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Need for Repetition

- An algorithm with get/give, assignment and conditional branch instructions processes one set of data (givens) each time it is executed
- What would we do if we needed to process multiple sets of data?
- To calculate the grades for 150 students (instead of a grade for one student)
- To determine the payroll for all employees

- We could execute the algorithm multiple times or once, as in the following example…..

Algorithm 3.1

- Write an algorithm to find the sum of 100 numbers.
- Name: SUM100
- Givens:N1, N2, N3, … N99, N100
- Change:None

- Results: Total
- Intermediates: None
- Definition: Total := SUM100(N1, N2, N3… N100)
- ---------------------------------
- Method
Get N1

Get N2

…

Get N100

Let Total = N1 + N2 + N3 + … + N99 + N100

Give Total

- What would we do for 1,000,000 numbers?

Loop Block

- Clearly, we need a better way
- We need an instruction that
- Executes a block of instructions multiple times, and
- After each execution, tests to see if the block of instructions should be executed again

- This instruction is called a loop (repetition) control instruction

Parts of a Loop

- There are four parts to a loop
- Setup
- The conditions to be set before we enter the loop

- Test (included in the loop control instruction)
- The Boolean test to determine if the loop should be executed again (the same type of test used in a conditional branch instruction)

- Instruction Block
- The instructions that are to be executed repeatedly

- Change
- A change to one of the variables in the test so that we can exit the loop

- Setup

Two Styles of Loops

- Usually we place the test before the instructions that are to be executed
- However, occasionally, we need to execute the instructions before we test; in this case, we place the test after the instructions that are to be executed repeatedly

Loop Block

Set Up

Set Up

TEST

Loop Block

----------------

Instruction 1

Instruction 2

Loop Block

----------------

Instruction 1

Instruction 2

Change

Change

TEST

Set Up

Loop When (Test)

Instruction 1

Instruction 2

……….

Change

Finish Loop

Semantic

After the Set Up, the Test is evaluated, if its value is true, instruction1, instruction2,… are executed, and Test is evaluated again. Instruction1, Instruction2 are repeated as long as the Test is true.

When the value of Test becomes false, the Loop ends and the instruction following the Loop is executed.

Setting up a Loop in an Algorithm (1)Setting up a Loop in an Algorithm (2)

Syntax

Set Up

Loop

- Instruction 1
- Instruction 2
- ……….
- Change
Finish Loop When (Test)

Semantic

After the Set Up, instruction1, instruction2, … are executed and then Test is evaluated, if its value is true, instruction1, instruction2,… are executed and the Test is evaluated again.

When the value of Test becomes false, the Loop ends and the instruction following the Loop is executed.

Loop Instructions

- Since a loop works on a block of instructions, any set of instructions can be repeated, not just assignment and conditional branch instructions
- For example, multiple Gets
Loop

Get X

Finish Loop

- Or multiple Gives
Loop

Give Answer

Finish Loop

Algorithm 3.1 (a)

- Write an algorithm to find the sum of 100 numbers. Place the test at the beginning of the loop.
- Loop Set Up
- Let Total = 0
- Let Count = 0

- Loop Test
- Continue until 100 numbers have been read (Count <100)

- Change
- Let Count = Count + 1

- Instructions
- Get N
- Let Total = Total + N

- Loop Set Up

Givens: N

Change: None

Results: Total

Intermediates: Count

Definition: Total := SUM100(N)

Method

Let Total = 0

Let Count = 0

Loop When (Count < 100)

Get N

Let Total = Total + N

Let Count = Count + 1

Finish Loop

Give Total

Algorithm 3.1 (a)Write an algorithm to find the sum of 100 numbers. Place the test at the beginning of the loop.

Givens: N

Change: None

Results: Total

Intermediates: Count

Definition: Total := SUM100(N)

Method

Let Total = 0

Let Count = 0

Loop

Get N

Let Total = Total + N

Let Count = Count + 1

Finish Loop When (Count = 100)

Give Total

Algorithm 3.1 (b)Write an algorithm to find the sum of 100 numbers. Place the test at the end of the loop.

