)
4
3276
7345
m
4
i
0
3
2
a3 a2
7
3
b3 b2
c4
Output
(screen)
c3 c2
carry
7
6
a1 a0
4
5
b1 b0
c1 c0
0
Algorithm for computing MPG
Write a pseudocode algorithm to compute the distance traveled and
the average miles per gallon on a trip when given as input the number
of gallons used and the starting and ending mileage readings on the
odometer.
Variables: response, gallons, start, end, distance, mpg
0
Set response to “Yes”
1
Repeat steps 2-8 until response = “No”
2
Get gallons, start, end
3
Set distance to end - start
4
Set mpg to distance ÷ gallons
5
Print mpg
6
if mpg > 25.0 then
print “You are getting good gas mileage”
else print “You are NOT getting good gas mileage”
7
Print “Do you want to do this again, Yes or No?”
8
Get response
9
Stop
So, how does this work?
For example, suppose we use 25 gallons, beginning at 12000 and ending at 13000
on the odometer. Then, after step 2, some variables have the following
values:
Yes
response
25
gallons
12000
13000
start
end
After step 4, the variables distance and mpg are computed.
distance
1000
mpg
40
Steps 5-9 displays these results on the output screen:
40
You are getting good gas mileage
Do you want to do this again, Yes or No?
Visualizing Fig
Computer
Input
(keyboard)
25
12000
13000
0 Set response …
…
11 Stop
response
start
distance
Yes
Output
(screen)
gallons
end
mpg
Designing Algorithms: A Methodology
1. Read the problem, identifying the input and the
output.
2. What variables are needed?
3. What computations are required to achieve the
output?
4. Usually, the first steps in your algorithm bring
input values to the variables.
5. Usually, the last steps display the output
6. So, the middle steps will do the computation.
7. If the process is to be repeated, add loops.
How was the MPG program designed?
Problem statement:
Write a pseudocode algorithm to compute the distance
traveled and the average miles per gallon on a trip when
given as input the number of gallons used and the starting
and ending mileage readings on the odometer.
Input: number of gallons, starting mileage, ending mileage
Output: distance traveled, average miles per gallon
Variables: gallons, start, end, distance, mpg
Calculate: distance = end - start
mpg = distance / gallons
Put the steps in order: input, calculate, output (steps 2-8)
Determine if a loop is needed (steps 0, 1, 9, 10)
A Search Algorithm
Problem statement: Write a pseudocode algorithm to find
the location of a target value in a list of values.
Input: a list of values and the target value
Output: the location of the target value, or else a message that
the value does not appear in the list.
Variables:
The (sequential) search algorithm
Variables: target, n, list of n values
Get the value of target, n, and the list of n values
Set index to 1
Set found to false
Repeat until found = true or index > n
If the value of listindex = target
then
Output the index
Set found to true
else
Increment the index by 1
If not found then
Output a message that target was not found
Stop