QUIZ on Ch.6: Write the pseudocode for adding two numbers together; our target language is PEP/8 assembly. 1
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QUIZ on Ch.6:
Write the pseudocode for adding two numbers together; our target language is PEP/8 assembly.
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•Input the first number - into memory location "A"
•Input the second number - into memory location "B"
•Load A from memory into Accumulator reg.
•Add B from memory; result goes into Accumulator reg.
•Store the AC into memory location "C"
•Output the value in memory location "C"
Solution
For more practice: Write the assembly code!
QUIZ on Ch.6:
Write a PEP assembly program that reads in a character, and then outputs ‘A’ if the character was A, nothing otherwise.
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What is the hex ASCII code for A?
Use this code as template:
Problem Solving
How to Solve It: A New Aspect of Mathematical Method by George Polya, 1945
The book is written within the context of solving mathematical problems, but the methods described are applicable to problem solving in general.
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We can use the methods described
by Polya to solve any computer-
related problem!
1. Understand the problem
– What are the hypotheses? Data? Unknowns?
2. Devise a plan
– Can we find a related problem? A sub-problem?
– Can we strengthen or relax the hypotheses to obtain an easier problem?
3. Carry out the plan
— Prove that each step is correct!
4. Look back
– Check result
– Find shortcuts and alternate solutions
– Generalize to related problems6
Strategies
Don’t reinvent the wheel!
Similar problems come up again and again in different guises.
A good programmer recognizes a task or subtask that has been solved before and reuses the solution.
Can you think of two similar problems we solved in Python?
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StrategiesDivide and Conquer!
Break up a large problem into smaller sub-problems and solve each separately
– Applies the concept of abstraction
– The divide-and-conquer approach can be applied over and over again until each subtask is manageable
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Polya’s 4 steps can be applied to Computer Problem Solving
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Analysis and Specification Phase
Ask questions to understand all the requirements
Explain in detail what needs to be achieved
Algorithm Development Phase
Develop algorithm
Test algorithm
Implementation Phase
Code algorithm
Test the code in various ways
Maintenance Phase
Use the code, find bugs
Fix bugs
Code new features, as requested by users
QUIZ: Match the steps in Polya’s method to the ones in the computer method for
problem solving
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Analysis and Specification
Implementation
Algorithm Development
Maintenance
Devise a plan
Look back
Understand
Carry out the plan
Phase Interactions
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Algorithms
Algorithm = A set of unambiguous instructions for solving a problem in a finite amount of time, using a finite amount of resources
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Algorithm = A set of unambiguous instructions for solving a problem in a finite amount of time, using a finite amount of data
13Image source: http://my-online-log.com/tech/archives/1214
Algorithm = A set of unambiguous instructions for solving a problem in a finite amount of time, using a finite amount of data
Abstract Step
An algorithmic step containing unspecified details
Concrete Step
An algorithm step in which all details are specified
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Whether the step is abstract or concrete depends on the
programming language we’re using!
7.2 Algorithms with simple variables
Variable = a means of storing intermediate results from one task to the next.
At the hardware level, a simple variable is just one or several adjacent Bytes in the computer memory.
Q: How many Bytes does a simple variable have in PEP/8?
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QUIZ
We have two variables a and b.
Create an algorithm to swap their values!
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What is wrong with this
algorithm?
a = b
b = a
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save = a
a = b
b = save
Solution
Source: AP CS Principles – Course and Exam Descriptions
Similar problem (Don’t reinvent the wheel!)
Algorithms with Selection Statements (a.k.a. decision or if)
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Flowchart of if statement
Figure is not in text
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Algorithm with Selection
Problem: Write the appropriate dress for a given
temperature.
Write "Enter temperature"
Read temperature
Determine Dress
Which statements are concrete?
Which statements are abstract?
Algorithm Determine Dress v.1
Computer language is Python from now on!
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Write “Enter temperature”
Read temperature
IF (temperature > 90)
Write “Texas weather: wear shorts”
ELSE IF (temperature > 70)
Write “Ideal weather: short sleeves are fine”
ELSE IF (temperature > 50)
Write “A little chilly: wear a light jacket”
ELSE IF (temperature > 32)
Write “Philadelphia weather: wear a heavy coat”
ELSE
Write “Stay inside”
Algorithm Determine Dress v.2
Is this concrete enough for implementation in Python?