Algorithm 3.2

Setup

- Let Sum = 0
- Let Value = 1
Test

- Value <= N
Change

- Let Value = Value + 1

Write an algorithm to calculate the sum from 1 to N. ie. (1+2+3+4+…N)

Givens: N

Change: None

Results: Sum

Intermediates: Value

Definition: Sum := SUMN(N)

Method

Get N

Let Value = 1

Let Sum = 0

Loop When (Value <= N)

Let Sum = Sum + Value

Let Value = Value + 1

Finish Loop

Give Sum

Algorithm 3.2Write an algorithm to calculate the sum from 1 to N. ie. (1+2+3+4+…N)

Trace algorithm 3.2 when N is 4

(1) Get N

(2) Let Value = 1

(3) Let Sum = 0

(4) Loop When (Value <= N)

(5) Let Sum = Sum + Value

(6) Let Value = Value + 1

(7) Finish Loop

(8) Give Sum

LN N Value Sum Test

1 4

2 1

3 0

4 (1<=4)

5 1

6 2

4 (2<=4)

5 3

6 3

4 (3<=4)

5 6

6 4

4 (4<=4)

5 10

6 5

4 (5<=4)

8 Output 10

Trace 3.1Complex Tests

- Sometimes, a simple test is not adequate for either a conditional branch (If) or repetition (loop) control structure.
- For more complicated tests, we use the Boolean operators
- AND
- OR
- NOT
- XOR

AND will return a TRUE only when both are TRUE

X | Y | X AND Y

---+---+---------

F | F | F

F | T | F

T | F | F

T | T | T

OR will return a TRUE when either is TRUE

X | Y | X OR Y

---+---+---------

F | F | F

F | T | T

T | F | T

T | T | T

Boolean OperatorsNOT will change a TRUE to a FALSE or a FALSE to a TRUE

X | NOT(X)

---+-------

F | T

T | F

XOR will return a TRUE when either is TRUE, but NOT BOTH

X | Y | X XOR Y

---+---+---------

F | F | F

F | T | T

T | F | T

T | T | F

Boolean OperatorsAlgorithm 3.3

Write an algorithm to find the average of up to 10 numbers entered by the user.

- Execution of the algorithm stops whenever one of the following conditions occurs
- 10 numbers have been entered
- The user decides to stop entering numbers

- Setup
- Let Count = 0, Let Total = 0, Let Again = True

- Test
- Again = True AND Count <= 10

- Change
- Let Count = Count + 1, Get Again

Algorithm 3.3

Write an algorithm to find the average of up to 10 numbers entered by the user.

Name: AVERAGE10

Givens: N

Change: None

Results: AVG

Intermediates: Count, Sum

Again

Definition:

AVG = AVERAGE10(N)

- Method
- Let Count = 0
- Let Sum = 0
- Let Again = True
- Loop When (Count < 10) AND
- (Again)
- Get N
- Let Sum = Sum + N
- Let Count = Count + 1
- Get Again

- Finish Loop
- Let AVG = Sum/Count
- Give AVG

1 0

2 0

3 Yes

4 (0<10)AND YES

5 1

6 1

7 1

8 Yes

4 (1<10)AND YES

5 3

6 4

7 2

8 Yes

4 (2<10)AND YES

5 5

6 9

7 3

8 Yes

4 (3<10)AND YES

5 3

6 12

7 4

8 No

4 (4<10) AND NO

10 3

11 Output 3

Trace 3.2- Trace algorithm 3.3 when the user enters only the numbers 1, 3, 5 and 3

- (1) Let Count = 0
- (2) Let Sum = 0
- (3) Let Again = True
- (4) Loop When
- (Count < 10) AND
- (Again)
- (5) Get N
- (6) Let Sum = Sum + N
- (7) Let Count = Count + 1
- (8) Get Again
- (9) Finish Loop
- (10) Let AVG = Sum/Count
- (11) Give AVG

Base Algorithm 1

Write an algorithm to perform a COUNT.

Let COUNT = 0

Loop

……….

Let COUNT = COUNT + 1

Finish Loop

Give COUNT

Base Algorithm 2

Write an algorithm to perform a SUM.

Let SUM = 0

Loop

Get N

……….

Let SUM = SUM + N

Finish Loop

Give SUM

Base Algorithm 3

Write an algorithm to perform an AVERAGE.

Let SUM = 0

Let COUNT = 0

Loop

Get N

……….

Let SUM = SUM + N

Let COUNT = COUNT + 1

Finish Loop

Let Average = SUM/COUNT

Give Average

Base Algorithms 4 and 5

Write algorithms to perform a MAX and MIN.

Let MAX = - infinity

Loop

Get N

If (N > MAX)

Let MAX = N

Finish Loop

Give MAX

Let MIN = infinity

Loop

Get N

If (N < MIN)

Let MIN = N

Finish Loop

Give MIN

Base Algorithm 6

Write an algorithm to perform a SEARCH.

Get Search

Let FOUND = False

LOOP Until (FOUND Or No_More_Values)

Get Value

If (Value = Search)

Let FOUND = True

Finish Loop

If FOUND

Give Value

Else

Give “Not Found”

Algorithm 3.4

Write an algorithm to find the average of all positive numbers given by the user.

- Name: AVGPOS
- Givens: N
- Change: None

- Results: Avg
- Intermediates:
- Again, Sum, Count

- Definition:
- Avg := AVGPOS(N)

Method

Let Sum = 0

Let Count = 0

Loop

Get N

If (N > 0)

Let Sum = Sum + N

Let Count = Count + 1

Get Again

Finish Loop When Not(Again)

Let Avg = Sum/Count

Give Avg

Trace 3.3

LN Sum Count N Avg Again Test

1,2 0 0

4 1

5 (1>0)

6 1

7 1

8 Yes

9 Not(Yes)

4 -5

5 (-5>0)

8 Yes

9 Not(Yes)

4 5

5 (5>0)

6 6

7 2

8 Yes

9 Not(Yes)

4 3

5 (3>0)

6 9

7 3

8 No

9 Not(No)

10 3

11 Output 3

- Trace algorithm 3.4 when the user enters only the numbers 1, -5, 5 and 3

(1) Let Sum = 0

(2) Let Count = 0

(3) Loop

(4) Get N

(5) If (N > 0)

(6) Let Sum = Sum + N

(7) Let Count = Count + 1

(8) Get Again

(9) Finish Loop When Not(Again)

(10) Let Avg = Sum/Count

(11) Give Avg

Algorithm 3.5

Write an algorithm to find the largest of 5 numbers (range 1 - 10).

- Name: MAX5
- Givens: N
- Change: None

- Results: Max
- Intermediates:
- Count
- LastCount (Constant)

- Definition:
- Max := Max5(N)

Method

Set LastCount = 5

Let Max = -1

Let Count = 1

Loop When (Count <= LastCount)

Get N

If (N > Max)

Let Max = N

Let Count = Count + 1

Finish Loop

Give Max

1,2,3 -1 1 5

4 (1<=5)

5 1

6 (1>-1)

7 1

8 2

4 (2<=5)

5 5

6 (5>1)

7 5

8 3

4 (3<=5)

5 8

6 (8>5)

7 8

8 4

4 (4<=5)

5 3

6 (3>8)

8 5

4 (5<=5)

5 2

6 (2>8)

8 6

4 (6<=5)

10 Output 8

Trace 3.4- Trace algorithm 3.5 when the user enters only the numbers 1,5,8,3 and 2
(1) Set LastCount = 5

(2) Let Max = -1

(3) Let Count = 1

(4) Loop When (Count <= 5)

(5) Get N

(6) If (N > Max)