Algorithms with Loops (a.k.a. repetition)
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Flow of control of while statement
Figure is not in text
QUIZ: There are loops that can execute 0 times and loops that must execute at least 1 time.Which type is this?
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Answer: It depends on whether the decision (diamond) is at the beginning or at the end of the loop!
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Figure is not in text
Loops in Python
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Not in text
Extra-credit question:
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Event-controlled Loops, a.k.a. WHILE loops
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They’re the most general type of loops!
Set sum to 0
Set allPositive to true
WHILE (allPositive)
Read number
IF (number > 0)
Set sum to sum + number
ELSE
Set allPositive to false
Write "Sum is " + sum
Counter-controlled Loops
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They are a particular case of event-controlled
loops: the event is that a counter variable has
reached a predetermined limit
Set sum to 0
Set limit to 42
Set count to 1
While (count <= limit)
Read number
Set sum to sum + number
Increment count
Write "Sum is " + sum
Counter-controlled Loops
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They are a particular case of event-controlled
loops: the event is that a counter variable has
reached a predetermined limit
Set sum to 0
Set limit to 42
Set count to 1
While (count <= limit)
Read number
Set sum to sum + number
Increment count
Write "Sum is " + sum
For loops are counter-
controlled!
EOL1
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Important application of looping: Successive approximation
algorithms
Read in square
Calculate the square root
Write out square and the square root
Algorithm Calculate Square Root v.1
Which steps are abstract and which concrete?
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Set epsilon to 1
WHILE (epsilon > 0.001)
Calculate new guess
Set epsilon to abs(square - guess * guess)
Which steps are abstract and which concrete?
Algorithm Calculate Square Root v.2
In Python usemath.fabs(…)
A more appropriate name for guess would be approximation
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Set newGuess to
(guess + (square/guess)) / 2.0
Set guess to square/4
Set epsilon to 1
What’s the mathematical formula for the new approximation?
How do we get the loop started?
Algorithm Calculate Square Root - Refinements in v.2
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Read in square
Set guess to square/4
Set epsilon to 1
WHILE (epsilon > 0.001)
Set guess to (guess + (square/guess)) / 2.0
Set epsilon to abs(square - guess * guess)
Write out square and guess
Which steps are abstract and which concrete?
Algorithm Calculate Square Root v.3
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Set newGuess to
(guess + (square/guess)) / 2.0
QUIZ: Square root approximation alg.
We want to calculate = 2.236…
Set your initial guess x0 = 1 and show the
next 3 approximations x1, x2, x3
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Solution
We want to calculate = 2.236…5
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Set newGuess to
(guess + (square/guess)) / 2.0
To do in notebook:
We want to calculate = 2.236…
Set your initial guess x0 = 42 and show the
next 3 approximations x1, x2, x3
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QUIZ: Square root algorithm
We want to calculate = 2.6457…
Set your initial guess to x0 = 7/4 = 1.75
and show the next 3 approximations x1, x2, x3
7
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QUIZ: Square root algorithm
We want to calculate = 2.645751…7
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Remember: The
algorithm converges
irrespective of the
initial guess x0!
= 2.645751…7
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QUIZ:
Re-write this program
using a for loop!
QUIZ:
Re-write this program
using a for loop!
Individual work for
next time:
Re-write this program
using a while loop!
QUIZ True/False
• Count-controlled loops repeat a specific number of times.
• Event-controlled loops repeat a specific number of times
• Count-controlled loops are controlled by a counter
• Count-controlled loops are more general than event-controlled loops
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Set newGuess to
(guess + (square/guess)) / 2.0
QUIZ: Square root algorithms
(approximations)
We want to calculate = 3.162…
Set your initial guess x0 = 10/4 and show
the next 3 approximations x1, x2, x3
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Set newGuess to
(guess + (square/guess)) / 2.0
QUIZ: Square root algorithms
(approximations)
We want to calculate = 3.16227…
Set your initial guess x0 = 10/4 and show
the next 3 approximations x1, x2, x3
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x0 = 2.5 x1 = 3.25 x2 = 3.16346… x3 = 3.16227…
7.3 Composite Data Types
They can be classified according to many criteria:
• homogeneous vs. heterogeneous
• accessed by index vs. accessed by name
• mutable vs. immutable
• linear vs. non-linear
• etc.