(7) Let Max = N

(8) Let Count = Count + 1

(9) Finish Loop

(10) Give Max

Flow Charts

- The Diamond symbol is reused, to test the condition.
- At the end of the block, the flow will reverse itself back to the top of the loop
- Note the Test can be at the bottom or the top as stated previously

Algorithm 3.1(a)

Name: SUM100

Givens: N

Change: None

Results: Total

Intermediates: Count

Definition: Total := SUM100(N)

Algorithm 3.1(b)

Name: SUM100

Givens: N

Change: None

Results: Total

Intermediates: Count

Definition: Total := SUM100(N)

Algorithm 3.2

Name: SUMN

Givens: N

Change: None

Results: Sum

Intermediates: Value

Definition: Sum := SUMN(N)

Algorithm 3.3

Name: AVERAGE10

Givens: N

Change: None

Results: AVG

Intermediates: Count, Sum

Again

Definition:

AVG = AVERAGE10(N)

Algorithm 3.4

Name: AVGPOS

Givens: N

Change: None

Results: Avg

Intermediates:

Again, Sum, Count

Definition:

Avg := AVGPOS(N)

Algorithm 3.5

Name: MAX5

Givens: N

Change: None

Results: Max

Intermediates:

Count

LastCount (Constant)

Definition:

Max := Max5(N)

NSD

- Create an L or Γ
- The test, while or until, goes in the row part
- The column controls what block is to be repeated
- NB this is done with two cells (1 row, 1 column) with no boarder between the two

Algorithm 3.1(a)

Name: SUM100

Givens: N

Change: None

Results: Total

Intermediates: Count

Definition: Total := SUM100(N)

Algorithm 3.1(b)

Name: SUM100

Givens: N

Change: None

Results: Total

Intermediates: Count

Definition: Total := SUM100(N)

Algorithm 3.2

Name: SUMN

Givens: N

Change: None

Results: Sum

Intermediates: Value

Definition: Sum := SUMN(N)

Algorithm 3.3

Name: AVERAGE10

Givens: N

Change: None

Results: AVG

Intermediates: Count, Sum

Again

Definition:

AVG = AVERAGE10(N)

Algorithm 3.4

Name: AVGPOS

Givens: N

Change: None

Results: Avg

Intermediates:

Again, Sum, Count

Definition:

Avg := AVGPOS(N)

Algorithm 3.5

Name: MAX5

Givens: N

Change: None

Results: Max

Intermediates:

Count

LastCount (Constant)

Definition:

Max := Max5(N)

- In your homework (Lecture 9), you developed algorithms, and traces
- reversing the digits in a three digit number and adding that number to 500
- determining the eldest of 2 people
- calculating a sales commissionFor each of these questions, revise the algorithm to:
- Run the reversing algorithm multiple times until the user enters the value 999
- Run the eldest algorithm until the user indicates “no” to the question “Do you want to continue?”
- Run the sales commission algorithm for exactly 10 sales representatives

Develop an algorithm to assist a clerk in determining some statistics about students. For each student, she enters the name, gender (M or F), age and marital status (M or S). She wants to determine the number of married men, married women, single women and eligible bachelors (single men over 25). Each time she has completed entry of data for a student, the algorithm should give her a chance to indicate whether she has entered the data for all of the students.

Write an algorithm to determine the closing balance for a teller. The teller enters his opening balance and then a number of transactions. Deposits are entered as positive numbers and withdrawals are entered as negative numbers. He also needs to calculate the number of deposits and the number of withdrawals. The teller indicates that all transactions have been entered by entering a transaction with a value of 0.

Write an algorithm to calculate and display the total gross earnings, tax payable, medical deduction and net earnings for all employees.

The user will input the hours worked and the hourly rate of pay for each employee. The gross earnings for each employee can be calculated from this information.

The net earnings are calculated by subtracting the tax payable and medical deductions from the gross earnings. Tax payable is 20% of gross pay for the first $300, 30% for the next $200 and 40% thereafter. The medical deduction is calculated as 1% of gross pay.

The user will indicate that all employee information has been entered by entering hours worked as 999.

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