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Composite Data Types
Records
A named heterogeneous collection of items in which individual items are accessed by name. For example, we could bundle name, age and hourly wage items into a record named Employee
Arrays
A named homogeneous collection of items in which an individual item is accessed by its position (index) within the collection
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Are these the lists from Python? Why not?
Python strings are arrays of charactersCan implement in Python w/lists …
Composite Data Types
Lists (will be covered in next chapter)
A named heterogeneous collection of items in which individual items are accessed by position (index).
We have them in Python, e.g.
>>> myList = [“dog”, 42, 51.375, [1,2]]
>>> myList[1]
42
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Composite Data Types - Records
Employee
name
age
hourly/Wage
Algorithm to store values into the fields of record:
Employee employee // Declare an Employee variableSet employee.name to “Frank Jones”Set employee.age to 32Set employee.hourlyWage to 27.50
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Composite Data Types - Arrays
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numbers[0]
numbers[4]
numbers
Some items in an array may be unused at a given time
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first
last
numbers
??
??
??
??
Useful Algorithms on Arrays
• Initializing all items
• Printing all items
• Searching for an item
• Sorting the array
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Initializing arrays
integer data[20]
Write “How many values?”
Read length
Set index to 0
WHILE (index < length)
Read data[index]
Set index to index + 1
Fill an array numbers with length values
that are being input from the keyboard
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QUIZ
integer data[20]
Write “How many values?”
Read length
Set index to 0
WHILE (index < length)
Read data[index]
Set index to index + 1
Modify this pseudocode to print the values
after initializing them.
An Unsorted Array
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data
A Sorted Array
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data
Sorted Arrays
• The values stored in an array have unique keysof a type for which the relational operators are defined.
• Sorting rearranges the elements into either ascending or descending order within the array.
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Reality check: In a real-life problem it’s very common
to encounter repeated keys!
7.4 Search algorithms
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Sequential Search inUnsorted Array
A sequential search examines each item in turn and compares it to the one we are searching.
If it matches, we have found the item. If not, we look at the next item in the array.
We stop either when we have found the item or when we have looked at all the items and not found a match.
Thus, we have a loop with two ending conditions.
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Set position to 0
Set found to FALSE
WHILE (position < length AND NOT found )
IF (numbers[position] equals searchItem)
Set found to TRUE
ELSE
Set position to position + 1
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The array’s name is numbersThe value we’re searching for is stored in searchItem
Set position to 0
Set found to FALSE
WHILE (position < length AND NOT found )
IF (numbers[position] equals searchItem)
Set found to TRUE
ELSE
Set position to position + 1
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QUIZ: When the loop exits, what do we need to do?
Set index to 0
Set found to FALSE
WHILE (index < length AND NOT found )
IF (data[index] equals searchItem)
Set found to TRUE
ELSE
Set index to index + 1
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QUIZ: Desk-check this algorithm for the array
The item searched is:
• 35
• 43
42
100
35
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Sequential Search in Sorted Array
Idea:
If items in an array are sorted, we can stop looking when we pass the place where the item would be if it were present in the array
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Is this better?
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Sequential Search in Sorted Array
Set index to 0
Set found to FALSE
WHILE (index < length AND NOT found)
IF (data[index] equals searchItem)
Set found to TRUE
ELSE IF (data[index] > searchItem)
Set index to length
ELSE
Set index to index + 1
This is the new part!
(Compare with previous alg. for unsorted array)
This alg. is called sequential search with early termination
QUIZ:End-of-chapter question 66 a
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SearchItem = 4
SearchItem = 49
SearchItem = 50
Binary Search in Sorted Array
Search begins at the middle and finds the item or eliminates half of the unexamined items; the process is then repeated on the half where the item might be
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24 30 31 42 44 90 92 94 99
Example: searchItem = 42
Binary Search Algorithm
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Set first to 0
Set last to length-1
Set found to FALSE
WHILE (first <= last AND NOT found)
Set middle to (first + last)/ 2
IF (item equals data[middle]))
Set found to TRUE
ELSE
IF (item < data[middle])
Set last to middle – 1
ELSE
Set first to middle + 1
RETURN found
IntegerDivision!
Binary Search example
68Figure 7.10 Trace of the binary search
rat
QUIZ Binary Search
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Search for deer
EOL 3
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